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 in diameter, what is the height of the tank in meters?

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]



Answer :

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]