Circular Lawn with Path

A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grassy area is:

  1. 36pi – 34
  2. 30pi – 15
  3. 36pi – 33
  4. 35pi – 9√3
  5. 30pi – 9√3


Show/Hide Solution

Area I = pi(62)/2 = 18pi.

Area II = Sector AOB – △AOB.
We see that △AOC is a 30°-60° right △ since OC = 3 and
OA = 6. Thus ∠AOC = 60° and ∠AOB = 120°.
So the area of the sector = 120/360 · (pi(62)) = 36pi/3 = 12pi.

△AOB has base AB = 6√3 and altitude OC = 3. Its area is
(6√3 · 3)/2 = 9√3.

So the grassy area = I + II = 18pi + (12pi – 9√3) = 30pi – 9√3. The answer is (e).

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