David recently visited Pop's Ice Cream Shoppe and ordered the $1 sugar cone with one scoop of vanilla ice cream placed firmly atop the cone.

Pop told David that he would receive a percentage discount in the cost of the treat equal to the closest integer value to the answer of the following question if he answered it correctly:

"If the sugar cone is a right circular cone with a height of 10 inches, and the scoop of vanilla ice cream is a perfect sphere with a diameter of 4 inches, and both the cone and sphere are equal in spatial volume, what percentage of ice cream is above the base of the cone?"

David, a bright student, gave a correct answer. How much did David pay for the ice cream cone?

The sphere's volume is 4*pi*r^3 / 3. As the radius is 2 inches, the volume is 33.5103216382911 cubic inches.

The cone's volume is to be the same and its formula is B*h/3 where B is the area of the base.

B*h/3 = 33.5103216382911
B = 33.5103216382911 * 3 / 10 = 10.05309649148733 sq.in.

10.05309649148733 = pi*r^2 where r is the radius of the cone's base, so this r = sqrt(3.2) = 1.788854381999832.

The slant height of the cone then makes an angle with the cones axis equal to arctan (1.788854381999832 / 10) = 10.14210615657398°. The length of the slant height is then 10 / cos(10.14210615657398°) = 10.15874007936024.

Then consider the triangle formed by the center of the sphere, the apex of the cone and one slant height element of the cone, to where the sphere contacts that slant height. The two inch radius of the sphere is sufficient to reach from the lip of the cone to the axis, so the triangle includes that lip point of contact with the sphere and the cone.

Using the law of sines and calling the acute angle at the center of the sphere C:

sin C / 10.15874007936024 = sin 10.14210615657398° / 2
sin C = .8944271909999157
C = 63.434948822922°

Then the angle that the radius of the sphere makes with its slant height at the point of touching is 180 - 63.434948822922 - 10.14210615657398 = 106.422945020504°.

Then to find the distance d from the center of the sphere to the apex of the cone we use the law of sines again:

d / sin 106.422945020504° = 2 / sin 10.14210615657398°
d = 10.89442719099992 inches.

So at maximum the sphere extends 12 - 10.89442719099992 = 1.105572809000083 inches into the cone.

Wolfram's Mathworld gives a formula for the volume of a spherical cap:

V = pi * h * (3*a^2 + h^2) / 6

where a is the radius of the cap's base and h is the cap's height.

Plugging in, we get pi * 1.105572809000083 * (3 * 1.788854381999832^2 + 1.105572809000083^2) / 6 = 6.26477082079138 cu. in. Out of the total volume of the sphere, 33.5103216382911, this is the fraction .1869504831500286 or 18.69504831500286 %. So 81.30495168499714 % is above the base.  The closest integer to this is 81%--the discount that David received. Then, 81% off of $1.00 is 19 cents, the amount David paid.

Posted by Charlie on 2010-03-01 16:40:24