To compare the estimated average rates of change of the functions \( b(x) \) and \( d(x) \) over the interval \([-12, -3]\), we need to define the functions and analyze their behaviors within that interval.
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Function Definitions:
- \( b(x) = \sqrt[3]{3x + 9} \)
- \( d(x) = \sqrt{-3x + 9} \)
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Assessing the Functions over the Interval:
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For \( b(x) = \sqrt[3]{3x + 9} \):
- At \( x = -12 \): \( b(-12) = \sqrt[3]{3(-12) + 9} = \sqrt[3]{-36 + 9} = \sqrt[3]{-27} = -3 \)
- At \( x = -3 \): \( b(-3) = \sqrt[3]{3(-3) + 9} = \sqrt[3]{-9 + 9} = \sqrt[3]{0} = 0 \)
The average rate of change of \( b(x) \) over the interval \([-12, -3]\) is: \[ \text{Average rate of change of } b(x) = \frac{b(-3) - b(-12)}{-3 - (-12)} = \frac{0 - (-3)}{-3 + 12} = \frac{3}{9} = \frac{1}{3} \]
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For \( d(x) = \sqrt{-3x + 9} \):
- At \( x = -12 \): \( d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \approx 6.7 \)
- At \( x = -3 \): \( d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.2 \)
The average rate of change of \( d(x) \) over the interval \([-12, -3]\) is: \[ \text{Average rate of change of } d(x) = \frac{d(-3) - d(-12)}{-3 - (-12)} = \frac{3\sqrt{2} - 3\sqrt{5}}{-3 + 12} = \frac{3(\sqrt{2} - \sqrt{5})}{9} = \frac{\sqrt{2} - \sqrt{5}}{3} \]
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Signs of Average Rates:
- \( b(x) \) is increasing over the interval, leading to a positive average rate of change.
- \( d(x) \), since \( \sqrt{2} < \sqrt{5} \), results in a negative rate of change, as it goes from a higher value to a lower value.
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Comparison: The average rate of change of \( b(x) \) is positive and greater than that of \( d(x) \), which is negative.
Conclusion:
The correct response to the comparison is: The estimated average rate of change of \( b(x) \) is greater than the estimated average rate of change of \( d(x) \) because \( b(x) \) is increasing over the interval but \( d(x) \) is decreasing.
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