To determine the height of the cylindrical fuel tank, we start with the volume formula for a cylinder. The volume [tex]\( V \)[/tex] of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
where:
- [tex]\( V \)[/tex] is the volume
- [tex]\( r \)[/tex] is the radius of the base of the cylinder
- [tex]\( h \)[/tex] is the height of the cylinder
- [tex]\( \pi \)[/tex] is a constant (approximately 3.14159)
Since [tex]\( r \)[/tex], the radius, is half the diameter [tex]\( d \)[/tex], we can express the radius as:
[tex]\[ r = \frac{d}{2} \][/tex]
Substituting [tex]\( r \)[/tex] into the volume formula, we get:
[tex]\[ V = \pi \left( \frac{d}{2} \right)^2 h \][/tex]
Simplifying inside the parentheses:
[tex]\[ V = \pi \left( \frac{d^2}{4} \right) h \][/tex]
[tex]\[ V = \frac{\pi d^2}{4} h \][/tex]
To isolate [tex]\( h \)[/tex], we solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{4V}{\pi d^2} \][/tex]
Examining the provided answer choices:
- A. [tex]\(\frac{2 V}{\pi d^2}\)[/tex]
- B. [tex]\(\frac{4 V}{d^2}\)[/tex]
- C. [tex]\(\frac{V}{\pi d^2}\)[/tex]
- D. [tex]\(\frac{4 V}{\pi d^2}\)[/tex]
- E. [tex]\(\frac{8 V}{\pi d^2}\)[/tex]
We see that the accurate solution for the height [tex]\( h \)[/tex] is:
[tex]\[ h = \frac{4V}{\pi d^2} \][/tex]
Thus, the correct answer is:
[tex]\[ \text{D.} \quad \frac{4 V}{\pi d^2} \][/tex]
