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

Part 1:
`(2\pi)/(3)` radians

Part 2: The minute hand travels
`(8\pi)/(3)` inches.

Part 3: The minute hand travels
`(3\pi)/(4)` radians.

Part 4: The coordinate point is
`(-(√(2))/(2),(√(2))/(2))`

Explanation:

Part 1:

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

So each minute represents
`(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
`(\pi)/(180)`

120° is equal to
`120*(\pi)/(180)=(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
  `(2\pi)/(3)`)

So,
`s=r\theta\\s=(4)((2\pi )/(3))=(8\pi)/(3)`

The minute hand travels
`(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=(3\pi)/(4)`

The minute hand travels
`(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((3\pi)/(4))sin((3\pi)/(4)))\\(x,y)=(-(√(2))/(2),(√(2))/(2))`

Thus the coordinate point is
`(-(√(2))/(2),(√(2))/(2))`