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?

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 Link

TaeKwonDoIsDead TaeKwonDoIsDead · Answer 2 · 2019-01-07T05:29:49-05:00

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 \((-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\)