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 9-foot pendulum to swing back and forth one time? Round your answer to one decimal place. seconds b. If it takes 4 seconds for a pendulum to swing back andforth one time, what is the length of the pendulum? Round your answer to one decimal place. feet

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**Step 1. Write Down the Given Function**

The time function for the pendulum is given by  
$$
S(x) = 2\pi \cdot \sqrt{\frac{x}{32}}
$$
where $$ S(x) $$ is the time in seconds and $$ x $$ is the length in feet.

---

**a. Finding the Time for a 9-ft Pendulum**

Substitute $$ x = 9 $$ into the function:

$$
S(9) = 2\pi \cdot \sqrt{\frac{9}{32}}
$$

**Step 2. Simplify Under the Square Root**

$$
\sqrt{\frac{9}{32}} = \frac{\sqrt{9}}{\sqrt{32}} = \frac{3}{\sqrt{32}}
$$

**Step 3. Write the Expression for $$ S(9) $$**

$$
S(9) = 2\pi \cdot \frac{3}{\sqrt{32}} = \frac{6\pi}{\sqrt{32}}
$$

**Step 4. Calculate the Numerical Value**

First, calculate $$\sqrt{32}$$:

$$
\sqrt{32} \approx 5.65685
$$

Then compute:

$$
S(9) \approx \frac{6\pi}{5.65685}
$$

Using $$\pi \approx 3.14159$$,

$$
6\pi \approx 18.8496
$$

Thus,

$$
S(9) \approx \frac{18.8496}{5.65685} \approx 3.333 \text{ seconds}
$$

Rounding to one decimal place:

$$
S(9) \approx 3.3 \text{ seconds}
$$

---

**b. Finding the Length $$ x $$ for a 4-Second Swing**

Given that $$ S(x) = 4 $$ seconds, we set:

$$
4 = 2\pi \cdot \sqrt{\frac{x}{32}}
$$

**Step 1. Isolate the Square Root Term**

Divide both sides by $$ 2\pi $$:

$$
\sqrt{\frac{x}{32}} = \frac{4}{2\pi} = \frac{2}{\pi}
$$

**Step 2. Square Both Sides**

$$
\frac{x}{32} = \left(\frac{2}{\pi}\right)^2 = \frac{4}{\pi^2}
$$

**Step 3. Solve for $$ x $$**

Multiply both sides by 32:

$$
x = 32 \cdot \frac{4}{\pi^2} = \frac{128}{\pi^2}
$$

**Step 4. Calculate the Numerical Value**

Using $$\pi^2 \approx 9.8696$$,

$$
x \approx \frac{128}{9.8696} \approx 12.976
$$

Rounding to one decimal place:

$$
x \approx 13.0 \text{ feet}
$$

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**Final Answers:**

a. The 9-foot pendulum takes approximately $$3.3$$ seconds to swing back and forth one time.  
b. A pendulum that takes $$4$$ seconds to swing back and forth one time is approximately $$13.0$$ feet long.
a. A 9-foot pendulum swings back and forth in approximately 3.3 seconds. b. A pendulum that swings back and forth in 4 seconds is about 13.0 feet long.

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