A rectangular park measures 30 m by 20 m. A quarter-circle flower bed of radius 7 m is placed in each of the four corners. A running track is built around the remaining area. How long is the track?

Step-by-step solution

1. 1

   Decide what the boundary is actually made of. Removing a quarter circle at a corner deletes the corner itself. The boundary of what is left is no longer four straight sides: it is four shortened straight sides plus four circular arcs. So the answer is not the rectangle's perimeter, and it is not the perimeter minus the flower beds' full outlines either.

2. 2

   Shorten each straight side. Each corner eats $7\ \text{m}$ off the side running one way and $7\ \text{m}$ off the side running the other way. Every side loses $7\ \text{m}$ at each of its two ends, so

   $\text{long sides: }2(30-2\cdot 7)=2(16)=32\ \text{m},\qquad\text{short sides: }2(20-2\cdot 7)=2(6)=12\ \text{m},$

   giving $32+12=44\ \text{m}$ of straight track. (Both $30-14$ and $20-14$ are positive, so the four beds do not overlap — worth checking before going further.)

3. 3

   Add the four arcs. Each arc is a quarter of a circle of radius $7$, and four quarters make one whole circle:

   $4\cdot\frac{1}{4}\left(2\pi r\right)=2\pi(7)=14\pi\ \text{m}\approx 43.98\ \text{m}.$

4. 4

   Total the perimeter.

   $P=44+14\pi\approx 44+43.98=87.98\ \text{m}.$

   Because the radius is $7$, the school-book approximation $\pi\approx\tfrac{22}{7}$ makes the arcs exactly $2\cdot\tfrac{22}{7}\cdot 7=44\ \text{m}$ and the total exactly $88\ \text{m}$ — which is why $88\ \text{m}$ is the intended answer.

5. 5

   Check the result against the plain rectangle. The original perimeter is $2(30+20)=100\ \text{m}$. At each corner, $14\ \text{m}$ of straight edge is replaced by an arc of $\tfrac{7\pi}{2}\approx 11\ \text{m}$, a loss of about $3\ \text{m}$; four corners lose about $12\ \text{m}$, and $100-12=88\ \text{m}$. Rounding the corners shortens the lap, as it should.