trigonometry help

Question

The minute hand of a clock is 6 inches long. What distance does its tip move in 24 minutes?

Answer 1

Ezequiel, this just needs the simple definition formula:

s = rθ . . . . . . . . . . where s = arc length; r = radius; θ = angle in radians

In this case r = 6 so we only need to figure out the angle in radians the clock hand moves in 24 minutes

We know, that in 12 hours or 360 mins, it moves 2π radians

∴ in 24 mins, it moves (24/360) x 2π radians

= 2π/15 radians

Now, plug into the above equation: s = 6 x 2π/15

= 4π/5 inches

Note that the answer is a distance so the units are inches (not radians.)

Jeff

Answer 2

You can use the arc length formula provided in your textbook. We already know that r= 6.

Look below (under all the dashes) if you're trying to understand how the problem can be solved logically, and to get a sense of how the formula comes about.

Arc Length Formula (if θ is in degrees)

s = 2 π r (θ/360°)

Recall that every hour, a minute hand makes one complete rotation, ie 2pi rotation for every 1hr.

However, since we are only traveling for 24 minutes, we need to come up with an equivalent relationship.

2pi = 1hr so how many radians of rotation do we get for 24 minutes? Why dont we call it X?

So 2pi/X = 60min/24min

Cross multiply and you get 48pi = 60X.

Divide by 60 and you get 48pi/60 = X.

Reduce fractions and you get 4pi/5 = X.

Remember that X is in radians. So we get X= 4pi/5 radians.

To use the formula, we need to convert radians into degrees, we can do so by using our conversion of 180/pi since 180/pi = 1.

4pi/5 (180/pi) = 144 degrees.

Plug that into

s = 2 π r (θ/360°)

and you get 24pi /5 inches.

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Long winded solution (A logical approach without formulas)

The equation of the circumference of a circle is C(r)=2 (pi) r, where r is the independent variable, and C(r) is circumference.

Given that the minute hand is 6 inches long, then the circle formed by going around for a complete rotation is 2pi (6in) = 12pi inches.

Recall that every hour, a minute hand makes one complete rotation, ie 2pi rotation for every 1hr.

However, since we are only traveling for 24 minutes, we need to come up with an equivalent relationship.

2pi = 1hr so how many radians of rotation do we get for 24 minutes? Why dont we call it X?

So 2pi/X = 60min/24min

Cross multiply and you get 48pi = 60X.

Divide by 60 and you get 48pi/60 = X.

Reduce fractions and you get 4pi/5 = X.

Remember that X is in radians. So we get X= 4pi/5 radians.

So now instead of using the equation for the circumference of the entire circle, we can substitute 4pi/5 instead of 2pi and come up with the smaller arc created by traveling for only 24 minutes.

C(r) = (4pi/5) 6 = 24pi/5 inches

Answer 3

d = ωrt = 2π/60·24·6inch = 15.08 inch