Question

There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is $5$ inches from each wall which that ball touches and $10$ inches from the floor, then the sum of the diameters of the balls is

$20$ inches

$30$ inches

$40$ inches

$60$ inches

not determined by the given information

Answer: C

Concepts:sphere coordinate geometry quadratic

Difficulty rating: 2040

Small Hint:

Place the corner at the origin with the walls and floor as coordinate planes

Big Hint:

A sphere of radius $r$ tangent to all three planes has center $(r,r,r)$

Solution:

For radius $r,$ the center is $(r,r,r)$ and the given point is $(5,5,10).$ Thus $2(5-r)^2+(10-r)^2=r^2,$ which simplifies to $r^2-20r+75=0,$ or $(r-5)(r-15)=0.$ The two radii are $5$ and $15,$ so the sum of the diameters is $2(5+15)=40$ inches.

Therefore, the correct answer is C.

1977 AMC 12 Problem 27

Answer: C

Difficulty rating: 2040

Small Hint:

Big Hint:

Solution:

Problem 27 in Other Years