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A freshly-painted circular emblem on a football field is completely covered by the smallest possible square tarp. The tarp covers an area of 196 square feet. In square feet, what is the area of the circular emblem? Express your answer in terms of pi

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Ashraf82

Answer:

The area of the circular emblem is 49π ft²

Step-by-step explanation:

  • The formula of the area of a square is A = s²,, where s is the side of the square
  • The formula of the area of a circle is A = Ï€r², where r is the radius of the circle

If a circle is inscribed in a square and touches its four sides, then the diameter of the circle is equal to the side of the square and they have the same center

∵ The circular emblem on a football field is completely covered

    by the smallest possible square tarp

- That means the circular emblem is inscribed in the square tarp

∴ The side of the trap = the diameter of the circular emblem

∵ The tarp covers an area of 196 square feet

∴ Area of the square = 194 ft²

∵ Area of the square = s²

∴ 196 = s²

- Take √  for both sides

∴ 14 = s

∴ The side of the trap is 14 feet

∵ The side of the trap = the diameter of the circular emblem

∴ The diameter of the circular emblem = 14 feet

- The radius of any circle is half its diameter

∴ The radius of the circular emblem = [tex]\frac{1}{2}[/tex] × 14 = 7 feet

∵ Area of the circle = πr²

∴ The area of the circular emblem = π(7)²

∴ The area of the circular emblem = 49π ft²

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