Select the correct answer.

The volume of a cylinder is given by the formula [tex]\( V = \pi r^2 h \)[/tex]. If the volume is 539 cubic inches, which equation represents the value of [tex]\( r \)[/tex] in terms of [tex]\( h \)[/tex]?

A. [tex]\( r = \left( \frac{\pi h}{599} \right)^2 \)[/tex]

B. [tex]\( r = \left( \frac{599}{\pi} \right)^2 \)[/tex]

C. [tex]\( r = \sqrt{ \frac{599}{\pi} } \)[/tex]

D. [tex]\( r = \sqrt{ \frac{4}{599} } \)[/tex]



Answer :

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.