\( 1 \leftarrow \) The period of a pendulum is the time it takes the pendulum to make one full swing back and forth. The period \( T \), in seconds, is given by the formula \( T=2 \pi \sqrt{\frac{L}{32}} \), where \( L \) is the length, in feet, of the pendulum. Suppose a pendulum used to show the rotation of the earth has a period of approximately 18.2 seconds. What is its length? Use 3.14 for \( \pi \). The length of a pendulum that has a period of 18.2 seconds is 269.12 feet. (Round to two decimal places as needed.)

Let's solve the problem step by step using the given formula and the specified value for \( \pi \). **Given:** - Period \( T = 18.2 \) seconds - \( \pi = 3.14 \) - Formula for the period of a pendulum: \[ T = 2\pi \sqrt{\frac{L}{32}} \] where \( L \) is the length in feet. **Objective:** Find the length \( L \) of the pendulum. **Step 1: Isolate the Square Root** Start with the formula: \[ 18.2 = 2 \times 3.14 \times \sqrt{\frac{L}{32}} \] Divide both sides by \( 2 \times 3.14 \) to isolate the square root: \[ \frac{18.2}{6.28} = \sqrt{\frac{L}{32}} \] Calculate the left side: \[ \frac{18.2}{6.28} \approx 2.899 \] So, \[ 2.899 = \sqrt{\frac{L}{32}} \] **Step 2: Remove the Square Root** Square both sides to eliminate the square root: \[ (2.899)^2 = \frac{L}{32} \] Calculate \( (2.899)^2 \): \[ 2.899^2 \approx 8.403 \] So, \[ 8.403 = \frac{L}{32} \] **Step 3: Solve for \( L \)** Multiply both sides by 32 to solve for \( L \): \[ L = 8.403 \times 32 \] Calculate the multiplication: \[ L \approx 268.896 \text{ feet} \] **Step 4: Round to Two Decimal Places** \[ L \approx 268.90 \text{ feet} \] **Conclusion:** Using \( \pi = 3.14 \), the length of the pendulum with a period of 18.2 seconds is approximately **268.90 feet** when rounded to two decimal places. **Note:** If you use a more precise value of \( \pi \) (e.g., \( \pi \approx 3.14159265 \)), the calculated length would be slightly larger, closer to the 269.12 feet you obtained. However, based on the instructions to use \( \pi = 3.14 \), the accurate length is **268.90 feet**.