the minute hand of the tower clock is 12 inches long and the hour hand is 9 inches long. on this clock what is the positive difference between the number of inches traveled by the tip of the minute hand and the tip of the hour hand in one hour? express your answer as a decimal in terms of pi.
consider 12:00 am , the hands will coincide.
In on hour the minute hand makes one rotation, or 2πr inches
= 2π(12) or 24π inches
in the same time the hour hand will make 1/12 of a rotation.
distance of hour hand = (1/12)(2π)(9) = 3π/2 inches
So the difference = 24π - 3π/2 = 45π/2 inches
circumference of hour hand circle = 2 pi *9
in one hour it goes circumference / 12
so
distance = 18 pi/12 = 1.5 pi
now the minute hand
circumference = 2 pi * 12
it goes all the way around so
answer = 24 pi - 1.5 pi = 22.5 pi
opoiolklo
To find the difference between the number of inches traveled by the tip of the minute hand and the tip of the hour hand in one hour, we need to calculate the respective distances.
The minute hand of the clock makes a full rotation every 60 minutes, while the hour hand makes a full rotation every 12 hours.
First, let's find the length of the arc covered by the minute hand in one hour:
The circumference of a circle is given by the formula C = 2πr, where r is the radius. In this case, the radius of the minute hand is 12 inches.
So, the distance covered by the minute hand in one hour is:
C_minute = (2π)(12) = 24π inches.
Next, let's find the length of the arc covered by the hour hand in one hour:
Since the hour hand completes a full rotation in 12 hours, in one hour, it covers 1/12 of a full rotation. Therefore, the angle covered is (1/12) × 360 degrees = 30 degrees.
To find the arc length, we need to find the circumference of the sector formed by the hour hand's movement. The radius of the hour hand is 9 inches.
The formula for the arc length of a sector is given as S = (θ/360) × 2πr, where θ is the angle in degrees and r is the radius.
So, the distance covered by the hour hand in one hour is:
S_hour = (30/360) × (2π)(9) = (1/12) × 18π = 3π/2 inches.
Finally, we can find the positive difference between the distances covered by the minute and hour hands in one hour:
Difference = C_minute - S_hour
= 24π - 3π/2
= (48π - 3π)/2
= 45π/2
Therefore, the positive difference between the number of inches traveled by the tip of the minute hand and the tip of the hour hand in one hour is 45π/2 inches.