To determine the height [tex]\( h \)[/tex] of the cylindrical fuel tank, we start by recalling the formula for the volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( V \)[/tex] is the volume of the cylinder, [tex]\( r \)[/tex] is the radius, and [tex]\( h \)[/tex] is the height.
Given:
- The volume of the tank [tex]\( V \)[/tex] (in cubic meters)
- The diameter of the tank [tex]\( d \)[/tex] (in meters)
First, we calculate the radius [tex]\( r \)[/tex]. The radius is half of the diameter:
[tex]\[ r = \frac{d}{2} \][/tex]
Substitute [tex]\( r = \frac{d}{2} \)[/tex] into the volume formula:
[tex]\[ V = \pi \left(\frac{d}{2}\right)^2 h \][/tex]
Simplify the equation:
[tex]\[ V = \pi \left(\frac{d^2}{4}\right) h \][/tex]
[tex]\[ V = \frac{\pi d^2}{4} h \][/tex]
To find the height [tex]\( h \)[/tex], solve for [tex]\( h \)[/tex] in terms of [tex]\( V \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[ h = \frac{4 V}{\pi d^2} \][/tex]
Thus, the height of the tank is:
[tex]\[ h = \frac{4 V}{\pi d^2} \][/tex]
Looking at the provided options:
A. [tex]\(\frac{2 V}{\pi^2}\)[/tex]
B. [tex]\(\frac{4 V}{d^2}\)[/tex]
C. [tex]\(\frac{V}{\pi^d}\)[/tex]
D. [tex]\(\frac{4 V}{\pi d^d}\)[/tex]
E. [tex]\(\frac{8 V}{\pi d^2}\)[/tex]
The correct answer is:
B. [tex]\(\frac{4 V}{d^2}\)[/tex]
