Find the function rule \(f(x) = \frac{a}{x}\) for the graph shown. Use a point whose coordinates can be read exactly to determine \(a\).

The graph shows a direct variation \(y=kx\). a) Find the constant of variation \(k\). b) Explain how the graph confirms that this is a direct variation.

Which of the two lines represents a direct variation? Explain using a feature of the graph.

Classify the three rules as direct variation, inverse variation, or neither. a) \(y=4x\) b) \(y=4x+2\) c) \(y=\frac{4}{x}\)

Two pairs from an inverse-variation table are \((2,9)\) and \((3,6)\), where each pair is \((x,y)\). Multiply \(x\) and \(y\) in each pair. What number stays the same?

A \(24\)-foot rope is cut into equal pieces with nothing left over. Each piece has length \(L\) feet when there are \(n\) pieces. If there are \(3\) pieces, how long is each one? What about \(6\) pieces? Explain what happens to \(L\) when \(n\) doubles.

A store sells markers for \(\$1.50\) each with no other fee. How much do \(4\) markers cost? How much do \(6\) markers cost? What number stays the same when you divide the total cost by the number of markers?

For each situation, decide whether the relationship is a direct variation, an inverse variation, or neither. Briefly justify each answer. a) Number of muffins purchased \(\rightarrow\) total cost, when every muffin has the same price. b) A child's age \(\rightarrow\) the child's height. c) Number of identical pumps \(\rightarrow\) time needed to drain a pool. Assume the pumps run at constant rates and do not interfere with one another. d) Side length of a square \(\rightarrow\) area of the square.

Determine whether the pairs in the table could represent an inverse variation. Justify your answer with calculations. <table> <tbody> <tr><td>\(x\)</td><td>\(0.5\)</td><td>\(1\)</td><td>\(2.5\)</td><td>\(5\)</td><td>\(10\)</td></tr> <tr><td>\(y\)</td><td>\(40\)</td><td>\(20\)</td><td>\(8\)</td><td>\(4\)</td><td>\(2\)</td></tr> </tbody> </table>

Examine each relationship. Decide whether it is an inverse variation, and briefly explain your reasoning. Assume that all people or machines work at the same rate. a) Number of painters \(\rightarrow\) time needed to paint a warehouse b) Number of concert tickets purchased at a fixed price per ticket \(\rightarrow\) total cost of the tickets c) A cyclist’s speed \(\rightarrow\) travel time for a fixed \(30\,\text{mi}\) route

The table represents an inverse variation. <table><tbody><tr><td>\(x\)</td><td>\(5\)</td><td>\(8\)</td><td>\(10\)</td><td>\(15\)</td><td>\(20\)</td><td>\(30\)</td><td>\(40\)</td></tr><tr><td>\(y\)</td><td></td><td></td><td>\(12\)</td><td></td><td></td><td></td><td></td></tr></tbody></table> a) Complete the table. b) Write the equation in the form \(y=\frac{k}{x}\). c) Explain what happens to \(y\) when a positive \(x\) doubles.

A vehicle must travel \(120\,\text{mi}\) on a closed test track at a constant speed \(v\), measured in miles per hour. The travel time \(t\), measured in hours, depends on the speed. a) Write a formula for \(t\) in terms of \(v\). b) Complete the table. <table> <tr><td>\(v\) (in \(\text{mi/h}\))</td><td>\(20\)</td><td>\(30\)</td><td>\(40\)</td><td>\(60\)</td><td>\(80\)</td><td>\(100\)</td><td>\(120\)</td></tr> <tr><td>\(t\) (in \(\text{h}\))</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr> </table> c) Without calculating, explain how \(t\) changes when \(v\) is cut in half.

Consider the following ordered pairs: \(A(4, 15)\), \(B(2, 30)\), \(C(10, 6)\), \(D(12, 5)\), and \(E(2.5, 24)\). Determine algebraically whether the ordered pairs represent an inverse variation. If they do, write the equation in the form \(y = \frac{a}{x}\).

The tables show relationships between \(x\) and \(y\). Table A: <table><tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td></tr><tr><td>\(y\)</td><td>\(5\)</td><td>\(7\)</td><td>\(9\)</td><td>\(11\)</td></tr></table> Table B: <table><tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(4\)</td><td>\(8\)</td></tr><tr><td>\(y\)</td><td>\(2.5\)</td><td>\(5\)</td><td>\(10\)</td><td>\(20\)</td></tr></table> For each table, decide whether the relationship is a direct variation, an inverse variation, or neither. Justify each decision by checking ratios or products.

A class is planning a cleanup project in the school garden. One student working alone would need exactly \(12\,\text{h}\) to complete the project. a) Create a table showing how long the project would take if \(1\), \(2\), \(3\), \(4\), or \(6\) students worked together at the same rate. b) What type of variation is this? Justify your answer using constant products. c) Write a formula for the time \(t\), in hours, in terms of the number of students \(n\).

In a factory, \(6\) identical robots need exactly \(10\,\text{h}\) to sort a batch of parts. Let \(t\) be the time in hours when \(n\) robots work at the same constant rate. a) Write the inverse-variation model relating \(t\) and \(n\). b) For a rush order, the same work must be completed in \(4\,\text{h}\). How many robots must be used in total?

A pool can be filled completely in \(18\,\text{min}\) by \(4\) identical pipes. Let \(t\) be filling time and \(n\) the number of open pipes. a) Write the inverse-variation model relating \(t\) and \(n\). b) How long would filling take if only \(3\) pipes were open? c) How many pipes must be open to fill the pool in \(12\,\text{min}\)?

Four friends need \(6\,\text{h}\) to clean up a park. Assume everyone works at the same constant rate. a) Let \(t(n)\) be the cleanup time for \(n\) people. Write the inverse-variation equation. b) One friend claims, “With \(8\) people we need \(3\,\text{h}\), and with \(12\) people we need \(1.5\,\text{h}\).” Check both claims using your model. c) The group size changes from \(4\) to \(12\), a factor of \(3\). Explain what inverse variation predicts for the time before doing any new arithmetic.

A supply of horse feed lasts exactly \(30\) days when \(12\,\text{lb}\) is used each day. Let \(a\) be average daily use and \(d\) the number of days the fixed supply lasts. a) Write the inverse-variation model relating \(a\) and \(d\). b) The owner wants the supply to last \(45\) days. What average amount of feed may be used each day?

Both situations are inverse variations. For each one, first identify the constant product and write an inverse-variation equation, then find the requested value. a) Three roofers need \(12\,\text{h}\) to complete a roofing job. Let \(t\) be the number of hours needed by \(r\) roofers working at the same rate. Write \(t\) as a function of \(r\), then find the time needed by \(4\) roofers. b) A juice producer fills \(24\) bottles, each holding \(24\,\text{fl oz}\), from one tank. Let \(N\) be the number of bottles that can be filled when each bottle holds \(v\,\text{fl oz}\). Write \(N\) as a function of \(v\), then find how many \(16\,\text{fl oz}\) bottles can be filled.

