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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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