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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?

Respuesta :

taskmasters
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
TaeKwonDoIsDead

Answer:

Part 1: [tex]\frac{2\pi}{3}[/tex] radians

Part 2: The minute hand travels [tex]\frac{8\pi}{3}[/tex] inches.

Part 3: The minute hand travels [tex]\frac{3\pi}{4}[/tex] radians.

Part 4: The coordinate point is  [tex](-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})[/tex]


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  [tex]\frac{360}{60}=6[/tex] degrees

From 3:35 to 3:55 is 20 minutes. Hence, 20 minutes is [tex]6*20=120[/tex] degrees.


To convert from degrees to radians, we multiply the degrees by [tex]\frac{\pi}{180}[/tex]

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


Part 2:

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

Formula for arc length is  [tex]s=r\theta[/tex]

Where,

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

So, [tex]s=r\theta\\s=(4)(\frac{2\pi }{3})=\frac{8\pi}{3}[/tex]

The minute hand travels [tex]\frac{8\pi}{3}[/tex] inches.


Part 3:

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

[tex]s=r\theta\\3\pi=(4)(\theta)\\\theta=\frac{3\pi}{4}[/tex]

The minute hand travels [tex]\frac{3\pi}{4}[/tex] radians.


Part 4:

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

[tex](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})[/tex]

Thus the coordinate point is  [tex](-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})[/tex]