The rate (in cubic feet per hour) that a spherical snowball melts is proportional to the snowball's volume raised to the 2/3 power. (this assumes that the rate snow melts is proportional to the area exposed to the air -- i. e. the surface area of the snowball.) if a snowball of radius 1 foot (volume 4/3 \[pi] cubic feet) melts in 6 hours, how long will it take a snowball of raduis 3 feet (volume 36 cubic feet) to melt?

A 3 feet radius snowball will melt in 54 hours.

Explanation:

As we can assume that the rate of  snowball takes to melt is proportional to the surface area, then the rate for a 3 feet radius will be:

T= A(3 ft)/A(1 ft) * 6 hr

A is the area of the snowballs. For a spherical geometry is computing as:

A=4.pi.R^2

Then dividing the areas:

A(3 feet)/A(1 foot) = (4 pi (3 ft)^2)/(4 pi (1 ft)^2) =  (36pi ft^2)/(4pi ft^2)= 9

Finally, the rate for the 3 feet radius snowball is:

T= 9 * 6 hr = 54 hr