Questions: The formula ω = θ/t can be rewritten as θ = ωt. Using ωt for θ changes s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable. s = 6π cm, r = 2 cm, ω = π/6 radian per sec t = sec (Type an integer or a fraction.)

Solution Steps

We are given the formula \( s = r \omega t \), where:

• \( s \) is the arc length,
• \( r \) is the radius,
• \( \omega \) is the angular velocity,
• \( t \) is the time.

We need to find the value of \( t \).

Substitute the given values into the formula:

• \( s = 6\pi \) cm,
• \( r = 2 \) cm,
• \( \omega = \frac{\pi}{6} \) rad/s.

The equation becomes: \[ 6\pi = 2 \cdot \frac{\pi}{6} \cdot t \]

First, simplify the right side of the equation: \[ 2 \cdot \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]

So the equation is: \[ 6\pi = \frac{\pi}{3} \cdot t \]

To solve for \( t \), divide both sides by \(\frac{\pi}{3}\): \[ t = \frac{6\pi}{\frac{\pi}{3}} \]

Simplify the division: \[ t = 6\pi \times \frac{3}{\pi} = 18 \]

Final Answer