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Difficulty: Very hardPerimeter and Area of 2D Figures

A circular swimming pool is surrounded by a concrete deck of uniform width. The circumference of the outside edge of the deck is 44π44\pi feet, and the circumference of the pool itself is 36π36\pi feet. What is the area of the concrete deck in square feet?

  1. A
    8π8\pi
  2. B
    16π16\pi
  3. 160π160\piAnswer
  4. D
    640π640\pi

Answer

The area of the concrete deck is 160π160\pi square feet.
To find the area of the concrete deck, we must calculate the area of the large circle (pool plus deck) and subtract the area of the small circle (pool). First, find the radius of the large circle by dividing its circumference by 2π2\pi: 44π/2π=2244\pi / 2\pi = 22. Its area is π(22)2=484π\pi(22)^2 = 484\pi. Next, find the radius of the pool: 36π/2π=1836\pi / 2\pi = 18. Its area is π(18)2=324π\pi(18)^2 = 324\pi. Finally, subtract the inner area from the outer area: 484π−324π=160π484\pi - 324\pi = 160\pi.

Step-by-Step Solution

1
Find the radius of the entire circular area (pool plus deck).
2πR=44π2\pi R = 44\pi gives R=22R = 22 feet.
The area formula requires the radius, which is found by dividing the circumference by 2π2\pi.
2
Find the radius of the pool itself.
2πr=36π2\pi r = 36\pi gives r=18r = 18 feet.
We need the inner radius to calculate the area of the pool.
3
Calculate the total area of the pool and deck combined.
Areatotal=π(22)2=484πArea_{total} = \pi (22)^2 = 484\pi square feet.
This is the area of the larger outer circle.
4
Calculate the area of the pool.
Areapool=π(18)2=324πArea_{pool} = \pi (18)^2 = 324\pi square feet.
This is the area of the inner circle.
5
Subtract the pool's area from the total area to find the area of the deck.
484π−324π=160π484\pi - 324\pi = 160\pi square feet.
The deck is the ring-shaped region between the outer and inner circles.

Key Concept

Area of a Circular Ring (Annulus)
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Topics

Perimeter and Area of 2D Figures

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