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Celia is staring at the clock waiting for school to end so that she can go to track practice. She 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 1:25 to 1:50? (Hint: Find the number of degrees per minute first.) Part 2: How far does the tip of the minute hand travel during that time? You must show all of your work.

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Part 1: How many radians does the minute hand move from 1:25 to 1:50? (Hint: Find the number of degrees per minute first.)


Answer: 5Ï€/6 radians.

Explanation:

From 1:25 to 1:50 there are 50 - 25 = 25 minutes.


The whole turn of the minute hand is 60 minutes.


That is a ratio of 25 minutes / 60 minutes.


A complete turn is 2Ï€ radians, so you have this ratio with the unknown number of radians the minute hand has moved from 1:25 to 1:50: x / (2Ï€).


Set a proportion with the two ratios:

25 / 60 = x / (2π) ⇒ x = (2π) × 25 / 60 = 5π/6 radians.


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

Answer: 10π/3 incjes ≈ 10.47 inches

Explanation: 

Being the minute hand 4 inch long, that is the radius of the circle described bt the minute hand.

Since 1 complete turn describes an arc of 2Ï€r, you can set a new proportion:

2Ï€r = 2Ï€(4in) = 8Ï€ in is the length of the full turn.

x / 8Ï€ is the ratio of the arc described to the length of the circle, and it is proportional to the ratio of minutes 25 / 60:

x / 8π = 25/60 ⇒ x = 8π × 25 / 60 = 10π/3 ≈ 10.47 

The units is inches