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The time it takes for a pendulum to swing back and forth can be represented by the function \( S(x)=2 \pi \cdot \sqrt{\frac{x}{32}} \) where \( S(x) \) is the time in seconds and \( x \) is the length of the pendulum in feet. a. How many seconds will take for a 9-foot pendulum to swing back and forth one time? Round your answer to one decimal place. seconds b. If it takes 4 seconds for a pendulum to swing back andforth one time, what is the length of the pendulum? Round your answer to one decimal place. feet

Ask by King West. in the United States
Mar 15,2025

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a. A 9-foot pendulum swings back and forth in approximately 3.3 seconds. b. A pendulum that swings back and forth in 4 seconds is about 13.0 feet long.

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To find the time it takes for a 9-foot pendulum to swing back and forth, we can plug \( x = 9 \) into the function \( S(x) \). \[ S(9) = 2 \pi \cdot \sqrt{\frac{9}{32}} \approx 2 \pi \cdot 0.5 = \pi \approx 3.1 \text{ seconds} \] So, a 9-foot pendulum takes approximately **3.1 seconds** for one complete swing. Next, to find the length of the pendulum that takes 4 seconds to swing back and forth, we set up the equation: \[ 4 = 2 \pi \cdot \sqrt{\frac{x}{32}} \] Dividing both sides by \( 2 \pi \): \[ \frac{4}{2\pi} = \sqrt{\frac{x}{32}} \] Squaring both sides gives: \[ \left(\frac{4}{2\pi}\right)^2 = \frac{x}{32} \] Multiplying both sides by 32: \[ x = 32 \cdot \left(\frac{4}{2\pi}\right)^2 \approx 32 \cdot \left(\frac{2}{\pi}\right)^2 \approx 32 \cdot 0.4053 \approx 12.9 \] Hence, the length of the pendulum is approximately **12.9 feet**.

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