Semicircle distance
Can you find the shortest distance between the semicircles given the area between them?
Age
14 to 16
Challenge level
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 $\pi$.
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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.
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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.
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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 $x$.
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$d=x-5-5=x-10$, so $d=12.5+2.5\pi-10=2.5+2.5\pi$ cm.