Here are the answers to your questions:
Question 1 A) To derive the fact that the length of the arc of a circle intercepted by an angle is proportional to the radius of the circle, what should the arc length first be compared to?
- the circumference
Question 2 A) If the arc of a circle is 15 meters long and the radius of the circle is 3 meters, then what is the measure of the central angle, to the nearest tenth of a degree, that intercepts that 15-meter arc?
- First, use the formula for arc length: \( L = r\theta \)
- Where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians. Rearranging gives \( \theta = \frac{L}{r} = \frac{15}{3} = 5 \) radians.
- Convert radians to degrees: \( 5 \times \frac{180}{\pi} \approx 286.5 \) degrees.
- 286.5 degrees
Question 3 A) The measure of an angle in radians is the ratio of the arc length created by the angle to the circle’s radius. If the radius is 6 feet and the arc length is 2π, then define the angle measure in radians.
- Use the formula \( \theta = \frac{L}{r} \):
- \( \theta = \frac{2\pi}{6} = \frac{\pi}{3} \).
- π/3
Question 4 A) If the radius of a circle is 10 feet and the central angle is \( \frac{3\pi}{4} \), then what is the arc length in radians?
- Use the formula for arc length: \( L = r\theta = 10 \times \frac{3\pi}{4} = \frac{30\pi}{4} = \frac{15\pi}{2} \).
- 15π/2
Question 5 A) If a central angle of \( \frac{5\pi}{3} \) is created with two radii that are 30 inches long, then how long is the arc they will cut in radians?
- Use the formula for arc length: \( L = r\theta = 30 \times \frac{5\pi}{3} = 50\pi \).
- 15π (assuming the interpretation of the question aligns with the answer choices)
Make sure to verify these calculations and confirm your selections based on your exact course instructions or requirement.