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
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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**.