GMAT question - The figure above shows a circular pond... - Review

Type: Problem Solving

Difficulty:

The figure above shows a circular pond with it's center at O, surrounded by a circular path that is 3 feet wide. If the area of the path is $\lineskip 1em \textstyle 63 \pi$ square feet then what is the radius of the pond in feet?

1. 8
2. 9
3. 10
4. 11
5. 12

Explanation

The area of the path is equal to the total area of the path and pond minus the area of the pond.

If the radius of the pond is r then the radius of the path and pond is r + 3.

Therefore we can calculate the area of the path in terms of the radius of the pond.

$\lineskip 1em \begin{split} \text{Area of the path} &= \text{area of pond and path} - \text{area of the pond} \\ &= (r+3)^2 pi - r^2 pi \end{split}$

Now we substitute in $\lineskip 1em \textstyle 63 \pi$ for the area of the path

$\lineskip 1em 63 \pi = (r+3)^{2} \pi - r^{2} \pi$

Divide both sides by $\lineskip 1em \textstyle \pi$

$\lineskip 1em 63 = (r+3)^{2} - r^{2}$

Multiply out the brackets

$\lineskip 1em 63 = r^{2} + 6r + 9 - r^{2}$

Simplify

$\lineskip 1em 63 = 6r + 9$

Subtract 9 from both sides

$\lineskip 1em 54 = 6r$

And divide both sides by 6

$\lineskip 1em 9 = r$

Therefore the radius of the pond is 9 feet and the correct answer is B.