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The period $T$ (in seconds) of a pendulum is given by $T=2\pi\sqrt{(\frac{L}{32})}$ , where $L$ stands for the length (in feet) of the pendulum. If $\pi=3.14$ , and the period is $9.42$ seconds, what is the length? The length of the pendulum is in feet.

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Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. The period

T

(in seconds) of a pendulum is given by

T=2\pi\sqrt{(\frac{L}{32})}

, where

L

stands for the length (in feet) of the pendulum. If

\pi=3.14

, and the period is

9.42

seconds, what is the length? The length of the pendulum is in feet.

Given Formula Application: We are given the formula for the period of a pendulum: $T = 2 \cdot \pi \cdot \sqrt{\frac{L}{32}}$ , where $T$ is the period in seconds, $L$ is the length in feet, and $\pi$ is approximately $3.14$ . We need to solve for $L$ when $T$ is $9.42$ seconds.
Substitution and Simplification: First, let's plug in the values we know into the formula. We have $T = 9.42$ and $\pi = 3.14$ . So the equation becomes $9.42 = 2 \times 3.14 \times \sqrt{L / 32}$ .
Isolating Square Root Term: To solve for $L$ , we first need to isolate the square root term. We do this by dividing both sides of the equation by $2 \times 3.14$ . This gives us $\frac{9.42}{2 \times 3.14} = \sqrt{\frac{L}{32}}$ .
Removing Square Root: Now we calculate the left side of the equation: $9.42 / (2 \times 3.14) = 9.42 / 6.28$ . This simplifies to approximately $1.5 = \sqrt{L / 32}$ .
Squaring Both Sides: Next, we square both sides of the equation to remove the square root. This gives us $(1.5)^2 = \frac{L}{32}$ .
Calculating L: Calculating the left side, we have $1.5^2 = 2.25$ . So, $2.25 = L / 32$ .
Final Calculation: To find $L$ , we multiply both sides of the equation by $32$ . This gives us $2.25 \times 32 = L$ .
Conclusion: Finally, we calculate $L$ : $2.25 \times 32 = 72$ . So, the length of the pendulum is $72$ feet.

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