Question

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Answer 1

230.38 mm

Explanation:

The distance traveled by the tip of the hands is (part of) the circumference of the circle with radius of the lengths of the hands.

`\textsf{Circumference of a circle}=\sf 2 \pi r\quad\textsf{(where r is the radius)}`

Radii

Larger circle (minute hand):

• r = 200% of 20 mm = 40 mm

Smaller circle (hour hand):

• r = 20 mm

Minute Hand

The minute hand does a full rotation of the circle in one hour.

Therefore, the distance it travels in one hour is the complete circumference of a circle with radius 40 mm:

`\begin{aligned} \implies \textsf{Distance minute hand travels} & = \sf 2 \pi (40)\\ & = \sf 80 \pi \: mm\end{aligned}`

Hour Hand

There are 12 numbers on a clock.

The hour hand travels from one number to the next in one hour.

Therefore, the distance it travels in one hour is 1/12th of the circumference of the circle:

`\begin{aligned}\implies \sf \textsf{Distance hour hand travels} & =\left((1)/(12)\right)2 \pi r\\ & = \sf \left((1)/(12)\right)2 \pi (40)\\& = \sf (20)/(3)\pi \: mm \end{aligned}`

To find how much farther the tip of the minute hand moves than the tip of the hour hand, subtract the latter from the former:

`\begin{aligned}\implies \textsf{distance} & = \textsf{minute hand distance}-\textsf{hour hand distance}\\& = \sf 80 \pi - (20)/(3) \pi \\& = \sf (220)/(3) \pi \\& = \sf 230.38\: mm \:(nearest\:hundredth) \end{aligned}`

Answer 2

• 240.73 mm

Explanation:

The hour hand is 20 mm and the minute hand is

• 200% of 20 mm = 2*20 mm = 40 mm long

The tip of the hour hand travels 1/12 of the circle and the tip of the minute hand travels full circle in one hour.

Find each length and their difference using circumference formula.

• C = 2πr

Hour hand

• 1/12*(2*3.14*20) = 10.47 mm (rounded)

Minute hand

• 2*3.14*40 = 251.2 mm

The difference between the two above

• 251.2 - 10.47 = 240.73 mm