Question

The minute hand of a clock is 6 inches long. how far does the tip of the minute hand move in 15 minutes? how far does it move in 25 minutes? round answers to two decimal places.

Answer 1

This problem involves finding arc lengths.  The formula for arc length 

is    s = r*theta, where r is the radius and theta the central angle.

In 15 minutes, the minute hand sweeps out 1/4 of a circle, or pi/2 radians.  This is the central angle.  The arc length (how far the minute hand moves in 15 min) is then

s = (6 inches)(pi/2 rad) = 3pi inches, or about 9.42 inches.

25 minutes is equivalent to a central angle of (25/60)pi rad, or 1.31 radians.  What is the associated arc length?  Calculate this in the same way as I did for a central angle of pi/2.

Answer 2

We can think that the tip of the minute hand moves in a circle of radius R = 6in.

We will find that for each case, 15 minutes and 25 minutes, the distance traveled is 9.42 in and 15.7 in correspondingly.

Remember that the length of an arc is defined by an angle θ in a circle of radius R is given by:

L = (θ/360°)×2×pi×R

where pi = 3.14

Here, we know that R = 6in.

Now we want to find how much does the tip of the minute hand moves in 15 minutes.

Then we need to find the angle θ that the tip moves in these 15 minutes.

Remember that a complete rotation in the clock has 60 minutes.

This means that there is an equivalence:

60 min = 360°

1 = (360°/60 min)

with this we can rewrite:

15 min = (15 min)×(360°/60 min) = 90°

Then, in 15 minutes, the tip of the minute hand moves:

L = (90°/360°)×2×3.14×6in = 9.42 in

For the case of 25 minutes the procedure is the same:

25 min =  (25 min)×(360°/60 min) = 150°

Then, in 25 minutes, the tip of the minute hand moves:

L = (150°/360°)×2×3.14×6in = 15.7 in

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