Mathematico | Mathematico

The image below shows a circular piece of grass surrounded by a concrete path. The width of the path is equal to $2 m$

Path diagram

Given that the area of the path is three times larger than the area of the grass, find the radius of the grass circle.

Hint

Let the radius of the grass part be $r$ . Then the radius of the outer circle is $r + 2$ . Recall the area of a circle is $π r^{2} .$

Hint

The area of the concrete is the area of the outer circle minus the area of the grass part.

Hint

" $A$ is three times bigger than $B$ " means $A = 3 B$ , not the other way round!

Solution

Let the radius of the grass be $r$ . Then the area of the grass is $π r^{2}$ .

The area of the path alone must be $π (r + 2)^{2} - π r^{2}$ .

Since the area of the path is $3$ times the area of the grass, we have

\begin{aligned} π (r + 2)^{2} - π r^{2} & = 3 π r^{2} \\ (r + 2)^{2} - 4 r^{2} & = 0 \\ r^{2} + 4 r + 4 - 4 r^{2} & = 0 \\ - 3 r^{2} + 4 r + 4 & = 0 \\ 3 r^{2} - 4 r - 4 & = 0 \\ (3 r + 2) (r - 2) & = 0 \\ r & \in {- \frac{2}{3}, 2} \end{aligned}

However, $r$ is a radius and cannot be negative, so $r = 2$ .