Question

Freddie is at chess practice waiting on his opponent's next move. He notices that the 4-inch-long minute hand is rotating around the clock and marking off time like degrees on a unit circle.
Part 1: How many radians does the minute hand move from 3:35 to 3:55? (Hint: Find the number of degrees per minute first.)
Part 2: How far does the tip of the minute hand travel during that time?
Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 3π inches?
Part 4: What is the coordinate point associated with this radian measure?

Answer 1

Part 1: How many radians does the minute hand move from 3:35 to 3:55?2π/3

Part 2: How far does the tip of the minute hand travel during that time?π/90

Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 3π inches?0.2193

Answer 2

Answer:

Part 1: $\frac{2\pi}{3}$ radians

Part 2: The minute hand travels $\frac{8\pi}{3}$ inches.

Part 3: The minute hand travels $\frac{3\pi}{4}$ radians.

Part 4: The coordinate point is  $(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$

Step-by-step explanation:

Part 1:

There are 60 minutes in an hour. 1 hour is 1 revolution (1 circle), which is 360°.

So each minute represents  $\frac{360}{60}=6$ degrees

From 3:35 to 3:55 is 20 minutes. Hence, 20 minutes is $6*20=120$ degrees.

To convert from degrees to radians, we multiply the degrees by $\frac{\pi}{180}$

120° is equal to  $120*\frac{\pi}{180}=\frac{2\pi}{3}$ radians

Part 2:

We want to find the "arc length" of this.

Formula for arc length is  $s=r\theta$

Where,

• s is the arc length
• r is the radius (here the minute hand was given as 4 inches)
• $\theta$ is the angle in radians (we found it to be $\frac{2\pi}{3}$)

So, $s=r\theta\\s=(4)(\frac{2\pi }{3})=\frac{8\pi}{3}$

The minute hand travels $\frac{8\pi}{3}$ inches.

Part 3:

Here we use the arc length formula where we want to find $\theta$ given that $s=3\pi$ and radius is 4 inches. So we have:

$s=r\theta\\3\pi=(4)(\theta)\\\theta=\frac{3\pi}{4}$

The minute hand travels $\frac{3\pi}{4}$ radians.

Part 4:

The coordinate point associated with a specif radian is given by the formula:

$(x,y)=(cos(\theta)sin(\theta))\\(x,y)=(cos(\frac{3\pi}{4})sin(\frac{3\pi}{4}))\\(x,y)=(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$

Thus the coordinate point is  $(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$