To determine the height of a cylindrical fuel tank that can hold [tex]\( V \)[/tex] cubic meters of fuel and has a diameter [tex]\( d \)[/tex] meters, we start by using the formula for the volume of a cylinder:
[tex]\[
V = \pi r^2 h
\][/tex]
Here, [tex]\( V \)[/tex] is the volume, [tex]\( r \)[/tex] is the radius of the base, and [tex]\( h \)[/tex] is the height.
Given that the diameter [tex]\( d \)[/tex] of the cylinder is [tex]\( d \)[/tex] meters, the radius [tex]\( r \)[/tex] can be expressed as:
[tex]\[
r = \frac{d}{2}
\][/tex]
Substitute this expression for [tex]\( r \)[/tex] into the volume formula:
[tex]\[
V = \pi \left(\frac{d}{2}\right)^2 h
\][/tex]
Simplify the term inside the parentheses:
[tex]\[
\left(\frac{d}{2}\right)^2 = \frac{d^2}{4}
\][/tex]
So, the formula for the volume becomes:
[tex]\[
V = \pi \cdot \frac{d^2}{4} \cdot h
\][/tex]
To isolate [tex]\( h \)[/tex], rearrange the equation to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{4V}{\pi d^2}
\][/tex]
Here, [tex]\( h \)[/tex] is expressed in terms of the volume [tex]\( V \)[/tex], the diameter [tex]\( d \)[/tex], and [tex]\(\pi\)[/tex]. According to the multiple-choice options given in the question, the correct answer is:
[tex]\[
D. \frac{4 V}{\pi d^2}
\][/tex]
This indicates that the height of the tank is:
[tex]\[
h = \frac{4 V}{\pi d^2}
\][/tex]
