6) A Ferris wheel at the 1904 St. Louis World’s Fair had a diameter of 250 ft and made one revolution
in 30 minutes
a) Find the exact angular speed in radians per hour
b) Find the linear speed of a person riding the Ferris wheel in miles per hour. Round to 2 decimal
a) 2 rev/hr = 4π rad/hr
b) 4π*125 ft/hr
now make that into mi/hr
To find the exact angular speed in radians per hour, we need to convert the time from minutes to hours and then determine how many radians the Ferris wheel rotates in that time.
a) Let's start by converting the time to hours:
30 minutes = 30/60 = 0.5 hours
Next, we need to determine the angle through which the Ferris wheel rotates in 0.5 hours. One revolution of the Ferris wheel corresponds to a complete circle, which is equal to 2π radians.
Therefore, the angular speed in radians per hour is given by:
angular speed = (2π radians) / (0.5 hours)
= (2π/0.5) radians per hour
= 4π radians per hour
So, the exact angular speed of the Ferris wheel is 4π radians per hour.
b) To find the linear speed of a person riding the Ferris wheel in miles per hour, we need to know the circumference of the Ferris wheel.
The circumference of a circle is given by the formula:
circumference = 2π × radius
In this case, the diameter of the Ferris wheel is given as 250 ft, so the radius is half of that:
radius = diameter / 2 = 250 ft / 2 = 125 ft
Now, we can calculate the circumference:
circumference = 2π × 125 ft
Next, we need to convert the circumference from feet to miles, since we want the linear speed in miles per hour.
There are 5280 feet in a mile, so:
circumference = (2π × 125) ft / 5280 ft/mile
≈ 0.47124 miles
Finally, we need to find out how far a person riding the Ferris wheel travels in 0.5 hours, which is the time for one revolution.
linear speed = distance / time
= 0.47124 miles / 0.5 hours
= 0.94248 miles per hour
Rounding to two decimal places, the linear speed of a person riding the Ferris wheel is approximately 0.94 miles per hour.