2004 AMC 10A Problem 25

Problem 25 of 25HarderGeometry

Three mutually tangent spheres of radius

1

rest on a horizontal plane. A sphere of radius

2

rests on them. What is the distance from the plane to the top of the larger sphere?

Show solution

Solution

The three small centers form an equilateral triangle of side

2,

each

1

unit above the plane. Its centroid

D

is at distance

\dfrac{2\sqrt{3}}{3}

from each vertex. The large sphere’s center

E

sits directly above

D,

and the distance between

E

and a small center is

1 + 2 = 3.

Thus

\begin{aligned} DE &= \sqrt{3^2 - \left(\dfrac{2\sqrt{3}}{3}\right)^2} \\ &= \sqrt{9 - \dfrac{4}{3}} = \dfrac{\sqrt{69}}{3}. \end{aligned}

Adding the

1

unit from the plane to

D

and the

2

units from

E

to the top of the large sphere gives

1 + \dfrac{\sqrt{69}}{3} + 2 = 3 + \dfrac{\sqrt{69}}{3}.

Thus, the correct answer is B.