QuestionAugust 26, 2025

19. Suppose the total cost in dollars to produce x widgets is given by the function C(x)=0.0002x^3+0.09x^2+12x+1500 A. Find the average rate of change of the total cost when the number of widgets produced increases from 100 to 300 items. B. Find the average rate of change of the total cost when the number of widgets produced increases from 200 to 500 items.

19. Suppose the total cost in dollars to produce x widgets is given by the function C(x)=0.0002x^3+0.09x^2+12x+1500 A. Find the average rate of change of the total cost when the number of widgets produced increases from 100 to 300 items. B. Find the average rate of change of the total cost when the number of widgets produced increases from 200 to 500 items.
19. Suppose the total cost in dollars to produce x widgets is given by the function C(x)=0.0002x^3+0.09x^2+12x+1500 A. Find the average rate of change of the total cost when the number of widgets produced increases from 100 to 300 items. B. Find the average rate of change of the total cost when the number of widgets produced increases from 200 to 500 items.

Solution
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Answer

A. 74 ### B. 153 Explanation 1. Identify the formula for average rate of change The average rate of change of a function C(x) over an interval [a, b] is given by **\frac{C(b) - C(a)}{b - a}**. 2. Calculate C(100) and C(300) C(100) = 0.0002(100)^3 + 0.09(100)^2 + 12(100) + 1500 = 200 + 900 + 1200 + 1500 = 3800. C(300) = 0.0002(300)^3 + 0.09(300)^2 + 12(300) + 1500 = 5400 + 8100 + 3600 + 1500 = 18600. 3. Compute the average rate of change from 100 to 300 Use the formula: \frac{C(300) - C(100)}{300 - 100} = \frac{18600 - 3800}{200} = \frac{14800}{200} = 74. 4. Calculate C(200) and C(500) C(200) = 0.0002(200)^3 + 0.09(200)^2 + 12(200) + 1500 = 1600 + 3600 + 2400 + 1500 = 9100. C(500) = 0.0002(500)^3 + 0.09(500)^2 + 12(500) + 1500 = 25000 + 22500 + 6000 + 1500 = 55000. 5. Compute the average rate of change from 200 to 500 Use the formula: \frac{C(500) - C(200)}{500 - 200} = \frac{55000 - 9100}{300} = \frac{45900}{300} = 153.

Explanation

1. Identify the formula for average rate of change<br /> The average rate of change of a function $C(x)$ over an interval $[a, b]$ is given by **$\frac{C(b) - C(a)}{b - a}$**.<br /><br />2. Calculate $C(100)$ and $C(300)$<br /> $C(100) = 0.0002(100)^3 + 0.09(100)^2 + 12(100) + 1500 = 200 + 900 + 1200 + 1500 = 3800$.<br /> $C(300) = 0.0002(300)^3 + 0.09(300)^2 + 12(300) + 1500 = 5400 + 8100 + 3600 + 1500 = 18600$.<br /><br />3. Compute the average rate of change from 100 to 300<br /> Use the formula: $\frac{C(300) - C(100)}{300 - 100} = \frac{18600 - 3800}{200} = \frac{14800}{200} = 74$.<br /><br />4. Calculate $C(200)$ and $C(500)$<br /> $C(200) = 0.0002(200)^3 + 0.09(200)^2 + 12(200) + 1500 = 1600 + 3600 + 2400 + 1500 = 9100$.<br /> $C(500) = 0.0002(500)^3 + 0.09(500)^2 + 12(500) + 1500 = 25000 + 22500 + 6000 + 1500 = 55000$.<br /><br />5. Compute the average rate of change from 200 to 500<br /> Use the formula: $\frac{C(500) - C(200)}{500 - 200} = \frac{55000 - 9100}{300} = \frac{45900}{300} = 153$.
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