To find the interval where the average rate of change of the function \( h(x) = -5x + 8 \) equals \(-30\), we can use the formula for the average rate of change over an interval \([a, b]\):
\[ \text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a} \]
We need to set this equal to \(-30\) and solve for \(a\) and \(b\).
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First, compute \(h(b)\) and \(h(a)\): \[ h(x) = -5x + 8 \]
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The average rate of change is given by: \[ \frac{(-5b + 8) - (-5a + 8)}{b - a} = \frac{-5b + 5a}{b - a} = \frac{5(a - b)}{b - a} = -5 \] This implies: \[ -5 = -30 \]
However, we notice we set the sections incorrectly as we should have worked towards making the average rate of change equal to \(-30\).
To find possible values, let's check different intervals provided.
Now let's compute:
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Interval \( x = -4 \) to \( x = 4 \): \[ h(-4) = -5(-4) + 8 = 20 + 8 = 28 \] \[ h(4) = -5(4) + 8 = -20 + 8 = -12 \] \[ \text{Average Rate} = \frac{-12 - 28}{4 - (-4)} = \frac{-40}{8} = -5 \]
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Interval \( x = 0 \) to \( x = 2 \): \[ h(0) = -5(0) + 8 = 8 \] \[ h(2) = -5(2) + 8 = -10 + 8 = -2 \] \[ \text{Average Rate} = \frac{-2 - 8}{2 - 0} = \frac{-10}{2} = -5 \]
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Interval \( x = 2 \) to \( x = 4 \): \[ h(2) = -5(2) + 8 = -10 + 8 = -2 \] \[ h(4) = -12 \] \[ \text{Average Rate} = \frac{-12 - (-2)}{4 - 2} = \frac{-12 + 2}{2} = \frac{-10}{2} = -5 \]
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Single point \( x = 4 \): Average rate cannot be computed.
Conclusion:
The average rate of change is consistently \(-5\) for all intervals provided in the options. Given the comparison needed to be with \(-30\), none of your intervals directly gives an average of \(-30\).
If you had to choose one based on these calculations and closest to a negative slope, you might prefer \(x = -4\) to \(x = 4\) or any other since they are equally valid, but the answer does not indicate achieving \(-30\) average.
Please clarify if the function or any constraints have additional context or specific precision, as per your question's requirement about targeting \(-30\).