A satellite launch rocket has a cylindrical fuel tank. The fuel tank can hold [tex]$V$[/tex] cubic meters of fuel. If the tank measures [tex]$d$[/tex] meters across, what is the height of the tank in meters?

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]



Answer :

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]