Two circles with equal radii are enclosed by a rectangle, as shown. The distance between their centres is $\frac{2 x}{3}$. The value of $x$ is
(A) $\frac{15}{4}$
(B) 5
(C) 6
(D) $\frac{60}{7}$
(E) $\frac{15}{2}$
Two circles with equal radii are enclosed by a rectangle, as shown. The distance between their centres is $\frac{2 x}{3}$. The value of $x$ is
(A) $\frac{15}{4}$
(B) 5
(C) 6
(D) $\frac{60}{7}$
(E) $\frac{15}{2}$

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-221.jpg?height=309&width=504&top_left_y=1683&top_left_x=1388)

## Solution

We observe first of all that the width of the rectangle is $x$ which corresponds to the diameter of a circle or $2 r=x$. If the distance between the centres is $\frac{2}{3} x$ the length of the rectangle is equal to two radii plus the distance between the two centres. Or, $x+\frac{2}{3} x=10$

$$
\begin{aligned}
\frac{5}{3} x & =10 \\
x & =6
\end{aligned}
$$

ANSWER: (C)