Questions: Solve the following problem using the formula P=2 π sqrt(L/32), where P, the period of a pendulum in seconds, depends on L, its length in feet. Determine the length if the period is 4.8 seconds. Round to two decimal places.

Solution Steps

To determine the length \( L \) of a pendulum given its period \( P \), we can rearrange the formula \( P = 2\pi \sqrt{\frac{L}{32}} \) to solve for \( L \). First, isolate the square root term by dividing both sides by \( 2\pi \). Then, square both sides to eliminate the square root and finally multiply by 32 to solve for \( L \).

We start with the formula for the period of a pendulum given by

\[ P = 2\pi \sqrt{\frac{L}{32}}. \]

To find the length \( L \), we first isolate the square root term:

\[ \frac{P}{2\pi} = \sqrt{\frac{L}{32}}. \]

Next, we square both sides to eliminate the square root:

\[ \left(\frac{P}{2\pi}\right)^2 = \frac{L}{32}. \]

Now, we multiply both sides by 32 to solve for \( L \):

\[ L = 32 \left(\frac{P}{2\pi}\right)^2. \]

Substituting \( P = 4.8 \) into the equation, we have:

\[ L = 32 \left(\frac{4.8}{2\pi}\right)^2. \]

Calculating the value gives us:

\[ L \approx 18.68. \]

Final Answer