How far does the tip of a minute hand on a clock travel in 45 minutes if the distance from the center to the tip is 10 in? Leave your answer in terms of π.
Show work.
The minute hand of a clock is a circle with radius 10 in.
The circumference of the circle is 2πr, where r is the radius.
So, the distance the tip of the minute hand travels in one minute is:
2π(10 in) / 60 min = π/3 in/min
Therefore, the distance the tip of the minute hand travels in 45 minutes is:
(π/3 in/min) * 45 min = 15π in
Answer: The tip of the minute hand travels 15π inches in 45 minutes.
To find the distance the tip of the minute hand travels, we need to calculate the circumference of the circle formed by the tip of the minute hand.
The formula for the circumference of a circle is C = 2πr, where C is the circumference, π is pi (approximately 3.14159), and r is the radius.
In this case, the radius is given as 10 inches. So the circumference of the circle is:
C = 2π(10)
C = 20π
Since the minute hand of a clock travels the circumference of the circle in 60 minutes, we need to find the distance it travels in 45 minutes.
To calculate the distance traveled in 45 minutes, we can use the proportion:
45 minutes is to 60 minutes as x is to 20π inches
This can be written as:
45/60 = x/20π
Simplifying, we get:
3/4 = x/20π
To solve for x, we can cross multiply:
3 * 20π = 4 * x
60π = 4x
x = 60π/4
x = 15π
Therefore, the tip of the minute hand on a clock travels 15π inches in 45 minutes.