A farmer has enough feed for \(15\) cows for exactly \(20\) days. Let \(D\) be the number of days the fixed supply lasts when \(n\) cows are fed the same amount per day. a) Write the inverse-variation model \(D(n)\). b) How many days would the feed last if the farmer sold \(3\) cows? c) The farmer wants the feed to last exactly \(30\) days. How many cows could be fed in total?

A batch of lemonade fills \(15\) bottles, each holding \(16\,\text{fl oz}\). Let \(N\) be the number of bottles needed when each bottle holds \(s\) fluid ounces. a) Write an inverse-variation model \(N(s)\) for the fixed batch. b) How many bottles are needed if each bottle holds \(12\,\text{fl oz}\)?

A pump fills a water tank. At a flow rate of \(10\,\text{gal/min}\), filling takes exactly \(60\,\text{min}\). At \(20\,\text{gal/min}\), the time is cut in half to \(30\,\text{min}\). a) How long will filling take at a flow rate of \(15\,\text{gal/min}\)? b) Explain why flow rate and filling time form an inverse variation.

A farmer has enough hay to feed \(18\) horses for exactly \(40\) days. a) What is the greatest number of horses the same supply could feed for \(60\) days? b) Briefly explain why the number of horses and the number of days form an inverse variation.

A crew of \(6\) painters needs \(10\,\text{h}\) to paint the outside of a house. Let \(t\) be completion time and \(n\) the number of painters, assuming equal constant rates. a) Write the inverse-variation model \(t(n)\). b) How many hours would \(4\) painters need for the same job?

Two identical pumps need \(15\,\text{h}\) to fill an empty swimming pool. a) How long would \(3\) identical pumps need to fill the same pool? b) Linnea claims, “Using more pumps increases the filling time because more water is being pumped at once.” Explain why this claim is incorrect and identify the actual relationship between the number of pumps and the filling time.

A group of \(6\) students needs \(4\,\text{h}\) to decorate the school auditorium for an event. a) How long would \(8\) students need for the same work if everyone worked at the same rate? b) Briefly explain why this situation represents an inverse variation.

The table represents an inverse variation. Complete the table and explain how you found the missing values. <table><tr><td>\(x\)</td><td>\(3\)</td><td>\(4\)</td><td>\(\square\)</td><td>\(12\)</td><td>\(24\)</td></tr><tr><td>\(y\)</td><td>\(16\)</td><td>\(\square\)</td><td>\(8\)</td><td>\(\square\)</td><td>\(2\)</td></tr></table>

The table represents an inverse variation between \(x\) and \(y\). <table> <tr><td>\(x\)</td><td>\(2.5\)</td><td>\(4\)</td><td>\(5\)</td><td>\(8\)</td><td>\(10\)</td></tr> <tr><td>\(y\)</td><td>\(16\)</td><td>\(10\)</td><td>\(8\)</td><td>\(5\)</td><td>\(4\)</td></tr> </table> 1. Find the constant of variation \(k\). 2. Write the equation of the function. 3. Find \(y\) when \(x = 20\).

An inverse variation is given by \(f(x) = \frac{a}{x}\). Its graph passes through \(P(1.25, 8)\). a) Find \(a\) and write the complete function rule. b) Find \(f(5)\) and \(f(0.1)\). c) For what value of \(x\) is \(f(x) = 20\)?

Decide whether each relationship is an inverse variation. Briefly explain your reasoning. a) Number of equal slices in a pizza \(\rightarrow\) area of one slice b) Pounds of apples purchased \(\rightarrow\) total price at a fixed price per pound c) A cyclist’s speed \(\rightarrow\) travel time for a fixed \(20\,\text{mi}\) route d) Age of a tree \(\rightarrow\) height of the tree

A tour boat travels from Dock A to Dock B and back. The one-way distance is \(s=36\) miles. The boat's speed in still water is \(v=15\) miles per hour, and the river current is \(c=3\) miles per hour. a) For a fixed one-way distance \(s\), explain why travel time \(T\) varies inversely with the boat's actual speed \(q\), and write \(T(q)\). b) Identify the downstream and upstream values of \(q\), then find each travel time. c) Find the total round-trip time.

A large swimming pool is drained using identical pumps. The draining time \(T\), in hours, varies inversely with the number of pumps \(n\). Three pumps need exactly \(16\,\text{h}\). 1) Find the constant of variation and write the function \(T(n)\). 2) Find the draining time when \(2\), \(4\), \(6\), and \(8\) pumps are used. 3) Without a new calculation, explain how \(T\) changes when the number of pumps is tripled.

A bread recipe uses \(w\) cups of water for \(f\) pounds of flour. A second batch uses \(m\) pounds of flour. Assume \(f>0\) and \(m>0\). Write a proportion to find the water amount \(x\), in cups, for the new batch. Then solve your proportion for \(x\) in terms of \(w\), \(f\), and \(m\).

At a construction site, \(6\) workers need exactly \(10\) days to build a wall. Assume everyone works at the same rate. Let \(D\) be completion time and \(n\) the number of workers. a) Write the inverse-variation model \(D(n)\). b) How many days would \(4\) workers need for the same job? c) How many workers would be needed in total to finish the wall in exactly \(4\) days?

Point \(P(4, -3)\) lies on the graph of a function of the form \(g(x) = \frac{k}{x}\). a) Find \(k\) and write the function rule. b) Find \(g(-1.5)\). c) For what value of \(x\) is \(g(x) = 12\)?

A class rents a bus for a trip at a total cost of \(\$360.00\). The cost is divided equally among all students who attend. 1. Find the amount each student pays if \(24\) students attend. 2. Write a function for the price per student \(y\), in dollars, in terms of the number of students \(x\). 3. Complete the table. <table> <tr><td>Number of students \(x\)</td><td>\(12\)</td><td>\(15\)</td><td>\(20\)</td><td>\(30\)</td><td>\(40\)</td></tr> <tr><td>Price per student \(y\)</td><td></td><td></td><td></td><td></td><td></td></tr> </table> 4. Explain how the price per student changes when the number of students doubles.

A youth group rents a bus for a total cost of \(G\) dollars. There are \(n\) people on the trip, including three adult leaders who do not pay. Let \(p=n-3\) be the number of paying people, so the cost per paying person is \(k=\frac{G}{p}\), with \(p>0\). 1) When \(p\) stays fixed, what type of variation relates \(k\) and \(G\)? Write the relationship in variation form. 2) When \(G\) stays fixed, what type of variation relates \(k\) and \(p\)? State the constant product. 3) With \(G\) fixed, is \(k\) inversely proportional to the total number of people \(n\)? Explain.

