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Proportional relationships in graphs

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5332117
At a farmers market, the cost of potatoes is directly proportional to their weight. Use the graph to write an equation of the form \(y = kx\), where \(x\) is the weight in pounds and \(y\) is the cost in dollars.
Figure for problem 533211

Hints

- A proportional relationship has a graph that passes through the origin. - Choose a point on the line that lies at a grid intersection. - Divide cost by weight to find the unit rate \(k\).

Solution

1. A directly proportional relationship has the form \(y = kx\). 2. The point \((4, 3)\) lies on the graph, so \(4\) pounds cost \(\$3\). 3. The constant of proportionality is \(k = \frac{y}{x} = \frac{3}{4} = 0.75\). 4. Therefore, the equation is \(y = 0.75x\).

Answer

\(y = 0.75x\)
5119087
A cyclist rides at a constant speed of \(12\,\text{mi/h}\). a) Make a table for the relationship between time, in hours, and distance, in miles, for \(1\), \(2\), \(3\), \(4\), and \(5\) hours. b) A coordinate graph of this relationship contains the points from your table. Describe the graph. Why does it make sense in this situation to connect the points with a line? c) How far has the cyclist traveled after \(150\) minutes? First estimate from the graph's pattern, and then verify by calculation.

Hints

- How many miles does the cyclist travel in one hour? - Does the cyclist also travel during parts of an hour? - Convert \(150\) minutes to hours. - Consider what happens to distance when time is doubled.

Solution

1. Use \(d=rt\). The distances are \(12\), \(24\), \(36\), \(48\), and \(60\) miles for \(1\) through \(5\) hours. 2. The graph is a straight line through \((0, 0)\), \((1, 12)\), \((2, 24)\), \((3, 36)\), \((4, 48)\), and \((5, 60)\). Connecting the points makes sense because time and distance vary continuously during the ride. 3. Convert the time: \(150\) minutes is \(2.5\) hours. 4. The graph gives about \(30\) miles. The calculation is \(12\cdot 2.5=30\).

Answer

a) \(1\,\text{h}\to 12\,\text{mi}\), \(2\,\text{h}\to 24\,\text{mi}\), \(3\,\text{h}\to 36\,\text{mi}\), \(4\,\text{h}\to 48\,\text{mi}\), and \(5\,\text{h}\to 60\,\text{mi}\). b) Connect the points because the cyclist travels during every moment between the listed times. c) The cyclist travels \(30\) miles.
5119107
A printer uses exactly \(12.5\,\text{mL}\) of ink to print \(500\) pages. a) Find the ink used for one page. b) Describe the graph of the relationship between the number of pages and the ink used, in milliliters. State an appropriate scale for each axis. c) A new ink cartridge contains \(40\,\text{mL}\) of ink. How many pages can it print? Briefly explain how to find the result by calculation and how it can be read from the graph.

Hints

- Find the amount of ink used for one page. - Think about the amount needed for \(1000\) pages when choosing a scale. - Divide the total ink in the cartridge by the ink used per page. - Decide which variable belongs on each axis.

Solution

1. The unit rate is \(12.5\div 500=0.025\,\text{mL}\) per page. 2. The graph is a line through the origin and \((500, 12.5)\). One reasonable scale uses increments of \(100\) pages on the x-axis and \(2.5\,\text{mL}\) on the y-axis. 3. Calculate \(40\div 0.025=1600\) pages. 4. On the graph, start at \(40\) on the y-axis, move horizontally to the line, and then move vertically to the x-axis to read about \(1600\) pages.

Answer

a) \(0.025\,\text{mL}\) per page. b) A line through \((0, 0)\) and \((500, 12.5)\); for example, use increments of \(100\) pages on the x-axis and \(2.5\,\text{mL}\) on the y-axis. c) The cartridge can print \(1600\) pages because \(40\div 0.025=1600\). On the graph, the point with y-coordinate \(40\) has x-coordinate about \(1600\).

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