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Zorluk: OrtaArea of Two-Dimensional Shapes

A circular fountain with a diameter of 88 feet is positioned in the center of a square lawn. The lawn has a side length of 2020 feet. A concrete walkway of width 22 feet is built directly around the fountain. What is the area, in square feet, of the remaining grass region of the lawn?

  1. 400−36π400 - 36\piCevap
  2. B
    400−16π400 - 16\pi
  3. C
    400−24π400 - 24\pi
  4. D
    400−100π400 - 100\pi

Cevap

400−36π400 - 36\pi
The area of the square lawn is 202=40020^2 = 400 square feet. The fountain has a radius of 44 feet (half of the 88-foot diameter), and the walkway adds another 22 feet to the radius, making the total radius of the combined circular area 66 feet. The area of this circular region is π×62=36π\pi \times 6^2 = 36\pi square feet. Subtracting this from the total area of the square lawn gives the remaining grass area of 400−36π400 - 36\pi square feet.

Adım Adım Çözüm

1
Calculate the total area of the square lawn.
Area of the lawn is 20×20=40020 \times 20 = 400 square feet.
The area of a square is calculated by squaring its side length.
2
Find the combined radius of the fountain and the walkway.
The radius of the fountain is 8÷2=48 \div 2 = 4 feet. Adding the 22-foot width of the walkway gives a combined radius of 4+2=64 + 2 = 6 feet.
The radius is half of the diameter. The walkway surrounds the fountain, so its width must be added to the fountain's radius to find the outer boundary's radius.
3
Calculate the combined area of the fountain and the walkway.
Area of the combined circular region is π×62=36π\pi \times 6^2 = 36\pi square feet.
The area of a circle is given by πr2\pi r^2, where rr is the radius.
4
Subtract the combined circular area from the total area of the lawn to find the remaining grass area.
The remaining area is 400−36π400 - 36\pi square feet.
The remaining grass area is the total square area minus the area of the inner circular region that contains the fountain and the walkway.

Anahtar Kavram

Area of composite shapes involving squares and circles, and scaling properties.
Tahmini Süre:1m 30s
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Area of Two-Dimensional Shapes

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