1. If a bicycle wheel has a diameter of 27 inches, find the radian measure of the angle through which the wheel turns as the bicycle travels a distance of one mile.
thanks!!
To find the radian measure of the angle through which the wheel turns, we need to use the formula:
θ = s / r
Where:
θ is the angle in radians
s is the arc length
r is the radius
In this case, we're given the diameter of the wheel, which is 27 inches. Since the radius is half the diameter, the radius would be 27 / 2 = 13.5 inches.
Now, we need to find the arc length and convert it to miles. We know that the circumference of the wheel (which is the arc length) is given by the formula:
C = 2πr
Substituting the value of the radius (13.5 inches), we get:
C = 2π × 13.5 = 27π inches
To convert this to miles, we need to multiply by the conversion factor (1 mile = 63,360 inches). So:
s = 27π inches × (1 mile / 63,360 inches) = π / 2,360 miles
Now, we can substitute the values into the formula:
θ = (π / 2,360 miles) / 13.5 inches
Simplifying the expression:
θ = π / (2,360 × 13.5) miles per inch
≈ π / 31,860 radians per inch
Finally, since we want to find the angle through which the wheel turns for one mile of travel, we multiply by the number of inches in a mile (12 inches × 5,280 feet):
θ = π / 31,860 radians per inch × (12 inches × 5,280 feet)
≈ π / 609,120 radians
Therefore, the radian measure of the angle through which the wheel turns as the bicycle travels one mile is approximately π / 609,120 radians.