Let's solve the problem step-by-step:
1. Understand the problem:
- We need to find the height of a cylindrical fuel tank that holds [tex]\( V \)[/tex] cubic meters of fuel.
- The diameter of the tank is [tex]\( d \)[/tex] meters.
2. Recall the formula for the volume of a cylinder:
- The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cylinder.
3. Express the radius in terms of the diameter:
- The radius [tex]\( r \)[/tex] is half of the diameter [tex]\( d \)[/tex]:
[tex]\[
r = \frac{d}{2}
\][/tex]
4. Substitute the radius into the volume formula:
- Substitute [tex]\( r = \frac{d}{2} \)[/tex] into the volume formula to get:
[tex]\[
V = \pi \left( \frac{d}{2} \right)^2 h
\][/tex]
5. Simplify the expression:
- Square the radius term:
[tex]\[
V = \pi \left( \frac{d^2}{4} \right) h
\][/tex]
- This simplifies to:
[tex]\[
V = \frac{\pi d^2}{4} h
\][/tex]
6. Solve for the height [tex]\( h \)[/tex]:
- To isolate [tex]\( h \)[/tex], multiply both sides of the equation by 4 and then divide by [tex]\( \pi d^2 \)[/tex]:
[tex]\[
h = \frac{4V}{\pi d^2}
\][/tex]
Therefore, the height of the tank is:
[tex]\[
\frac{4V}{\pi d^2}
\][/tex]
So, the correct answer is option E: [tex]\(\frac{4V}{\pi d^2}\)[/tex].
