# A gas tank has ends that are hemispheres of radius r ft. the cylindrical midsection is 6 ft long. express the volume of the tank as a function of r?

Oct 15, 2015

${v}_{\text{all}} \left(f {t}^{3}\right) = \left(6 \pi {r}^{2} + \frac{4}{3} \pi {r}^{3}\right) f {t}^{3}$

#### Explanation:

Let volume be v then:
Let length of cylinder be L

Volume of two ends when put together form a sphere
${v}_{\text{sphere}} = \frac{4}{3} \pi {r}^{3}$ ........ ( 1 )

Volume of cylinder is circle area times length of cylinder
${v}_{\text{cylinder}} = \pi {r}^{2} L$
But $L$ = 6 (feet) giving
${v}_{\text{cylinder}} = 6 \pi {r}^{2}$ ........ ( 2 )

Both r and L are both measured in feet. so you can add the volumes directly, giving:

Putting the two together: ( 1 ) + ( 2 )

${v}_{\text{all}} = 6 \pi {r}^{2} + \frac{4}{3} \pi {r}^{3}$

Let feet be ft
${v}_{\text{all}} \left(f {t}^{3}\right) = \left(6 \pi {r}^{2} + \frac{4}{3} \pi {r}^{3}\right) f {t}^{3}$