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abruzzese [7]
3 years ago
13

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?
Mathematics
2 answers:
timurjin [86]3 years ago
6 0
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
ale4655 [162]3 years ago
3 0

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:


<u>Part 1:</u>

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.


<u><em>To convert from degrees to radians, we multiply the degrees by \frac{\pi}{180}</em></u>

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


<u>Part 2:</u>

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.


<u>Part 3:</u>

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.


<u>Part 4:</u>

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})

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