The graph shows two functions of the form \(y = \frac{a}{x}\). a) Find the value of \(a\) for the blue graph \(f\). b) Find the value of \(a\) for the red graph \(g\).

A farm has a fixed amount of feed. Every sheep eats the same amount each day. The graph shows how many days \(y\) the feed lasts for each marked number \(x\) of sheep. a) Complete the table using the marked points on the graph. b) Find \(xy\) for each pair. What do you notice? c) How many one-animal daily portions of feed are there in all? <table><tr><td>Sheep \(x\)</td><td>\(2\)</td><td>\(4\)</td><td>\(5\)</td><td>\(8\)</td><td>\(10\)</td></tr><tr><td>Days \(y\)</td><td></td><td></td><td></td><td></td><td></td></tr></table>

The graph shows the first-quadrant branch of \(f(x) = \frac{a}{x}\). a) Use the graph to find \(a\). b) Determine algebraically whether \(P(2.5, 4.8)\) lies on the graph. c) Complete the table. <table> <tr><td>\(x\)</td><td>\(0.5\)</td><td>\(1.5\)</td><td>\(5\)</td><td>\(10\)</td></tr> <tr><td>\(f(x)\)</td><td>\(\square\)</td><td>\(\square\)</td><td>\(\square\)</td><td>\(\square\)</td></tr> </table>

The coordinate plane shows three functions \(f\), \(g\), and \(h\), each of the form \(y = \frac{a}{x}\). a) Find the value of \(a\) for each graph. b) For function \(f\), find the value of \(f(6)\).

The graph shows a function of the form \(f(x) = \frac{a}{x}\). a) Read the coordinates of a point on the graph and use them to find \(a\). b) Check your result using a second point on the graph.

Four workers can clean a large pool in exactly \(6\) hours. The time \(t\) is inversely proportional to the whole-number count of workers \(n\). Write a function \(t(n)\), and use the plotted worker-count points to determine how many workers are needed to finish the job in \(3\) hours.

A car travels a fixed distance of \(60\,\text{mi}\). The graph shows travel time \(t\), in hours, as a function of average speed \(v\), in miles per hour. a) Write the inverse-variation model \(t(v)\) and identify its constant of variation. b) Use the graph to read the times at \(20\,\text{mi/h}\) and \(40\,\text{mi/h}\). Explain how these two points illustrate inverse variation. c) Use the model to predict the travel time at \(80\,\text{mi/h}\).

Which of the four graphs represents \(f(x) = \frac{4}{x}\)? Give the corresponding letter.

The table represents a direct variation between \(x\) and \(y\). <table><tr><td>\(x\)</td><td>\(2\)</td><td>\(5\)</td><td>\(\square\)</td></tr><tr><td>\(y\)</td><td>\(7\)</td><td>\(17.5\)</td><td>\(35\)</td></tr></table> a) Find the constant of variation and write the equation relating \(x\) and \(y\). b) Find the missing \(x\)-value.

A custom framing shop charges \(\$2.40\) per foot of trim, with no fixed fee. Let \(C\) be the cost in dollars for \(x\) feet of trim. a) Explain why \(C\) varies directly with \(x\). b) Identify the constant of variation \(k\) and write the direct-variation equation. c) Find the cost of \(7.5\) feet of trim.

Which table shows direct variation? For each table, compare \(y/x\) for all three pairs. For the direct-variation table, write the rule. Table A: <table><tr><th>\(x\)</th><td>\(1\)</td><td>\(2\)</td><td>\(4\)</td></tr><tr><th>\(y\)</th><td>\(3\)</td><td>\(6\)</td><td>\(12\)</td></tr></table> Table B: <table><tr><th>\(x\)</th><td>\(1\)</td><td>\(2\)</td><td>\(4\)</td></tr><tr><th>\(y\)</th><td>\(3\)</td><td>\(6\)</td><td>\(13\)</td></tr></table>

A cyclist travels at a constant speed. After \(3\) hours, the cyclist has gone \(36\) miles. Write a direct-variation model \(d(t)=kt\) for distance \(d\) in miles after \(t\) hours. How far does the model predict after \(5\) hours?

Two triangular flags have the same base length but different heights. The area varies directly with height. One flag has height \(4\,\text{cm}\) and area \(18\,\text{cm}^2\). Find the height of a flag with area \(45\,\text{cm}^2\) and the same base.

A garden center sells fertilizer for \(\$7\) per bag with no other charge. A soil delivery costs \(\$5\) per bag plus a \(\$20\) fee on every order. Which total cost varies directly with the number of bags purchased? Explain why the other one does not.

The graph shows \(y=\frac{k}{x}\). Find \(k\), state the quadrants containing the graph, and explain why the output is negative when the input is positive.

Stickers cost \(\$5\) each for the first two stickers and \(\$4\) for each sticker after that. A student says, “The total cost must be a direct variation because buying twice as many stickers doubles the cost.” Is that enough evidence? Use the price of three stickers to explain.

Each bag of trail mix holds \(3\) cups. At camp, \(a\) bags are shared equally among \(b\) hikers. Each hiker gets \(Q=\frac{3a}{b}\) cups of trail mix. a) What happens to the cups per hiker if the number of bags doubles but the number of hikers stays the same? b) What happens if the number of hikers triples but the number of bags stays the same? c) What happens if both changes occur together?

Each identical printer makes \(6\) posters per hour. With \(m\) printers working for \(t\) hours, the total is \(P=6mt\) posters. A school doubles the number of printers but runs them for half as many hours. How does the number of posters change? Explain using the formula.

The table shows input-output pairs. <table><tr><th>\(x\)</th><td>\(0\)</td><td>\(2\)</td><td>\(6\)</td></tr><tr><th>\(y\)</th><td>\(0\)</td><td>\(5\)</td><td>\(15\)</td></tr></table> A student says, “This cannot be direct variation because \(y/x\) is undefined at \(x=0\).” Is the student correct? Find the ratio for each nonzero input, write the rule, and explain the role of \((0,0)\).

A submersible starts at sea level and goes down \(4\) meters each minute for its first \(5\) minutes. Let \(h(t)\) be its signed height relative to sea level, so underwater heights are negative. Write a rule \(h(t)\) for the submersible's height. Find \(h(3)\). Is this direct variation even though the heights are negative? Explain.

A square poster is \(12\) inches long on each side. It is cut into smaller equal squares, each with side length \(s\) inches. Assume \(s\) divides \(12\) evenly. a) How many small squares are there when \(s=2\) and when \(s=4\)? b) If \(s\) doubles, is the number of squares cut in half or to one-fourth? Explain using \(N=(12/s)^2\).

