Given the volume of the cylinder [tex]\( V = 539 \)[/tex] cubic inches, we can use the volume formula for a cylinder, which is given by [tex]\( V = \pi r^2 h \)[/tex]. We need to express the radius [tex]\( r \)[/tex] in terms of the height [tex]\( h \)[/tex].
1. Start with the given volume formula:
[tex]\[
V = \pi r^2 h
\][/tex]
Substitute [tex]\( V \)[/tex] with 539:
[tex]\[
539 = \pi r^2 h
\][/tex]
2. To isolate [tex]\( r^2 \)[/tex], divide both sides of the equation by [tex]\( \pi h \)[/tex]:
[tex]\[
r^2 = \frac{539}{\pi h}
\][/tex]
3. Finally, take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt{\frac{539}{\pi h}}
\][/tex]
So, the correct equation that represents the value of [tex]\( r \)[/tex] in terms of [tex]\( h \)[/tex] is:
[tex]\[
r = \sqrt{\frac{539}{\pi h}}
\][/tex]
Given the list of provided options, none of them seem to correctly represent this derived expression. However, remember that the answer must match the form [tex]\( r = \sqrt{\frac{539}{\pi h}} \)[/tex]; it appears that there might have been a typographical or conceptual discrepancy in the options you provided.
Based on the given numerical result:
The root is consistent with the options provided, and the correct form should match:
[tex]\[
r = 7\sqrt{11}\sqrt{\frac{1}{\pi h}}
\][/tex]
Given our expression simplifies directly as it matches none exactly it shows the mismatch error.
Thus there might be a conceptual error with the interpretations given provided options does not include correct answer specifically.
Thus we know correct form remains not matching provided errors.
