To find the height [tex]\( h \)[/tex] of a cylindrical fuel tank given its volume [tex]\( V \)[/tex] and diameter [tex]\( d \)[/tex], we can follow these steps:
1. Understand the Volume Formula for 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 of the base of the cylinder and [tex]\( h \)[/tex] is its height.
2. Relate Diameter to Radius:
Since the diameter [tex]\( d \)[/tex] is twice the radius [tex]\( r \)[/tex], we have:
[tex]\[
r = \frac{d}{2}
\][/tex]
3. Substitute the Radius into the Volume Formula:
Replace [tex]\( r \)[/tex] in the volume formula with [tex]\( \frac{d}{2} \)[/tex]:
[tex]\[
V = \pi \left( \frac{d}{2} \right)^2 h
\][/tex]
4. Simplify the Equation:
Simplify the equation for volume:
[tex]\[
V = \pi \left( \frac{d^2}{4} \right)h
\][/tex]
[tex]\[
V = \frac{\pi d^2}{4} h
\][/tex]
5. Solve for the Height [tex]\( h \)[/tex]:
Rearrange the equation to solve for [tex]\( h \)[/tex]:
[tex]\[
V = \frac{\pi d^2}{4} h \implies h = \frac{4V}{\pi d^2}
\][/tex]
Thus, the height of the tank in meters is given by:
[tex]\[
h = \frac{4V}{\pi d^2}
\][/tex]
From the given choices, this corresponds to option B:
[tex]\[
\boxed{\frac{4V}{d^2}}
\][/tex]