A park crew covers \(p\) square patios with small square tiles. Each patio is \(8\) feet on a side. If each tile is \(s\) feet on a side and fits evenly, the number of tiles is \(N=p(8/s)^2\). The crew triples the number of patios and doubles the side length of the tiles. By what factor does \(N\) change? Explain each change separately, then check with \(p=2\) and \(s=1\).

A school packs \(72\) pencils equally into \(c\) cartons, with \(p\) packs in every carton. The same number \(n\) of pencils goes in each pack, so \(n=\frac{72}{cp}\). At first there are \(2\) cartons with \(3\) packs in each. Later the number of cartons doubles and the packs per carton triple. Explain the effect of each change on \(n\), then find both amounts to check.

The graph shows a line. Does it represent direct variation? Use the marked point to find the constant \(k\), write its rule, and find the output when \(x=4\).

Four identical excavators can dig a swimming-pool foundation in \(15\) hours. Assume each excavator works at the same constant rate and that the excavators do not interfere with one another. a) What type of relationship connects the number of excavators and the time required? b) Create a table showing how long \(1\), \(2\), \(6\), and \(10\) excavators would take to complete the same job. c) Write a formula for the time \(t\), in hours, in terms of the number of excavators \(n\).

The table is intended to represent an inverse variation, but one pair is incorrect. a) Which pair does not fit the pattern? b) What should its \(y\)-value be so that the entire table represents an inverse variation? <table> <tbody> <tr><td>\(x\)</td><td>\(3\)</td><td>\(5\)</td><td>\(6\)</td><td>\(12\)</td><td>\(15\)</td></tr> <tr><td>\(y\)</td><td>\(20\)</td><td>\(12\)</td><td>\(10\)</td><td>\(6\)</td><td>\(4\)</td></tr> </tbody> </table>

A landscaping crew is replanting a city park. The table shows how long the job would take for different crew sizes. <table> <tr><td>Number of landscapers (\(x\))</td><td>\(3\)</td><td>\(6\)</td><td>\(12\)</td></tr> <tr><td>Time in hours (\(y\))</td><td>\(16\)</td><td>\(8\)</td><td>\(4\)</td></tr> </table> 1. Show by calculation that the relationship is an inverse variation. 2. Find the constant product \(xy\), and explain what it means in this situation. 3. Calculate how long the job would take a crew of \(8\) landscapers.

The points \(A(2, 24)\), \(B(4, 12)\), \(C(6, 8)\), and \(D(16, 3)\) are given. a) Determine algebraically whether all four points belong to the same inverse variation. b) Write the equation of the inverse variation. c) Find the missing \(x\)-coordinate of \(E(x, 1.5)\) so that this point also lies on the graph.

An inverse variation is described by \(y = \frac{k}{x}\). When \(x = 4\), the corresponding value is \(y = 18\). a) Find the constant of variation \(k\). b) Make a value table for \(x = 2, 3, 6, 9,\) and \(12\). c) In general, what happens to \(y\) when an \(x\)-value is tripled? Briefly justify your answer.

A supply of hay will feed one horse for exactly \(120\) days. Assume each horse eats hay at the same rate. a) Let \(d(n)\) be the number of days the supply lasts for \(n\) horses. Write the inverse-variation equation. b) Use your equation to create a table for \(2\), \(4\), \(6\), \(8\), \(10\), and \(12\) horses. c) The owner wants the hay to last at least \(18\) days. What is the greatest whole number of horses that can be fed?

A marketing agency has a batch of letters to stuff into envelopes. Working at the same rate, \(3\) employees can complete the job together in \(8\,\text{h}\). a) Let \(t(n)\) be the completion time in hours for \(n\) employees. Write the inverse-variation equation. b) Use your equation to find the time with \(2\), \(4\), \(5\), and \(6\) employees. c) For a rush order, the job must be completed in less than \(3\,\text{h}\). What is the minimum whole number of employees needed?

A cyclist is planning a fixed \(45\,\text{mi}\) ride. a) Let \(t(v)\) be the travel time in hours at average speed \(v\) miles per hour. Write the inverse-variation equation. b) Use your equation to find the travel time at \(10\), \(12\), \(15\), and \(18\,\text{mi/h}\). c) The cyclist wants to finish in at most \(2.5\,\text{h}\). What minimum average speed is needed? d) The cyclist usually averages \(12\,\text{mi/h}\). How many minutes are saved by increasing the average speed by \(3\,\text{mi/h}\)?

A picnic has \(384\,\text{fl oz}\) of punch. Every cup is filled, with none left over. a) How does the number of cups depend on the amount poured into each cup? What type of variation is this? b) Write a formula for the number of cups \(n\) when each cup contains \(v\,\text{fl oz}\). c) Find how many full cups can be filled when each contains \(6\,\text{fl oz}\) or \(8\,\text{fl oz}\).

A rectangle has a fixed area of \(48\,\text{cm}^2\). a) Describe the relationship between its length \(a\) and width \(b\). What type of variation is it? b) Give three possible ordered pairs \((a, b)\), in centimeters, that produce this area. c) What happens to the width \(b\) if the length \(a\) is tripled while the area remains fixed? Justify your answer mathematically.

A school copier prints at a constant rate. a) The copier prints \(45\) pages per minute. How many pages does it print in \(4\) minutes, \(10\) minutes, and \(15\) minutes? b) Is the relationship between time in minutes and the number of pages printed a direct variation, an inverse variation, or neither? c) A print job has \(540\) pages. How long will the job take on copiers that print \(30\), \(45\), and \(60\) pages per minute? d) Make a table for part c showing print rate in pages per minute and time in minutes. Is this relationship a direct variation, an inverse variation, or neither?

A flooring company estimates that tiling a gym requires \(120\) worker-hours. Assume all workers work at the same constant rate and do not slow one another down. a) How many hours will the job take if \(2\), \(3\), \(5\), or \(8\) workers tile at the same time? b) Write an equation for the duration \(d\), in hours, as a function of the number of workers \(p\). c) For another project, one room requires \(15\) worker-hours. How many total worker-hours are needed for \(4\), \(6\), or \(10\) identical rooms? d) For part c, classify the relationship between the number of rooms and the total worker-hours.

A class rents a bus for a trip in Europe at a flat cost of \(\text{€}480\). The cost is divided equally among all students who attend. a) Find the cost per student if \(20\), \(24\), or \(30\) students attend. b) Write a formula for the cost per student \(p\) in terms of the number of students \(n\). c) No student should pay more than \(\text{€}18\). What is the minimum number of students who must attend?

