A new sport utility vehicle has tires with a diameter of 27.1 inches. The tires have a tread-life warranty of 61,000 miles.
(a)
How many radians will the tires rotate through within the full warranty length?
The total linear distance in this case gives the total circumference length the tire rolls through. How is an angle in radians related to circumference and radius? Be careful of unit conversions. rad
(b)
How many revolutions is this equivalent to?
61000mi * 5280ft/mi * 12in/ft * 1rev/27.1πin) * 2πrad/rev = ____rad
To find the number of radians the tires will rotate through within the full warranty length, we need to understand the relationship between radians, circumference, and radius.
The formula for calculating the circumference of a circle is:
Circumference = 2 * π * radius
In this case, the diameter of the tires is given as 27.1 inches. The radius of the tires would be half of the diameter, which is 27.1/2 = 13.55 inches.
To convert inches to radians, we need to convert the circumference from inches to radians using this formula:
Radians = Circumference / radius
So, the number of radians the tires will rotate through within the full warranty length can be calculated as follows:
Radians = (2 * π * radius) / radius
= 2 * π
≈ 6.2832 radians
Therefore, the tires will rotate through approximately 6.2832 radians within the full warranty length.
Now, let's calculate the number of revolutions that this is equivalent to.
We know that one revolution is equal to 2π radians.
So, the number of revolutions is calculated as follows:
Revolutions = Radians / (2 * π)
= 6.2832 / (2 * π)
≈ 1.0000 revolutions
Therefore, the number of revolutions to which this is equivalent is approximately 1.0000 revolution.
To calculate the number of radians the tires will rotate through within the full warranty length, we first need to find the circumference of the tires.
(a) The circumference of a circle is given by the formula:
C = 2πr
where C is the circumference and r is the radius. Since we have the diameter of the tires, we can find the radius by dividing the diameter by 2:
r = 27.1 inches / 2 = 13.55 inches
Now we can calculate the circumference:
C = 2π * 13.55 inches
To convert the circumference from inches to radians, we need to keep in mind that 1 revolution is equal to 2π radians, since a complete revolution is equivalent to the full 360 degrees of a circle.
So, the distance in radians is:
Distance in radians = (C / 2π) * 2π = C
Therefore, the tires will rotate through a distance equal to the circumference.
(b) To find the number of revolutions the tires are equivalent to, we divide the distance traveled by the circumference. Since the tires have a tread-life warranty of 61,000 miles, we need to convert this distance to the same units as the circumference:
Distance in miles = 61,000 miles
We can then convert the distance in miles to inches by multiplying by the conversion factor:
Distance in inches = 61,000 miles * 5280 feet/mile * 12 inches/foot
Now, we can calculate the number of revolutions:
Number of revolutions = Distance in inches / Circumference
This gives us the number of full rotations or revolutions that the tires will make within the full warranty length.