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?

1 Answer
Jul 30, 2018

color(brown)(V_2 = 8 " gallons"V2=8 gallons

Explanation:

"Volume of cylinder " = V = pi r^2 hVolume of cylinder =V=πr2h

Let r_1, h_1r1,h1 be the radius and height of the first cylinder and r_2, h_2r2,h2 the second.

color(purple)(r_1 = 6 " inch", r_2 = 24 " inch", V_1 = 1 " pint"r1=6 inch,r2=24 inch,V1=1 pint

V_1 = pi r_1^2 h_1, V_2 = pi r_2 ^2 h_2V1=πr21h1,V2=πr22h2

V_2 / V_1 = r_2 ^3 / r_1 ^3 = 24^3 / 6^3, " given"V2V1=r32r31=24363, given

https://www.ck12.org/measurement/convert-between-ounces-cups-pints-quarts-and-gallons/enrichment/Measuring-capacity-in-Cups-Pints-Quarts-and-Gallons-Overview/

1 gallon = 8 pints1gallon=8πnts

:. V_2 = (24/6)^3 * V_1 = (24/6)^3 * 1 = 4^3 = 64 " pints"

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