A rectangular flower bed must have an area of exactly \(24\,\text{m}^2\). Its length \(x\) and width \(y\) can vary. a) Give four different ordered pairs \((x, y)\), in whole meters, that satisfy the area requirement. b) Describe the relationship between \(x\) and \(y\) mathematically. What happens to the width when the length is tripled? c) A landscaper claims, “If I double the length and halve the width, the perimeter always stays the same.” Test the claim with an example. Is the landscaper correct?

A student group wants to buy a ping-pong table. If \(12\) people split the cost equally, each person pays \(\$15.00\). Let \(c\) be the contribution per person and \(n\) the number of people sharing the fixed cost. a) Write the inverse-variation model \(c(n)\). b) How much would each person pay if \(18\) people split the cost equally? c) Three of the original \(12\) people decide not to participate. By how many dollars does each remaining person's contribution increase compared with the original \(\$15.00\)?

A backpacking group of \(6\) people packs enough food for a \(20\)-day trip. Assume each person eats the same amount each day. a) Let \(D\) be the number of days the full supply lasts for \(n\) people. Write the inverse-variation model \(D(n)\). b) After \(4\) days, they begin sharing the remaining food with \(2\) additional hikers. Write an inverse-variation model for the remaining food and find how many more days it lasts. c) How many days would the original full supply have lasted if the group had included \(8\) people from the beginning?

A crew of \(4\) landscapers plans to replant a city park in \(9\,\text{h}\). After they work together for \(3\,\text{h}\), \(2\) more landscapers join. Assume everyone works at the same rate. a) Write the inverse-variation model for the time \(T(n)\) needed to complete the entire job with \(n\) landscapers from the start. b) Determine the amount of work remaining after the first \(3\) hours and write an inverse model for the remaining time as a function of crew size. c) How many hours will the entire project take?

A construction company estimates that \(3\) identical trucks will need \(20\) days to haul away all the excavated soil from a site. After \(4\) days, one truck breaks down and cannot be replaced. Assume each truck works at the same constant rate. a) Write the inverse-variation model for the full job time as a function of truck count. b) After the breakdown, write an inverse model for the remaining hauling time as a function of the number of working trucks. c) By how many days is completion delayed compared with the original plan?

During an apple harvest, \(5\) workers need \(8\,\text{h}\) to pick all the apples in an orchard. Let \(t\) be harvest time and \(n\) the number of workers, assuming equal constant rates. a) Write the inverse-variation model \(t(n)\). b) The owner wants the next harvest completed in exactly \(5\,\text{h}\). How many workers are needed in total? c) How many additional workers must the owner hire?

Three landscapers need \(12\,\text{h}\) to plant flowers throughout a large park. Let \(t\) be completion time and \(n\) the number of landscapers. a) Write the inverse-variation model \(t(n)\). b) How many hours would \(4\) landscapers need if everyone worked at the same rate? c) A planner claims, “If we use \(40\) landscapers, we will finish in less than one hour.” Use the model to evaluate the claim, then explain why the prediction could be problematic in the real world.

A school cafeteria is studying two situations. 1. Each serving of pasta uses the same amount of water. 2. All helpers work at the same rate and do not interfere with one another. As more helpers cut vegetables, the preparation time decreases. a) For each situation, decide whether the relationship is a direct variation or an inverse variation. Briefly justify your answer. b) In situation 2, four helpers need \(60\) minutes. How long would six helpers need to prepare the same amount of vegetables?

A crew of \(4\) landscapers needs \(12\,\text{h}\) to trim the hedges in a city park. a) How long would \(6\) landscapers need for the same job if everyone worked at the same rate? b) How many landscapers would be needed in total to complete the job in \(4\,\text{h}\)? c) Explain why this relationship is an inverse variation.

Analyze each situation and identify the type of variation. Situation 1: Under constant driving conditions, a car travels \(25\,\text{mi}\) on each gallon of gasoline. How many gallons does it use to travel \(375\,\text{mi}\)? Situation 2: A supply of feed lasts \(10\) horses exactly \(12\) days. Assume every horse eats the same amount each day. How long will the feed last after \(2\) horses leave the stable? For each situation, state whether the relationship is a direct variation or an inverse variation and show your work.

For each situation, decide whether it represents a direct variation or an inverse variation. Then find the requested value. a) At a farm stand, \(3\,\text{lb}\) of apples cost \(\$5.40\). How much will \(7\,\text{lb}\) of the same apples cost? b) Four painters can paint a warehouse in exactly \(6\) hours. How long would three painters take to complete the same job? Assume all painters work at the same constant rate and do not interfere with one another.

A community center is mixing a fruit drink for a summer event. a) The recipe uses \(2\) cups of concentrate for every \(10\) cups of water. How much water is needed to use all \(5\) cups of concentrate? b) Independently of part a, the center has \(120\,\text{fl oz}\) of prepared drink. It can fill \(60\) sample cups that each hold \(2\,\text{fl oz}\). How many sample cups can be filled if each cup holds \(3\,\text{fl oz}\)? c) For each part, state whether the relationship is a direct variation or an inverse variation and briefly justify your answer.

A farmer has enough feed for \(12\) horses for exactly \(20\) days. After \(5\) days, the farmer sells \(3\) horses. How many more days will the remaining feed last for the horses that remain? Explain whether your calculation uses a direct or an inverse variation.

A charter bus for a class trip costs a flat \(\$540.00\). The cost is divided equally among all students who attend. a) Find the cost per student if \(20\), \(25\), or \(30\) students attend. Organize your results in a table. b) Originally, \(27\) students signed up. Shortly before the trip, \(3\) students cancel. By how many dollars does the cost per remaining student increase?

A gardener has a roll of twine. Cutting it into \(25\) pieces of \(8\,\text{ft}\) each uses the whole roll. Let \(N\) be the number of equal pieces obtainable when each piece has length \(\ell\) feet. a) Write the inverse-variation model \(N(\ell)\). b) How many pieces can be cut if each piece is \(3\,\text{ft}\) shorter than before?

A farmer observes that a supply of hay lasts \(12\) horses for \(20\) days and would last \(15\) horses for \(16\) days. a) Find the product of the number of horses and the number of days. What does this value represent in context? b) The farmer says, “If I had \(80\) horses, the hay would last exactly \(3\) days.” Check the calculation. c) Give practical reasons the hay might last for a slightly different amount of time in reality.

A heating-oil supply lasts exactly \(80\) days when \(12\,\text{gal}\) is used each day. Let \(D\) be duration in days and \(u\) be daily use in gallons. a) Write the inverse-variation model \(D(u)\). b) How many days will the supply last if daily use increases to \(16\,\text{gal}\)? c) What is the greatest daily use that will allow the supply to last \(120\) days?

