To find the average rate of change of the function [tex]\( f(x) = -5 \sqrt{x} + 8 \)[/tex] over the interval [tex]\([7, 17]\)[/tex], we follow these steps:
1. Evaluate the function at the endpoints of the interval.
First, we find [tex]\( f(7) \)[/tex]. Plugging [tex]\( x = 7 \)[/tex] into the function:
[tex]\[
f(7) = -5 \sqrt{7} + 8
\][/tex]
This gives us:
[tex]\[
f(7) \approx -5.228756555322953
\][/tex]
Next, we find [tex]\( f(17) \)[/tex]. Plugging [tex]\( x = 17 \)[/tex] into the function:
[tex]\[
f(17) = -5 \sqrt{17} + 8
\][/tex]
This gives us:
[tex]\[
f(17) \approx -12.615528128088304
\][/tex]
2. Calculate the change in the function values over the interval.
The change in the function values is:
[tex]\[
f(17) - f(7) = -12.615528128088304 - (-5.228756555322953)
\][/tex]
Simplifying this, we get:
[tex]\[
f(17) - f(7) \approx -12.615528128088304 + 5.228756555322953 = -7.386771572765352
\][/tex]
3. Calculate the length of the interval.
The length of the interval is:
[tex]\[
17 - 7 = 10
\][/tex]
4. Compute the average rate of change.
The average rate of change is given by the change in the function values over the change in [tex]\( x \)[/tex]:
[tex]\[
\frac{f(17) - f(7)}{17 - 7} = \frac{-7.386771572765352}{10} \approx -0.7386771572765352
\][/tex]
5. Round the result to the nearest tenth.
Rounding [tex]\(-0.7386771572765352\)[/tex] to the nearest tenth, we get:
[tex]\[
-0.7
\][/tex]
Hence, the average rate of change of [tex]\( f(x) = -5 \sqrt{x} + 8 \)[/tex] over the interval [tex]\([7, 17]\)[/tex] is [tex]\(\boxed{-0.7}\)[/tex].
