2024 AMC 10B Problem 21

Problem 21 of 25HarderGeometry

Two straight pipes (circular cylinders), with radii

1

and

\tfrac14,

lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?

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Solution

Two circles of radii

R

and

r

resting on the floor and touching each other have contact points a horizontal distance

2\sqrt{Rr}

apart. So the radius-

1

and radius-

\tfrac14

pipes touch the floor

2\sqrt{1 \cdot \tfrac14} = 1

apart. A third pipe of radius

r

sits

2\sqrt{r}

from the big pipe’s contact point and

2\sqrt{\tfrac14 r} = \sqrt{r}

from the small pipe’s. Nestled between them,

2\sqrt r + \sqrt r = 1,

\sqrt r = \tfrac13

and

r = \tfrac19.

Sitting past the small pipe,

2\sqrt r - \sqrt r = 1,

r = 1.

(Past the big pipe can’t happen.) The sum is

\tfrac19 + 1 = \tfrac{10}{9}.

Thus, C is the correct answer.

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