Question

A Ferris wheel at a carnival has a radius of 23 feet. Suppose a passenger is traveling at 9 miles per hour. (A useful fact: 1 mi-5280 ft.)

(a) Find the angular speed of the wheel in radians per minute.
(b) Find the number of revolutions the wheel makes per hour. (Assume the wheel does not stop.)

Answer 1

(a) To find the angular speed of the wheel in radians per minute, we first need to convert the speed from miles per hour to feet per minute. We can use the conversion factor of 1 mile = 5280 feet.

9 miles/hour * (5280 feet/mile) = 48640 feet/hour

To convert to feet per minute, divide by 60:

48640 feet/hour / 60 minutes/hour = 810 feet/minute

The formula for angular speed is:

ω = v/r

where ω is the angular speed (in radians per minute), v is the linear speed (in feet per minute), and r is the radius of the wheel (in feet).

So we can plug in the values:

ω = 810 feet/minute / 23 feet = 35.217391304347826 radians/minute

(b) To find the number of revolutions per hour, we use the formula:

R = v/(2πr)

where R is the number of revolutions per hour, v is the linear speed (in feet per minute), and r is the radius of the wheel (in feet).

We already have v from part (a), so we can plug in the values:

R = 810 feet/minute / (2 * π * 23 feet) = 810/(2π * 23) = 810/141.3716694115407 = 5.717791411042944

So the wheel makes about 5.717791411042944 revolutions per hour.