A circular pond with a radius of r feet is surrounded by a... | SAT Area and volume Question, Answer Explained

Math · Geometry and TrigonometryHard

A circular pond with a radius of $r$ feet is surrounded by a walkway that is $3$ feet wide all the way around. The area of the walkway alone is $75\pi$ square feet. What is the value of $r$ ?

Student-produced response: enter your answer.

Show answer and explanation

Answer: 11

The pond and walkway together form a circle with radius $r + 3$ . The walkway's area is the difference of the two circles' areas, so $\pi(r + 3)^2 - \pi r^2 = 75\pi$ . Expanding gives $r^2 + 6r + 9 - r^2 = 75$ , so $6r + 9 = 75$ , $6r = 66$ , and $r = 11$ .

Using $r + 6$ for the outer radius, because the walkway is on both sides of the pond, gives $r = 3.25$ ; a radius crosses the walkway only once, so the outer radius is $r + 3$ .

Ask Owl about this question Next question

Got it wrong? On SAT Owl, Owl looks at the answer you chose and explains the exact mistake, then gives you similar questions until it sticks. Get your predicted SAT score free, or take a free full-length practice test.