Question

1. the formula (s = 2pisqrt{\frac{l}{32}}) represents the swing of a pendulum. s is the time in seconds to swing back and forth, and l is the length of the pendulum in feet.

a. how long does it take for a 3 ft pendulum to swing back and forth?

Explanation:

⚡ Using what you learned: simplifying radical expressions

Step 1: Substitute the given value

Substitute \( L = 3 \) into the formula:

$$ s = 2\pi\sqrt{\frac{3}{32}} $$

Step 2: Simplify the radical expression

Simplify the fraction inside the square root:

$$ s = 2\pi\frac{\sqrt{3}}{\sqrt{32}} $$

$$ \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} $$

$$ s = 2\pi\frac{\sqrt{3}}{4\sqrt{2}} = \pi\frac{\sqrt{3}}{2\sqrt{2}} $$

Rationalize the denominator:

$$ s = \pi\frac{\sqrt{3} \times \sqrt{2}}{2\sqrt{2} \times \sqrt{2}} = \frac{\pi\sqrt{6}}{4} $$

Step 3: Calculate the decimal value

Using \(\pi \approx 3.14159\) and \(\sqrt{6} \approx 2.44949\):

$$ s \approx \frac{3.14159 \times 2.44949}{4} \approx 1.92\text{ seconds} $$

Answer:

The exact time is:

$$ \frac{\pi\sqrt{6}}{4}\text{ seconds} $$

Which is approximately:

$$ 1.92\text{ seconds} $$