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Difficulty: HardCircle Geometry: Arc Length and Sector Area

An automated lawn sprinkler sweeps through a central angle of θ\theta radians and waters a sector-shaped region of radius rr feet. Due to a mechanical adjustment, the central angle θ\theta is increased by 20%20\%, while the water pressure is reduced such that the radius rr is decreased by 10%10\%. What is the net percentage change in the area of the region watered by the sprinkler?

  1. Decreases by 2.8%2.8\%Answer
  2. B
    Increases by 2.8%2.8\%
  3. C
    Increases by 8.0%8.0\%
  4. D
    Increases by 10.0%10.0\%
  5. E
    Decreases by 19.0%19.0\%

Answer

Decreases by 2.8%2.8\%
The area of a circular sector is proportional to the square of its radius and linearly proportional to its central angle (A=12r2θA = \frac{1}{2} r^2 \theta). Decreasing the radius by 10%10\% scales the radius by 0.900.90, which scales r2r^2 by (0.90)2=0.81(0.90)^2 = 0.81. Increasing the central angle by 20%20\% scales θ\theta by 1.201.20. The overall scaling factor for the sector area is 0.81×1.20=0.9720.81 \times 1.20 = 0.972. This corresponds to 97.2%97.2\% of the original area, which is a net decrease of 2.8%2.8\%.

Step-by-Step Solution

1
Express the original sector area in terms of rr and θ\theta.
A1=12r2θA_1 = \frac{1}{2} r^2 \theta
The area of a circular sector with radius rr and central angle θ\theta in radians is given by A=12r2θA = \frac{1}{2} r^2 \theta.
2
Express the new radius and central angle after the percentage adjustments.
rnew=0.90rr_{new} = 0.90 r and θnew=1.20θ\theta_{new} = 1.20 \theta
A 10%10\% decrease in radius leaves 90%90\% of rr, and a 20%20\% increase in angle yields 120%120\% of θ\theta.
3
Substitute the new variables into the sector area formula and simplify.
A2=12(0.90r)2(1.20θ)=12(0.81r2)(1.20θ)=0.972(12r2θ)=0.972A1A_2 = \frac{1}{2} (0.90 r)^2 (1.20 \theta) = \frac{1}{2} (0.81 r^2) (1.20 \theta) = 0.972 \left(\frac{1}{2} r^2 \theta\right) = 0.972 A_1
Squaring 0.900.90 gives 0.810.81, and multiplying 0.81×1.200.81 \times 1.20 yields 0.9720.972.
4
Calculate the net percentage change from A1A_1 to A2A_2.
Percentage Change =(0.972−1)×100%=−2.8%= (0.972 - 1) \times 100\% = -2.8\%
A factor of 0.9720.972 means the new area is 97.2%97.2\% of the original area, which is a decrease of 2.8%2.8\%.

Key Concept

Circular Sector Area under proportional changes of parameters
Estimated Time:2m 0s
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Topics

Circle Geometry: Arc Length and Sector Area

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