A class rents a bus for a flat cost of \(\$480.00\). The cost is divided equally among all students who attend. 1. Create a table showing the cost per student \(p\), in dollars, for \(n = 12\), \(16\), \(20\), \(24\), \(32\), and \(40\) students. 2. Identify the type of variation and state whether it is characterized by a constant product or a constant ratio. 3. Describe how the cost per student changes when the number of students doubles.

A camp group of \(24\) people packs enough food for exactly \(10\) days. After \(4\) days, \(6\) people leave early. a) Express the remaining food after day \(4\) in person-days. b) Let \(R(n)\) be the number of additional days that remaining fixed supply lasts for \(n\) people. Write the inverse-variation model \(R(n)\). c) How many days longer than originally planned will the food last for the remaining group?

A flour supply at a large bakery lasts exactly \(12\) days when \(20\,\text{lb}\) is used each day. Let \(u\) be the amount of flour used per day, in pounds, and let \(d\) be the number of days the fixed supply lasts. a) Find the constant product and write an inverse-variation model for \(d\) in terms of \(u\). b) How long will the same supply last if daily use is reduced to \(15\,\text{lb}\)? c) The bakery wants the supply to last at least \(20\) days. What is the greatest amount of flour that may be used each day?

A team of \(8\) forestry workers plans to replant an area in \(15\) days. Everyone works at the same rate. After \(3\) days, \(2\) workers become ill and cannot return. a) Determine the remaining work in worker-days after the first \(3\) days. b) Let \(R(n)\) be the remaining completion time for \(n\) workers. Write the inverse-variation model \(R(n)\). c) How many more days will the remaining workers need to finish the project?

Examine the two tables. Which table represents an inverse variation? Justify your answer mathematically, and explain why the observation “as \(x\) increases, \(y\) decreases” is not enough by itself. Table A: <table> <tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(5\)</td><td>\(8\)</td></tr> <tr><td>\(y\)</td><td>\(120\)</td><td>\(60\)</td><td>\(40\)</td><td>\(24\)</td><td>\(15\)</td></tr> </table> Table B: <table> <tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td></tr> <tr><td>\(y\)</td><td>\(120\)</td><td>\(60\)</td><td>\(30\)</td><td>\(15\)</td></tr> </table>

Examine the two data sets. Data Set A: <table> <tr><td>\(x\)</td><td>\(3\)</td><td>\(5\)</td><td>\(10\)</td></tr> <tr><td>\(y\)</td><td>\(20\)</td><td>\(12\)</td><td>\(6\)</td></tr> </table> Data Set B: <table> <tr><td>\(x\)</td><td>\(3\)</td><td>\(5\)</td><td>\(10\)</td></tr> <tr><td>\(y\)</td><td>\(7.5\)</td><td>\(12.5\)</td><td>\(25\)</td></tr> </table> 1. Test both data sets for a constant product \(xy\) and a constant ratio \(\frac{y}{x}\). 2. Decide which data set represents an inverse variation, and justify your answer. 3. Write the equation for that data set.

A function \(f\) represents an inverse variation, and \(f(4) = 9\). a) Use the inverse-variation relationship to find \(f(12)\) without first calculating the constant of variation. Briefly explain your reasoning. b) Find the constant of variation \(a\), and use it to check your answer to part a). c) In general, how does the output change when the input is multiplied by \(4\)? How does the output change when the input is divided by \(10\)?

During an exchange program, a class has a fixed budget of \(\text{€}120\) for paint. The price per liter determines how many liters the class can buy. a) Complete the table for this inverse variation. <table> <tr><td>Price per liter</td><td>\(\text{€}4\)</td><td>\(\text{€}6\)</td><td>\(\text{€}8\)</td><td>\(\text{€}10\)</td><td>\(\text{€}12\)</td></tr> <tr><td>Paint in liters</td><td>\(\square\)</td><td>\(\square\)</td><td>\(\square\)</td><td>\(\square\)</td><td>\(\square\)</td></tr> </table> b) Write the equation in the form \(y=\frac{k}{x}\). What does \(k\) represent in this context?

Examine the two tables. Table A: <table><tbody><tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(6\)</td><td>\(8\)</td></tr><tr><td>\(y\)</td><td>\(12\)</td><td>\(6\)</td><td>\(4\)</td><td>\(2\)</td><td>\(\square\)</td></tr></tbody></table> Table B: <table><tbody><tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td></tr><tr><td>\(y\)</td><td>\(2\)</td><td>\(4\)</td><td>\(6\)</td><td>\(8\)</td></tr></tbody></table> a) Which table represents an inverse variation? Show how you know. b) Find the missing value in that table.

Lucas and Maya live in towns that are \(d\) miles apart. They leave at the same time and bicycle toward each other. Lucas rides at \(v\) miles per hour. Maya rides \(4\) miles per hour faster than Lucas. a) Let \(q\) be their closing rate. Write \(q\) in terms of \(v\). For fixed \(d\), explain why the meeting time \(t\) varies inversely with \(q\), and write \(t\) as a function of \(q\). b) Rewrite the model in terms of \(d\) and \(v\), then find \(t\) when \(d=48\) and \(v=10\). c) If Maya instead rides at half of Lucas's speed, write the new closing rate and the corresponding inverse-variation model for \(t\).

An expedition team has enough water for exactly \(t\) days when it uses \(v\) liters per day. The total supply is fixed. Let \(D\) be the number of days the supply lasts and let \(u\) be the daily use in liters. a) Show that \(D\) varies inversely with \(u\), and write the inverse-variation model in terms of \(t\) and \(v\). b) The team reduces its daily use by \(w\) liters, where \(0<w<v\). Write an expression for the additional number of days the water will last. c) Evaluate the additional time for \(t=20\), \(v=60\), and \(w=12\).

A pump is expected to fill a water tank in \(t\) hours at a rate of \(r\) gallons per hour. The tank volume is fixed. Let \(T\) be filling time and \(q\) be pumping rate. a) Write the inverse-variation model relating \(T\) and \(q\). b) The pump is upgraded so that its rate becomes \(r+s\). Write an expression for the number of hours saved. c) Find the time saved when \(t=10\), \(r=1200\), and \(s=300\), and explain what \(\frac{tr}{r+s}\) represents.

A group of \(24\) campers has enough meal packs for a \(10\)-day camp. Each camper uses one pack per day. a) Let \(p\) be the number of campers actually attending and \(T\) the number of days the food lasts. Write an inverse-variation model for \(T\) as a function of \(p\). b) How many additional days will the food last if \(4\) campers cancel? c) In general, let \(n\) be the original number of campers, \(t\) the planned number of days, and \(k\) the number who do not attend, with \(0\le k<n\). Write \(T\) in terms of \(n\), \(t\), and \(k\).

