2024 AMC 10B Problems/Problem 21

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Problem

Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, 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?

[asy] size(6cm); draw(circle((0,1),1), linewidth(1.2)); draw((-1,0)--(1.25,0), linewidth(1.2)); draw(circle((1,1/4),1/4), linewidth(1.2)); [/asy]

$\textbf{(A)}~\frac{1}{9} \qquad\textbf{(B)}~1 \qquad\textbf{(C)}~\frac{10}{9} \qquad\textbf{(D)}~\frac{11}{9} \qquad\textbf{(E)}~\frac{19}{9}$

Solution 1

In general, let the left and right outer circles and the center circle have radii $r_1,r_2,r_3$. When three circles are tangent as described in the problem, we can deduce $\sqrt{r_3}=\frac{\sqrt{r_1r_2}}{\sqrt{r_1}+\sqrt{r_2}}$ by Pythagorean theorem.

Setting $r_1=1, r_2=\frac14$ we have $r_3=\frac19$, and setting $r_1=1,r_3=\frac14$ we have $r_2=1$. Thus our answer is $\boxed{\textbf{(C)}}$.

~Mintylemon66


Solution 2

sum of radii = distances from center. Set the center of the big circle be (0,1). The center of the smaller circle is at (x_2, 1/4). (1+1/4)^2 = x_3^2 + (3/4)^2 -> x_2 = 4/4 = 1.

let (x_2,r_2) be the coordinates of the new circle. Then you have (x_2-0)^2+(r_3-1)^2 = (1+r_3)^2.

You also have (x_2-1)^2+(r_3-1/4)^2=(1/4 + r_3)^2.

These two you should get a quadratic for r_3, and you get sols 1 and 1/9 >>> 10/9 C

~mathboy282

Solution 3 (Descartes’s Theorem)

Descartes’s Theorem states that for curvatures $k_1,k_2,k_3,k_4$ we have

\[(k_1+k_2+k_3+k_4)^2=2(k_1^2+k_2^2+k_3^2+k_4^2)\]

with a curvature being the reciprocal of the radius of a circle, being positive if it is externally tangent to other circles, negative if it is internally tangent to other circles, and zero if we consider a line as a degenerate circle. Here our curvatures are $k_1=1,k_2=4,k_3=0$, and we wish to find $k_4$. Plugging these values into our formula yields:

\[(1+4+0+k_4)^2=2(1^2+4^2+0^2+k_4^2)\]

\[(5+k_4)^2=2(17+k_4^2)\]

\[k_4^2+10k_4+25=2k_4^2+34\]

\[k_4^2-10k_4+9=0\]

\[k_4=1,9\]

The curvature and the radius are reciprocals, so our radii must be $1$ and $\frac{1}{9}$, and their sum is $\boxed{\textbf{(C) }\frac{10}{9}}$.

~eevee9406

Video Solution 1 by Pi Academy (In Less Than 5 Mins ⚡🚀)

https://youtu.be/5fID8UOohr0?feature=shared

~ Pi Academy

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

See also

2024 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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