The volumes of two cylindrical cans of the same shape vary directly as the cubes of their radii. If a can with a six-inch radius holds pints, how many gallons will a similar can with a 24-inch radius hold?

Jul 30, 2018

color(brown)(V_2 = 8 " gallons"

Explanation:

$\text{Volume of cylinder } = V = \pi {r}^{2} h$

Let ${r}_{1} , {h}_{1}$ be the radius and height of the first cylinder and ${r}_{2} , {h}_{2}$ the second.

color(purple)(r_1 = 6 " inch", r_2 = 24 " inch", V_1 = 1 " pint"

${V}_{1} = \pi {r}_{1}^{2} {h}_{1} , {V}_{2} = \pi {r}_{2}^{2} {h}_{2}$

${V}_{2} / {V}_{1} = {r}_{2}^{3} / {r}_{1}^{3} = {24}^{3} / {6}^{3} , \text{ given}$

$1 g a l l o n = 8 \pi n t s$

$\therefore {V}_{2} = {\left(\frac{24}{6}\right)}^{3} \cdot {V}_{1} = {\left(\frac{24}{6}\right)}^{3} \cdot 1 = {4}^{3} = 64 \text{ pints}$

color(brown)(V_2 = 64 / 8 = 8 " gallons, as 8 pints = 1 gallon"