A car wheel has a 13-inch radius. Through what angle does the wheel turn when the car rolls forward 4 ft?
To determine the angle through which the wheel turns, we need to calculate the circumference of the wheel. We can then use the formula relating linear distance traveled to the angle of rotation.
1. First, convert the radius of the wheel from inches to feet:
- 13 inches = 13/12 = 1.0833 feet (approximately, rounded to four decimal places).
2. Next, calculate the circumference of the wheel using the formula:
- Circumference = 2 * π * radius
- Circumference = 2 * 3.14159 * 1.0833 = 6.807 feet (approximately, rounded to three decimal places).
3. Now, we can use a proportion to determine the angle through which the wheel turns. The linear distance traveled by the car is 4 ft, which represents the arc length of the wheel's rotation.
- The full circle represents 360 degrees (or 2π radians).
- The ratio of the arc length to the circumference of the wheel equals the ratio of the angle (in degrees or radians) of rotation to a full circle.
Using the formula: Angle (in degrees) = (Arc length / Circumference) * 360, we can calculate the angle:
- Angle (in degrees) = (4 ft / 6.807 ft) * 360 = 210.344 degrees (approximately, rounded to three decimal places).
So, the wheel turns through an angle of approximately 210.344 degrees when the car rolls forward 4 ft.