An aquarium with a capacity of \(V\) liters contains \(k\) grams of dissolved salt. Let \(w\) be the current water volume in liters and \(C\) the salt concentration in grams per liter. a) With the amount of salt fixed at \(k\) grams, show that \(C\) varies inversely with \(w\) and write \(C(w)\). b) The aquarium is one-third full. Find the concentration, then find the new concentration after it is filled to capacity with pure water. c) Assume \(mV\ge k\). How many grams of salt must be added after the aquarium is full so that the concentration is exactly \(m\) grams per liter?

Two robots, Alpha and Beta, move on a circular track with circumference \(C\). Alpha's speed is \(v_{\alpha}\), and Beta's speed is \(v_{\beta}\), where \(v_{\alpha}>v_{\beta}\). Let \(q\) denote the relative speed at which one full circumference of separation is closed. a) Explain why the meeting/lapping time \(t\) varies inversely with \(q\), and write \(t(q)\). b) If the robots start together and move in opposite directions, find \(q\) and write the first-meeting time \(t_O\). c) If they instead move in the same direction, find \(q\) and write the first-lapping time \(t_S\). d) Explain why the two relative speeds are different.

The quantity \(T\) is defined by \(T=\frac{xy}{z^2}\), where \(x\), \(y\), and \(z\) are positive. a) What happens to \(T\) if \(x\) is doubled while \(y\) is halved? b) Explain what happens to \(T\) if only \(z\) increases. c) By what factor does \(T\) change if \(z\) is doubled while \(x\) and \(y\) stay fixed?

A supply of hay lasts \(p\) horses for exactly \(d\) days. a) Write an equation for the number of days \(x\) the supply will last if the number of horses changes to \(n\). Assume every horse eats the same amount each day. b) How does \(x\) change if the number of horses is reduced to half the original number, so \(n = \frac{p}{2}\)? Briefly justify your answer using the type of variation.

The graph of the inverse variation \(y = \frac{k}{x}\) passes through \(P(4, 1.5)\). a) Find \(k\). b) Determine algebraically whether \(Q(-0.5, -12)\) lies on the graph. c) A point \(R\) on the graph has input \(x_R\). Another point \(S\) has input \(x_S = 4x_R\). In general, how does the \(y\)-coordinate of \(S\) compare with the \(y\)-coordinate of \(R\)? Explain.

A prize amount \(G\) is divided equally among \(n\) winners, and each person receives \(x\). a) Write a formula for the amount \(y\) each person receives when the same prize \(G\) is divided among \(m\) people. b) Suppose the new number of winners is four times the original number, so \(m = 4n\). Use your formula to determine how the amount per person changes.

The variables in \(m=\frac{kn}{p}\) are positive. a) How does \(m\) change if \(n\) is doubled while \(p\) is tripled? b) The value of \(m\) must remain unchanged. How must \(p\) change if \(n\) is reduced to one-fourth of its original value? c) Show algebraically why \(m\) does not change when both \(n\) and \(p\) are multiplied by the same positive number \(c\).

A rectangular flower bed must have an area of \(18\,\text{m}^2\). Its length \(x\) and width \(y\) can vary. 1. Find \(y\) when \(x=4.5\,\text{m}\). 2. Write a function for \(y\) in terms of \(x\). What values of \(x\) make sense? 3. Which quadrant contains the graph for this situation? Why? 4. Check the constant product for \((2,9)\) and \((6,3)\), with all lengths in meters. What does it represent?

Consider \(f(x)=\frac{4}{x}\). 1) For which inputs is \(f(x)\) positive? For which is it negative? 2) Find \(f(0.1)\), \(f(100)\), and \(f(-1000)\). What happens to the output as positive \(x\) grows larger and larger? 3) Find \(x\) when \(f(x)=0.5\). 4) Check whether \(P(2,2)\) and \(Q(-2,-2)\) lie on the graph. Explain how their coordinates are related.

Consider the function \(f(x)=\frac{36}{x}\). 1) Find all points on the graph whose \(x\)- and \(y\)-coordinates are positive integers. 2) Find where the graph meets the line \(y=x\). 3) A point \(R\) on the graph has an \(x\)-coordinate four times its \(y\)-coordinate. Find \(R\) if \(x>0\).

A crew of \(n\) workers can renovate a building in \(d\) days. Assume all workers have the same constant rate. Let \(N(T)\) be the number of workers required when the completion time is \(T\) days. a) Write the inverse-variation model \(N(T)\) in terms of \(n\) and \(d\). b) The city wants the project completed \(k\) days earlier, where \(0<k<d\). Write an expression for the total number of workers needed. c) Find the total number of workers when \(n=12\), \(d=20\), and \(k=5\). d) How many additional workers must be hired?

A hiking group is planning a trip on a trail. The graph shows the required time \(t\), in hours, as a function of the average speed \(v\), in kilometers per hour. a) Read the times for \(v = 2\,\text{km/h}\), \(v = 3\,\text{km/h}\), \(v = 4\,\text{km/h}\), and \(v = 6\,\text{km/h}\) from the graph. Record the values in a table. b) Use constant products to verify that the relationship is an inverse variation. Find the total trail distance represented by the constant of variation. c) The group wants to finish in exactly \(1.5\,\text{h}\). Calculate the required average speed and verify it on the graph.

A group of friends wants to buy a farewell gift. The total cost is divided equally among everyone who participates. The graph shows the amount \(y\), in dollars, each person pays as a function of the whole-number count of participants \(x\). a) Read the amount each person pays when \(3\) people participate and when \(6\) people participate. b) Find the parameter \(a\) in \(f(x)=\frac{a}{x}\). What does \(a\) represent in context? c) Use two ordered pairs from the graph to verify the constant product. Describe how the amount per person changes when the number of participants is tripled.

A portable lamp has a battery with \(24\,\text{Wh}\) (watt-hours) of energy. Watt-hours measure stored energy. If the lamp uses \(p\) watts steadily, its run time \(t\) in hours follows an inverse variation. Jordan says, “The graph falls, so the run time decreases by the same number of hours whenever power doubles.” Maya says, “Doubling power cuts the run time in half.” Use the graph to decide whether each claim is correct. Support your answers with two examples.

The graphs \(p\), \(q\), and \(r\) show the relationship between length \(x\) and width \(y\) for three rectangles with different fixed areas. a) Find the area represented by each graph. Use one clearly readable point from each graph. b) A fourth rectangle has area \(12\,\text{cm}^2\). Describe where its graph would lie relative to \(p\), \(q\), and \(r\).

A youth group plans a weekend trip and rents a cabin. The cost per person \(y\), in dollars, depends on the whole-number count of participants \(x\) who share the fixed rental cost. The graph shows the possible participant counts in the displayed range. a) Read the cost per person when \(4\) people attend and when \(5\) people attend. b) Find the total cabin rental cost. c) What is the minimum number of participants needed so that the cost per person is no more than \(\$8\)?

