A water wheel has a radius of 15 ft. A river passing through the water wheel turns the wheel at 11 revolutions per minute(rpm). What is the speed of the river in miles per hour?
what's the circumference of the wheel? 30pi
So, 30pi ft flows by 11 times per minute
That makes a river speed of 30pi*11 ft/min
Now convert that to mi/hr
To find the speed of the river in miles per hour, we need to calculate how far the water wheel travels in one minute and then convert it to miles per hour.
First, let's calculate the circumference of the water wheel using its radius. The formula for calculating the circumference of a circle is C = 2πr.
C = 2 * π * r
C = 2 * 3.14 * 15
C ≈ 94.2 ft
Next, let's calculate the distance traveled by the water wheel in one minute. Since the water wheel makes 11 revolutions per minute, we can multiply the circumference by the number of revolutions.
Distance = C * Number of revolutions
Distance = 94.2 ft * 11
Distance ≈ 1036.2 ft
Now, we need to convert the distance from feet to miles. There are 5,280 feet in a mile.
Distance in miles = Distance in feet / 5280
Distance in miles = 1036.2 ft / 5280
Distance in miles ≈ 0.20 miles
Finally, we need to convert the time from minutes to hours. There are 60 minutes in an hour.
Speed = Distance / Time
Speed = 0.20 miles / (1 minute / 60)
Speed ≈ 0.20 miles / 0.017 hours
Speed ≈ 11.76 miles per hour
Therefore, the speed of the river is approximately 11.76 miles per hour.
To find the speed of the river in miles per hour, we need to calculate the distance covered by a point on the circumference of the water wheel in one minute.
The formula to calculate the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
Given that the radius of the water wheel is 15 ft, we can substitute this value into the equation to find the circumference:
C = 2π(15) = 30π ft.
Since the water wheel makes 11 revolutions per minute, the distance covered by a point on the circumference in one minute would be:
Distance = Circumference × Number of Revolutions
Distance = 30π × 11 ft
Now, we need to convert this distance from feet to miles. There are 5,280 feet in a mile, so we can use this conversion factor:
Distance (in miles) = (30π × 11 ft) / 5280
To find the speed in miles per hour, we need to convert the distance from minutes to hours. There are 60 minutes in an hour, so we can use this conversion factor:
Distance (in miles per hour) = Distance (in miles) / 1 hour
So, the speed of the river in miles per hour can be calculated as:
Speed (in miles per hour) = [(30π × 11 ft) / 5280] / 1 hour
Now, let's calculate the value of the expression:
Speed (in miles per hour) ≈ [(30 × 3.14 × 11) / 5280] / 1 ≈ 0.2166 mph
Therefore, the speed of the river passing through the water wheel is approximately 0.2166 miles per hour.