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5139249
The function \(T(x) = x^2 + 5\) is given. Find the mean of the function values for \(x = 1, 2, 3, 4,\) and \(5\).

Hints

- First evaluate the function at each given input. - Add all five function values. - Divide the sum by the number of values.

Solution

1. Evaluate the function: \(T(1) = 6\), \(T(2) = 9\), \(T(3) = 14\), \(T(4) = 21\), and \(T(5) = 30\). 2. Add the function values: \(6 + 9 + 14 + 21 + 30 = 80\). 3. Divide by the number of values: \(80 \div 5 = 16\).

Answer

The mean of the function values is \(16\).
5139259
The function \(T(x) = 1.5x^2 - 4.5\) is given. Use a spreadsheet or calculate step by step to find the mean of the function values for \(x \in \{-2, -1.5, -0.5, 0.5, 1.5, 2\}\).

Hints

- Can you use the symmetry of \(x^2\) to reduce the work? - Pay close attention to signs when squaring negative numbers. - How many function values are included in the mean?

Solution

1. Evaluate the function at the six inputs: \(T(-2) = 1.5\), \(T(-1.5) = -1.125\), \(T(-0.5) = -4.125\), \(T(0.5) = -4.125\), \(T(1.5) = -1.125\), and \(T(2) = 1.5\). 2. Add the values: \(1.5 + (-1.125) + (-4.125) + (-4.125) + (-1.125) + 1.5 = -7.5\). 3. Divide by \(6\): \(-7.5 \div 6 = -1.25\).

Answer

The mean of the function values is \(-1.25\).
5351319
A survey asked \(200\) students how many minutes their trip to school takes. The histogram shows the relative frequencies in \(10\)-minute intervals. a) State the relative frequency for each of the five intervals. b) How many students have a travel time from \(10\) to \(20\) minutes? c) Estimate the mean travel time. Use each interval midpoint, such as \(5\) minutes for the interval from \(0\) to \(10\) minutes.
Figure for problem 535131

Hints

- Read each bar height as a percent. - Multiply the relative frequency by the total number of students to find a count. - For the estimated mean, multiply each interval midpoint by its relative frequency written as a decimal, then add the products. - Find the number halfway between the endpoints of each interval.

Solution

1. The relative frequencies are \(20\%\), \(35\%\), \(25\%\), \(15\%\), and \(5\%\) for the intervals from \(0\) to \(10\), \(10\) to \(20\), \(20\) to \(30\), \(30\) to \(40\), and \(40\) to \(50\) minutes, respectively. 2. The number in the \(10\)- to \(20\)-minute interval is \(0.35 \cdot 200 = 70\) students. 3. The interval midpoints are \(5, 15, 25, 35,\) and \(45\) minutes. The estimated mean is \((5 \cdot 0.20) + (15 \cdot 0.35) + (25 \cdot 0.25) + (35 \cdot 0.15) + (45 \cdot 0.05) = 20\) minutes.

Answer

a) \(0\)–\(10\) minutes: \(20\%\); \(10\)–\(20\) minutes: \(35\%\); \(20\)–\(30\) minutes: \(25\%\); \(30\)–\(40\) minutes: \(15\%\); \(40\)–\(50\) minutes: \(5\%\) b) \(70\) students c) Approximately \(20\) minutes

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