Question

There are two sprinklers that spread water in a circular sector (90 degrees). They only spray in an 18-foot radius. (See the diagram: One sprinkler is in the top left corner and the other is in the bottom right corner. Part of the lawn is not covered by either sprinkler.)

The lawn measures 18' by 36', and each sprinkler has a spray radius of 18'.
What is the area of the section of lawn that is not watered by either sprinkler?

Answer 1

139.1 ft² (nearest tenth)

Explanation:

Formula

• Area of a rectangle = width × length
• Area of a quarter circle = ¹/₄πr² (where r is the radius)

The two areas of the lawn that are watered by the sprinklers are quarter circles with radius equal to the width of the lawn (18 foot).

Therefore, the area of the section of the lawn that is not watered is the area of the rectangle minus the two quarter circles.

`\begin{aligned}\textsf{Area of section}&=\textsf{Area of rectangle}-2 * \textsf{Area of quarter circle}\\\\& = (18 * 36)-\left(2 * (1)/(4) \pi (18)^2\right)\\\\& = (18 * 36)-\left((1)/(2) \pi (324)\right)\\\\& = 648-162 \pi \\\\& = 139.0619901...\sf \; ft^2\end{aligned}`

Therefore, the area of the section of lawn that is not watered by either sprinkler is 139.1 ft² (nearest tenth).

Answer 2

• 139.32 ft²

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Area of the rectangular lawn

• A = lw = 18 × 36 = 648 ft²

The area covered by sprinklers, two quarter-circles

• A = 2 × 1/4 × πr² = 1/2 × 3.14×18² = 508.68 ft²

Area of not watered section

• 648 - 508.68 = 139.32 ft²