To determine the average rate of change of the functions \( r(x) \) and \( s(x) \) over the interval \([2, 17]\), we need to follow these steps for both functions:
- Calculate the values of \( r(2) \) and \( r(17) \) from the curve (in the image).
- Use the formula for average rate of change:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
where \( f \) is the function, and \( a \) and \( b \) are the endpoints of the interval.
For the function \( s(x) = \sqrt[3]{x - 2} \):
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Calculate \( s(2) \) and \( s(17) \):
- \( s(2) = \sqrt[3]{2 - 2} = \sqrt[3]{0} = 0 \)
- \( s(17) = \sqrt[3]{17 - 2} = \sqrt[3]{15} \)
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Using the average rate of change formula: \[ \text{Average rate of change of } s(x) = \frac{s(17) - s(2)}{17 - 2} = \frac{\sqrt[3]{15} - 0}{15} \approx \frac{2.466}{15} \approx 0.1644 \]
Next, without the specific values of \( r(2) \) and \( r(17) \) from the image, I can't directly compute the average rate of change for \( r(x) \).
To compare the rates:
- If calculations reveal that \( r(x) \) has a greater average rate of change than \( s(x) \), you would select the option stating that the average rate of change of \( r(x) \) is greater than that of \( s(x) \).
- Conversely, if \( r(x) \) has a lower average rate of change than \( s(x) \), you would select the option that the average rate of change of \( r(x) \) is less than that of \( s(x) \).
- If both rates are approximately equal, you would select the option stating that the average rate of change of both functions is approximately 0.164 (or whatever the estimate for \( r(x) \) is derived from the graph).
Based on your observation of the graph and comparison of the calculated average rates, you will choose the appropriate response. If you have specific values from the graph, please provide them for precise comparison!
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