Semicircle distance
Problem
The diagram shows a rectangle and two semicircles.
The height of the rectangle 10 cm, and the area of the shaded region is 125 cm$^2$.
What is the shortest distance between the two semicircles?
Give your answer in terms of $\pi$.
This problem is adapted from the World Mathematics Championships
Student Solutions
Let the length of the rectangle be $x$. Then by considering the area of the whole rectangle, we can find $x$.
The area of the whole rectangle can be found using length $\times$ width = $10x$, as shown below.
This must be equal to the shaded area added to the total area of the two semicircles. The radius of each semicircle must be half of 10 cm, which is 5 cm.
So the red area is equal to $\pi\times 5^2$ cm$^2=25\pi$ cm$^2$, because the two red semicircles could be stuck together to make a circle of radius 5 cm.
That means that the total area of the rectangle must be $125 + 25\pi$ cm$^2$, so $10x=125+25\pi$, so $x=\dfrac{125+25\pi}{10}=12.5+2.5\pi$.
Now, as shown below, the shortest distance can be found using $x$.
$d=x-5-5=x-10$, so $d=12.5+2.5\pi-10=2.5+2.5\pi$ cm.