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QuestionA clock's minute hand is 12 cm12 \mathrm{~cm} long. What distance does it cover in 15 minutes?

Studdy Solution

STEP 1

Assumptions1. The minute hand of a clock is12 cm long. . The minute hand moves uniformly around the clock.
3. The clock is a perfect circle, and the minute hand is a radius of this circle.
4. The distance the top of the minute hand travels is the circumference of the circle it traces out.
5. The minute hand completes a full circle (360 degrees) in60 minutes.

STEP 2

First, we need to calculate the total angle the minute hand moves in15 minutes. Since the minute hand completes a full circle (360 degrees) in60 minutes, we can find the angle for15 minutes by dividing360 by4.
Angle for15 minutes=3604\text{Angle for15 minutes} = \frac{360}{4}

STEP 3

Calculate the angle the minute hand moves in15 minutes.
Angle for15 minutes=360=90 degrees\text{Angle for15 minutes} = \frac{360}{} =90 \text{ degrees}

STEP 4

Now, we need to calculate the total distance the top of the minute hand travels. This is a fraction of the circumference of the circle traced out by the minute hand. The formula for the circumference of a circle is 2πr2\pi r, where rr is the radius (length of the minute hand). The fraction of the circumference the minute hand travels is equal to the fraction of the full circle (360 degrees) the minute hand moves.
Distance travelled=Angle for15 minutes360×2πr\text{Distance travelled} = \frac{\text{Angle for15 minutes}}{360} \times2\pi r

STEP 5

Plug in the values for the angle for15 minutes and the length of the minute hand (r) to calculate the distance travelled.
Distance travelled=90360×2π×12\text{Distance travelled} = \frac{90}{360} \times2\pi \times12

STEP 6

implify the fraction.
90360=0.25\frac{90}{360} =0.25Distance travelled=0.25×2π×12\text{Distance travelled} =0.25 \times2\pi \times12

STEP 7

Calculate the distance travelled.
Distance travelled=0.25×2π×12=6π cm\text{Distance travelled} =0.25 \times2\pi \times12 =6\pi \text{ cm}The top of the minute hand travels a distance of 6π6\pi cm in15 minutes.

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