To determine the height [tex]\( h \)[/tex] of a cylindrical fuel tank given its volume [tex]\( V \)[/tex] and diameter [tex]\( d \)[/tex], we can use the formula for the volume of a cylinder. The volume [tex]\( V \)[/tex] of a cylinder is given by:
[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.
Since the diameter [tex]\( d \)[/tex] of the tank is given, we can express the radius [tex]\( r \)[/tex] as:
[tex]\[ r = \frac{d}{2} \][/tex]
Now, substituting [tex]\( r \)[/tex] into the volume formula gives:
[tex]\[ V = \pi \left( \frac{d}{2} \right)^2 h \][/tex]
Simplify the expression:
[tex]\[ V = \pi \left( \frac{d^2}{4} \right) h \][/tex]
Rearranging to solve for the height [tex]\( h \)[/tex]:
[tex]\[ V = \frac{\pi d^2 h}{4} \][/tex]
Multiplying both sides by 4:
[tex]\[ 4V = \pi d^2 h \][/tex]
Finally, dividing both sides by [tex]\(\pi d^2\)[/tex]:
[tex]\[ h = \frac{4V}{\pi d^2} \][/tex]
Therefore, the correct expression for the height [tex]\( h \)[/tex] of the fuel tank is:
[tex]\[ \boxed{\frac{4 V}{\pi d^2}} \][/tex]
Among the given options, the correct answer is:
D. [tex]\(\frac{4V}{\pi d^2}\)[/tex]
