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Difficulty: Very hardCircle Geometry: Arc Length and Sector Area

A pendulum of a certain length swings back and forth such that the tip of the pendulum traces an arc of length 6π6\pi inches. If the length of the pendulum is increased by 44 inches and it swings through the same central angle, the tip of the pendulum traces an arc of length 8π8\pi inches. What is the area, in square inches, of the circular sector swept out by the original pendulum?

  1. A
    12π12\pi
  2. B
    18π18\pi
  3. 36π36\piAnswer
  4. D
    72π72\pi
  5. E
    144π144\pi

Answer

The correct answer is 36π36\pi square inches.
The correct answer is 36π36\pi square inches. By using the arc length formula s=rθs = r\theta where θ\theta is in radians, we establish the system of equations rθ=6πr\theta = 6\pi and (r+4)θ=8π(r+4)\theta = 8\pi. Subtracting the first equation from the second gives 4θ=2π4\theta = 2\pi, which simplifies to θ=π2\theta = \frac{\pi}{2}. Substituting this back into the first equation yields r(π2)=6πr(\frac{\pi}{2}) = 6\pi, so the original radius r=12r = 12. Finally, we calculate the area of the original sector using the formula A=12r2θ=12(12)2(π2)=36πA = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2(\frac{\pi}{2}) = 36\pi.

Step-by-Step Solution

1
Write down the arc length equations for both pendulums using the formula s=rθs = r\theta, where rr is the length of the pendulum (radius) and θ\theta is the central angle in radians.
For the original pendulum: rθ=6πr\theta = 6\pi. For the extended pendulum: (r+4)θ=8π(r+4)\theta = 8\pi.
To establish the mathematical relationships between the given arc lengths, the pendulum lengths, and the central angle.
2
Solve the system of equations for the central angle θ\theta.
Distributing the second equation gives rθ+4θ=8πr\theta + 4\theta = 8\pi. Substituting rθ=6πr\theta = 6\pi into this yields 6π+4θ=8π  ⟹  4θ=2π  ⟹  θ=π26\pi + 4\theta = 8\pi \implies 4\theta = 2\pi \implies \theta = \frac{\pi}{2} radians.
To find the constant central angle of the pendulum's swing.
3
Substitute θ=π2\theta = \frac{\pi}{2} back into the first equation to solve for the original radius rr.
r(π2)=6π  ⟹  r=12r\left(\frac{\pi}{2}\right) = 6\pi \implies r = 12 inches.
To find the length of the original pendulum, which serves as the radius of the sector.
4
Calculate the area of the sector swept out by the original pendulum using the formula A=12r2θA = \frac{1}{2}r^2\theta.
A=12(12)2(π2)=12(144)(π2)=36πA = \frac{1}{2}(12)^2\left(\frac{\pi}{2}\right) = \frac{1}{2}(144)\left(\frac{\pi}{2}\right) = 36\pi square inches.
To find the final area of the sector as requested by the question.

Key Concept

Calculating sector area using arc length relationships to determine radius and angle.

Alternative Method

Instead of solving for θ\theta first, one can note that the ratio of the arc lengths is equal to the ratio of the radii because the central angle is constant: rr+4=6π8π=34\frac{r}{r+4} = \frac{6\pi}{8\pi} = \frac{3}{4}. Solving for rr gives 4r=3r+12  ⟹  r=124r = 3r + 12 \implies r = 12. Since the arc length of the original sector is s=6πs = 6\pi, we can use the sector area formula A=12rs=12(12)(6π)=36πA = \frac{1}{2}rs = \frac{1}{2}(12)(6\pi) = 36\pi square inches.
Estimated Time:3m 0s
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Topics

Circle Geometry: Arc Length and Sector Area

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