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Difficulty: MediumCircles

An automated lawn sprinkler rotates through an angle of 120∘120^\circ and sprays water outward to a maximum distance of 1818 feet. Which of the following represents the total area of the lawn covered by this sprinkler? (Leave your answer in terms of π\pi.)

  1. 108π sq ft108\pi \text{ sq ft}Answer
  2. B
    27π sq ft27\pi \text{ sq ft}
  3. C
    12π sq ft12\pi \text{ sq ft}
  4. D
    6π sq ft6\pi \text{ sq ft}

Answer

108π sq ft108\pi \text{ sq ft}
The sprinkler covers a sector of a circle. The radius of this circle is the maximum spray distance, which is 1818 feet. The full area of a circle with this radius would be π(18)2=324π\pi(18)^2 = 324\pi square feet. Since the sprinkler rotates 120∘120^\circ, which is 120360\frac{120}{360} or 13\frac{1}{3} of a full circle, the covered area is 13\frac{1}{3} of 324π324\pi, resulting in 108π108\pi square feet.

Step-by-Step Solution

1
Identify the geometric shape and the relevant area formula.
The area covered by the sprinkler is a sector of a circle. The formula for the area of a sector is Area=θ360∘×πr2Area = \frac{\theta}{360^\circ} \times \pi r^2.
The sprinkler covers a fractional portion of a full circle defined by its rotation angle.
2
Determine the fraction of the circle covered.
120∘360∘=13\frac{120^\circ}{360^\circ} = \frac{1}{3}
The sprinkler rotates 120∘120^\circ out of a total possible 360∘360^\circ for a complete circle.
3
Substitute the radius into the formula and solve.
13×π×(18)2=13×π×324=108π\frac{1}{3} \times \pi \times (18)^2 = \frac{1}{3} \times \pi \times 324 = 108\pi
The maximum spray distance (1818 feet) is the radius. Squaring 1818 gives 324324, and finding one-third of this yields the sector's area.

Key Concept

Area of a Circular Sector
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Topics

Circles

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