Question

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?

$\dfrac{1}{9}$

$1$

$\dfrac{10}{9}$

$\dfrac{11}{9}$

$\dfrac{19}{9}$

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, $2\sqrt r + \sqrt r = 1,$ so $\sqrt r = \tfrac13$ and $r = \tfrac19.$ Sitting past the small pipe, $2\sqrt r - \sqrt r = 1,$ so $r = 1.$ (Past the big pipe can't happen.) The sum is $\tfrac19 + 1 = \tfrac{10}{9}.$ Thus, C is the correct answer.

2024 AMC 10B Problem 21

Solution:

Problem 21 in Other Years