Question

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sophia visited a science museum with her school and observed a large pendulum. from her physics class, she remembered that the time t (in seconds) it takes for a pendulum to complete a full cycle is determined by its length l (in feet). the formula for the period of a pendulum is:
$t = 2\pi\sqrt{\frac{l}{32}}$
sophia timed the pendulum and found that it took about 10 seconds to complete a full swing. what is the length of the pendulum in feet, rounded to the nearest whole number?
72 feet
71 feet
82 feet
81 feet

Explanation:

Step1: Substitute given value of T

Given $T = 10$, substitute into $T = 2\pi\sqrt{\frac{L}{32}}$. So, $10=2\pi\sqrt{\frac{L}{32}}$.

Step2: Isolate the square - root term

First, divide both sides by $2\pi$. We get $\frac{10}{2\pi}=\sqrt{\frac{L}{32}}$. Simplify $\frac{10}{2\pi}$ to $\frac{5}{\pi}$. So, $\frac{5}{\pi}=\sqrt{\frac{L}{32}}$.

Step3: Square both sides

Squaring both sides gives $(\frac{5}{\pi})^2=\frac{L}{32}$. Since $(\frac{5}{\pi})^2=\frac{25}{\pi^{2}}$, then $\frac{25}{\pi^{2}}=\frac{L}{32}$.

Step4: Solve for L

Cross - multiply: $L = \frac{25\times32}{\pi^{2}}$. Calculate $25\times32 = 800$. And $\pi\approx3.14$, so $\pi^{2}\approx9.8596$. Then $L=\frac{800}{9.8596}\approx81.14$.

Answer:

81 feet