Certainly! Let's determine the height of a cylindrical fuel tank given its volume and diameter.
We know the following:
- The tank has a volume [tex]\( V \)[/tex] cubic meters.
- The diameter of the tank is [tex]\( s \)[/tex] meters.
The formula for the volume of a cylinder is:
[tex]\[ V = \pi r^2 h \][/tex]
where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius,
- [tex]\( h \)[/tex] is the height,
- and [tex]\( \pi \)[/tex] is a constant (approximately 3.14159).
First, express the radius [tex]\( r \)[/tex] in terms of the diameter [tex]\( s \)[/tex]:
[tex]\[ r = \frac{s}{2} \][/tex]
Substitute [tex]\( r \)[/tex] into the volume formula:
[tex]\[ V = \pi \left(\frac{s}{2}\right)^2 h \][/tex]
Simplify the equation:
[tex]\[ V = \pi \frac{s^2}{4} h \][/tex]
To solve for the height [tex]\( h \)[/tex], rearrange the equation:
[tex]\[ h = \frac{4V}{\pi s^2} \][/tex]
Therefore, the height of the tank is:
[tex]\[ h = \frac{4V}{\pi s^2} \][/tex]
However, our derivation needs to match the given final form. Let's re-express it properly:
[tex]\[ h = \frac{V}{ (\pi r^2)} = \frac{V}{( \pi (\frac{s}{2})^2)} = \frac{V}{ \pi \frac{s^2}{4}} = \frac{4V}{\pi s^2}\][/tex]
To simplify further:
[tex]\[ h = \frac{1.27323954473516 V}{s^2} \][/tex]
This was the derived result.
Therefore:
The height of the cylindrical fuel tank, given its volume [tex]\( V \)[/tex] cubic meters and diameter [tex]\( s \)[/tex] meters, is
[tex]\[ h = \frac{1.27323954473516 V}{s^2} \][/tex]
