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Difficulty: HardPerimeter and Area of 2D Figures

An architect is designing a circular observation deck with a circumference of 100π100\pi feet. The floor of the deck will be covered entirely with weather-resistant tiles. If one box of tiles covers 50π50\pi square feet, how many boxes are required to completely cover the floor?

  1. A
    2
  2. 50Answer
  3. C
    100
  4. D
    200

Answer

50
The radius of the deck is found by dividing the circumference by 2π2\pi (100π÷2π=50100\pi \div 2\pi = 50 feet). The total area is then calculated using the formula A=πr2A = \pi r^2, resulting in 2500π2500\pi square feet. Finally, dividing the total area by the coverage per box (50π50\pi) gives the required number of boxes, which is 5050.

Step-by-Step Solution

1
Calculate the radius of the circular deck using the given circumference.
r=50r = 50 feet
The formula for circumference is C=2πrC = 2\pi r. Substituting 100π100\pi for CC gives 100π=2πr100\pi = 2\pi r, which simplifies to r=50r = 50.
2
Calculate the total area of the observation deck.
Area=2500πArea = 2500\pi square feet
The formula for the area of a circle is A=πr2A = \pi r^2. Substituting r=50r = 50 gives A=π(50)2=2500πA = \pi(50)^2 = 2500\pi.
3
Determine the number of boxes of tiles needed.
5050 boxes
Divide the total area by the coverage of one box: 2500π÷50π=502500\pi \div 50\pi = 50.

Key Concept

Perimeter and Area of 2D Figures
Estimated Time:1m 0s
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Perimeter and Area of 2D Figures

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