To find the height of the cylindrical fuel tank, we need to use the formula for the volume of a cylinder. The volume \( V \) of a cylinder can be expressed with the formula:
[tex]\[ V = \pi r^2 h \][/tex]
where \( r \) is the radius of the base of the cylinder, \( h \) is the height, and \( \pi \) (pi) is a constant approximately equal to 3.14159.
Given:
- The volume \( V \) of the cylinder is known (in cubic meters).
- The diameter \( d \) of the base of the cylinder is known.
First, let's relate the diameter \( d \) to the radius \( r \). The radius is half of the diameter:
[tex]\[ r = \frac{d}{2} \][/tex]
Now, we substitute \( r \) in the volume formula:
[tex]\[ V = \pi \left(\frac{d}{2}\right)^2 h \][/tex]
Next, simplify \(\left(\frac{d}{2}\right)^2\):
[tex]\[ \left(\frac{d}{2}\right)^2 = \frac{d^2}{4} \][/tex]
So, our volume formula now looks like:
[tex]\[ V = \pi \left(\frac{d^2}{4}\right) h \][/tex]
Simplify further:
[tex]\[ V = \frac{\pi d^2}{4} h \][/tex]
Now, solve for \( h \) (the height of the cylinder):
[tex]\[ h = \frac{4V}{\pi d^2} \][/tex]
Thus, the height of the tank is given by:
[tex]\[ h = \frac{4V}{\pi d^2} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D. \frac{4 V}{\pi d^2}} \][/tex]
