The time it takes for a pendulum to swing back and forth can be represented by the function
\[
S(x)=2 \pi \cdot \sqrt{\frac{x}{32}}
\]
where $S(x)$ is the time in seconds and $x$ is the length of the pendulum in feet.
a. How many seconds will take for a 5-foot pendulum to swing back and forth one time? Round your answer to one decimal place.
seconds
b. If it takes 6 seconds for a pendulum to swing back and forth one time, what is the length of the pendulum? Round your answer to one decimal place.
feet
Answer
Final Answer: The time it takes for a 5-foot pendulum to swing back and forth one time is approximately \(\boxed{2.5}\) seconds.
Step 1 ：The problem has two parts. Let's first solve part a. The question asks for the time it takes for a 5-foot pendulum to swing back and forth one time. This can be calculated by substituting \(x=5\) in the given function \(S(x)=2 \pi \cdot \sqrt{\frac{x}{32}}\).
Step 2 ：Substitute \(x=5\) into the function to get \(S(5)=2 \pi \cdot \sqrt{\frac{5}{32}}\).
Step 3 ：Solving the above expression gives approximately 2.5.
Step 4 ：Final Answer: The time it takes for a 5-foot pendulum to swing back and forth one time is approximately \(\boxed{2.5}\) seconds.
