Question

how many complete rotations does the minute hand of a clock make as it moves from 12:00 noon to 4:00 p.m.?
rotations
what is the total radian measure of the angle the minute hand travels in 4 full rotations?

the minute hand of the clock is 5 inches long. how far does the tip of the minute hand move in 4 full rotations? calculate your answer using the π button on your calculator and round to the nearest tenth of an inch.
inches

Explanation:

First Question:

Step1: Calculate the number of hours from 12:00 noon to 4:00 p.m.

The time - duration from 12:00 noon to 4:00 p.m. is \(t = 4\) hours.

Step2: Determine the number of rotations of the minute hand per hour

The minute hand of a clock makes 1 complete rotation in 1 hour.

Step3: Find the number of rotations in 4 hours

Using the formula \(n=\text{number of hours}\times\text{rotations per hour}\), we have \(n = 4\times1=4\) rotations.

Second Question:

Step1: Recall the radian measure of one full rotation

One full rotation of a circle has a radian measure of \(\theta_1 = 2\pi\) radians.

Step2: Calculate the radian measure for 4 full rotations

Using the formula \(\theta=\text{number of rotations}\times\theta_1\), we get \(\theta = 4\times2\pi=8\pi\) radians.

Third Question:

Step1: Recall the formula for the circumference of a circle

The formula for the circumference of a circle is \(C = 2\pi r\), where \(r\) is the radius of the circle. Given \(r = 5\) inches, the circumference of the circle traced by the minute - hand is \(C=2\pi\times5 = 10\pi\) inches.

Step2: Calculate the distance for 4 full rotations

Since the distance \(d\) for \(n = 4\) rotations is \(d=n\times C\), substituting \(n = 4\) and \(C = 10\pi\) inches, we have \(d=4\times10\pi=40\pi\) inches.

Step3: Evaluate using a calculator

Using a calculator with the \(\pi\) button, \(d = 40\pi\approx40\times3.14159 = 125.7\) inches.

Answer:

1. \(4\)
2. \(8\pi\)
3. \(125.7\)