The figure shows the graphs of two relationships, \(f\) and \(g\). a) Find the equation of the inverse variation \(f\) in the form \(y = \frac{k}{x}\). Use a clearly readable point. b) Determine whether \(g\) is also an inverse variation. Calculate \(xy\) for every marked point on \(g\), and justify your conclusion.

A juice factory makes cylindrical cans that hold the same amount of juice but have different radii. The area of a can's curved side (not its top or bottom) is \(M(r)=\frac{710}{r}\), where \(r>0\) is the radius in centimeters and \(M\) is in square centimeters. a) Does the curved-side area vary directly or inversely with the radius? Find the constant of variation. b) Compare the areas when the radii are \(2\,\text{cm}\) and \(4\,\text{cm}\). How does doubling the radius change the area? c) What radius gives a curved-side area of \(142\,\text{cm}^2\)?

Three functions \(f\), \(g\), and \(h\) are shown on the same coordinate plane. Classify each function as a direct variation, an inverse variation, or neither. Justify each classification using a defining graph feature.

The table defines a relationship between \(x\) and \(y\). <table><tr><td>\(x\)</td><td>\(-3\)</td><td>\(-1\)</td><td>\(0\)</td><td>\(2\)</td></tr><tr><td>\(y\)</td><td>\(6\)</td><td>\(2\)</td><td>\(0\)</td><td>\(-4\)</td></tr></table> The graph also shows this relationship as \(f\), together with \(g(x)=-2x+3\). a) Show that \(y\) varies directly with \(x\), and find the constant of variation. b) Write the direct-variation equation. c) Explain why \(g\) is not a direct variation even though it has the same slope as \(f\).

At a bottling plant, \(4\) identical bottling lines empty a large supply tank by bottling its contents in exactly \(90\,\text{min}\). Assume the idealized model that all operating lines have the same rate and do not interfere. a) Let \(T(n)\) be the time in minutes when \(n\) lines operate. Show that this is an inverse variation and write \(T(n)\). b) How long does the model predict if only \(3\) lines operate? c) A trainee says, “If we could operate \(100\) identical lines at the same time, the tank would be empty in less than \(4\,\text{min}\).” Check the claim using your model. d) Give one reason the prediction in part c) probably would not be exact in the real plant.

A project team of \(4\) students plans to paint the classroom walls in \(5\,\text{h}\). After working for \(2\,\text{h}\), they have painted only \(\frac14\) of the total wall area. Assume everyone continues at the observed constant rate. a) Use the observed progress to determine the equivalent total number of student-hours required for the whole job. b) Let \(R(n)\) be the remaining time after the first \(2\) hours when \(n\) students work. Write the inverse-variation model \(R(n)\). c) How many people must work on the project in total from that point forward so the walls are finished exactly at the end of the planned \(5\,\text{h}\)?

Lucas is reading a book for school. If he reads \(20\) pages per day, he needs \(15\) days. If he reads \(30\) pages per day, he finishes in \(10\) days. Let \(p\) be the number of pages Lucas reads per day and \(d\) the number of days needed to finish the book. a) Verify that the two schedules have the same constant product. Use that constant to write an inverse-variation model for \(d\) in terms of \(p\), and explain what the constant represents. b) Lucas wants to finish the book in \(4\) days. How many pages per day must he read? c) He says, “If I read \(600\) pages per day, the model says I will finish in half a day. That means I can still finish if I start only two hours before the assignment is due.” Explain both the mathematical and real-world errors in his reasoning.

A rectangle has a fixed area of \(72\,\text{cm}^2\), so its side lengths \(a\) and \(b\) form an inverse variation. a) Write the function \(b = f(a)\) that gives \(b\) in terms of \(a\). b) Find \(b\) when \(a = 12\,\text{cm}\). c) If \(a\) is increased by \(20\%\), by what percent does \(b\) decrease?

In an inverse variation, increasing \(x\) from \(2\) to \(10\) causes the corresponding \(y\)-value to decrease by \(8\). 1. Find the constant of variation \(k\). 2. Write the function equation. 3. Find the \(y\)-value that corresponds to \(x = 0.5\).

A motorboat travels \(s=120\) miles to a destination and then returns. Its speed in still water is \(v=25\) miles per hour. 1) For one fixed \(120\)-mile leg, write the inverse-variation model \(T(q)\) relating travel time to actual speed \(q\). 2) Use the model to find the total round-trip time on a lake with no current. 3) Use the model to find the total round-trip time on a river with a current of \(c=5\) miles per hour. 4) Does the time gained downstream exactly offset the time lost upstream? Explain using the reciprocal relationship between time and actual speed.

A cleaning concentrate is \(k\%\) active ingredient by volume. A cleaner mixes \(1\,\text{L}\) of concentrate with \(n\,\text{L}\) of water. Assume the volumes add. a) Write \(c(n)\), the percent of active ingredient in the mixture. b) Rewrite your expression as \(k\cdot r(n)\%\). What does \(r(n)\) tell you about the original concentration? c) The cleaner doubles the water from \(n\) liters to \(2n\) liters. Write the new concentration. Explain why it is not generally half of the concentration in part a). Check with \(k=20\) and \(n=1\).

A train travels a fixed route of \(180\,\text{mi}\). Its travel time \(t\), in hours, depends on its average speed \(v\), in miles per hour. 1. Write the function for \(t\) in terms of \(v\). 2. Find the travel time at average speeds of \(45\,\text{mi/h}\), \(60\,\text{mi/h}\), and \(90\,\text{mi/h}\). 3. If the train increases its speed by \(25\%\), by what percent does the travel time decrease? Justify your answer.

A quantity \(W\) is defined by \(W=\frac{x}{yz}\), where \(x\), \(y\), and \(z\) are positive. a) How does \(W\) change if \(x\) increases by \(20\%\) while \(y\) decreases by \(20\%\)? Express the new value as a multiple of \(W\). b) The value of \(W\) must double. By what factor must \(z\) change if \(y\) is reduced to one-third of its original value and \(x\) remains unchanged?

Consider the function \(g(x) = -\frac{9}{x}\). 1) In which quadrants does the graph lie? Explain your reasoning. 2) Find the coordinates of the intersection points of the graph of \(g\) and the line \(y = -x\). 3) Show algebraically that the graph of \(g\) has no real intersection with the line \(y = x\). 4) A graph of the form \(y = \frac{k}{x}\) is symmetric about the origin. Verify this for \(g\) by showing that whenever \((x, y)\) lies on the graph, \((-x, -y)\) also satisfies the equation.