High School

A clock has a radius of 6.2 inches. The minute hand is 5.9 inches long, and the hour hand is 5.5 inches long.

a. How much longer is the circumference of the clock than the circumference of the path of the tip of the minute hand? Round to the nearest hundredth.

b. How much longer is the circumference of the path of the tip of the minute hand than the circumference of the path of the tip of the hour hand? Round to the nearest hundredth.

Answer :

a. the circumference of the clock is approximately 1.884 inches longer than the circumference of the path traced by the tip of the minute hand. b. the circumference of the path traced by the tip of the minute hand is approximately 2.512 inches longer than the circumference of the path traced by the tip of the hour hand.

To solve these questions, let's calculate the circumferences involved:

a. Circumference of the clock:

The circumference of a circle is given by the formula C = 2πr, where r is the radius.

C_clock = 2π(6.2) = 12.4π inches

Circumference of the path of the tip of the minute hand:

The tip of the minute hand traces a circle with a radius of 5.9 inches.

C_minute_hand = 2π(5.9) = 11.8π inches

The difference in circumferences is:

C_difference = C_clock - C_minute_hand

C_difference = 12.4π - 11.8π

C_difference = 0.6π

To find the numerical value of C_difference, we can approximate π as 3.14:

C_difference ≈ 0.6 * 3.14 ≈ 1.884 inches

Therefore, the circumference of the clock is approximately 1.884 inches longer than the circumference of the path traced by the tip of the minute hand.

b. Circumference of the path of the tip of the minute hand:

C_minute_hand = 11.8π inches

Circumference of the path of the tip of the hour hand:

The tip of the hour hand traces a circle with a radius of 5.5 inches.

C_hour_hand = 2π(5.5) = 11π inches

The difference in circumferences is:

C_difference = C_minute_hand - C_hour_hand

C_difference = 11.8π - 11π

C_difference = 0.8π

Approximating π as 3.14:

C_difference ≈ 0.8 * 3.14 ≈ 2.512 inches

Therefore, the circumference of the path traced by the tip of the minute hand is approximately 2.512 inches longer than the circumference of the path traced by the tip of the hour hand.

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