Question

1. a pendulum swings through an angle of 25° each second. if the pendulum is 60 inches long, how far does its tip move each second? if necessary, round the answer to two decimal places.
2. a wheel with a 34 - inch radius is marked at two points on the rim. the distance between the marks along the wheel is found to be 10 inches. what is the angle (to the nearest tenth of a degree) between the radii to the two marks?
3. a carousel has a radius of 17 feet and takes 32 seconds to make one complete revolution. what is the linear speed of the carousel at its outside edge? express the answer in feet per second. if necessary, round the answer to two decimal places.

Explanation:

Problem 7

Step1: Recall arc length formula

The formula for the length of an arc \( s \) is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. First, convert \( 25^\circ \) to radians. We know that \( \theta_{\text{radians}}=\theta_{\text{degrees}}\times\frac{\pi}{180} \). So, \( 25^\circ\times\frac{\pi}{180}=\frac{5\pi}{36} \) radians.

Step2: Calculate arc length

Given \( r = 60 \) inches and \( \theta=\frac{5\pi}{36} \) radians. Substitute into the arc length formula: \( s = 60\times\frac{5\pi}{36}=\frac{300\pi}{36}=\frac{25\pi}{3}\approx\frac{25\times3.1416}{3}\approx26.18 \) inches.

Step1: Recall arc length formula

The arc length formula is \( s = r\theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians. We need to find \( \theta \) (in degrees), so first solve for \( \theta \) in radians: \( \theta=\frac{s}{r} \).

Step2: Calculate \( \theta \) in radians and convert to degrees

Given \( s = 10 \) inches and \( r = 34 \) inches. So, \( \theta=\frac{10}{34}=\frac{5}{17} \) radians. Now convert radians to degrees: \( \theta_{\text{degrees}}=\theta_{\text{radians}}\times\frac{180}{\pi} \). So, \( \frac{5}{17}\times\frac{180}{\pi}\approx\frac{900}{17\times3.1416}\approx\frac{900}{53.4072}\approx16.8 \) degrees.

Step1: Find the circumference and angular speed

The circumference of the carousel (a circle) is \( C = 2\pi r \), where \( r = 17 \) feet. So, \( C = 2\pi\times17 = 34\pi \) feet. The carousel makes one revolution ( \( 34\pi \) feet) in 32 seconds, so the linear speed \( v \) is the distance traveled (circumference) divided by time.

Step2: Calculate linear speed

\( v=\frac{34\pi}{32}=\frac{17\pi}{16}\approx\frac{17\times3.1416}{16}\approx3.34 \) feet per second.

Answer:

\( \approx 26.18 \) inches

Problem 8