Questions: The period T (in seconds) of a pendulum is given by T=2π√(L/32), where L stands for the length (in feet) of the pendulum. If π=3.14, and the period is 9.42 seconds, what is the length? The length of the pendulum is feet.

The period T (in seconds) of a pendulum is given by T=2π√(L/32), where L stands for the length (in feet) of the pendulum. If π=3.14, and the period is 9.42 seconds, what is the length?
The length of the pendulum is feet.

Transcript text: The period $T$ (in seconds) of a pendulum is given by $T=2 \pi \sqrt{\left(\frac{L}{32}\right)}$, 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 $\qquad$ feet.

Solution

Solution Steps

Step 1: Substitute Given Values

Substitute the given values into the formula $ T = 2 \pi \sqrt{\left(\frac{L}{32}\right)} $: \[ 9.42 = 2 \times 3.14 \times \sqrt{\left(\frac{L}{32}\right)} \]

Step 2: Simplify the Equation

Simplify the equation to isolate the square root term: \[ 9.42 = 6.28 \times \sqrt{\left(\frac{L}{32}\right)} \] \[ \frac{9.42}{6.28} = \sqrt{\left(\frac{L}{32}\right)} \]

Step 3: Solve for the Square Root Term

Calculate the left side of the equation: \[ \frac{9.42}{6.28} \approx 1.5 \] \[ 1.5 = \sqrt{\left(\frac{L}{32}\right)} \]

Step 4: Square Both Sides

Square both sides to eliminate the square root: \[ 1.5^2 = \frac{L}{32} \] \[ 2.25 = \frac{L}{32} \]

Step 5: Solve for $ L $

Multiply both sides by 32 to solve for $ L $: \[ L = 2.25 \times 32 \] \[ L = 72 \]

The length of the pendulum is $ 72 $ feet.

The solution is $ 72 $