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Zorluk: KolayCircle Geometry: Arc Length and Sector Area

A circular archery target has a radius of 12 inches12\text{ inches}. A sector of this target has a central angle of 150∘150^\circ. What is the area, in square inches, of this sector?

  1. A
    10π10\pi
  2. B
    15π15\pi
  3. 60π60\piCevap
  4. D
    120π120\pi
  5. E
    240π240\pi

Cevap

The correct area of the sector is 60π60\pi square inches.
To find the area of a sector, first calculate the total area of the circle, which is πr2=π(12)2=144π\pi r^2 = \pi (12)^2 = 144\pi square inches. Then, multiply this total area by the fraction of the circle that the sector represents: 150∘360∘=512\frac{150^\circ}{360^\circ} = \frac{5}{12}. Calculating the product gives 512×144π=60π\frac{5}{12} \times 144\pi = 60\pi square inches.

Adım Adım Çözüm

1
Find the total area of the circular archery target using the area formula A=πr2A = \pi r^2 with a radius of 12 inches12\text{ inches}.
The total area of the circle is π(12)2=144π\pi (12)^2 = 144\pi square inches.
The area of a sector is a fractional part of the circle's total area.
2
Calculate the fraction of the circle represented by a central angle of 150∘150^\circ.
The fraction is 150∘360∘=512\frac{150^\circ}{360^\circ} = \frac{5}{12}.
A complete circle has a central angle of 360∘360^\circ.
3
Multiply the total area of the circle by the fraction of the circle to determine the sector area.
The sector area is 512×144π=60π\frac{5}{12} \times 144\pi = 60\pi square inches.
Applying the fraction to the total area yields the area of the sector.

Anahtar Kavram

Calculating the area of a circle sector using the formula A=θ360∘πr2A = \frac{\theta}{360^\circ} \pi r^2.
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Circle Geometry: Arc Length and Sector Area

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