Question

As you stop your car at a traffic light, a pebble becomes wedged between the tire treads. When you start moving again, the distance between the pebble and the pavement varies sinusoidally with the distance you have gone. The period is the circumference ( 2πr ) of the tire. The diameter of the tire is 24 in. What is the pebble's distance from the pavement when you have gone 15 in?

Answer 1

The pebble's distance from the pavement as the car moves is sinusoidally related to the distance traveled by the car. It requires knowing the amplitude and initial phase of the sinusoidal movement to give a specific answer.

Step-by-step explanation:

The question involves finding the pebble's distance from the pavement when a car has moved a certain distance. Given that the diameter of the tire is 24 inches, the radius r is 12 inches (or 1 foot), so the circumference (period T) of the tire, which is 2πr, is approximately 24π inches.

As the car has moved 15 inches, we need to determine the angular distance covered by the tire and consequently the position of the pebble.

Since the distance traveled by the car is less than the circumference of the tire, we can calculate the corresponding angle θ as a ratio of the distance traveled (15 inches) to the total circumference (24π inches). We then use a sinusoidal function to find the pebble's distance from the pavement at that angle.

However, since there is not enough information given to completely resolve the problem (such as specifying the amplitude of the sine function or the initial phase, which would determine the position of the pebble when the car started), an exact numerical answer cannot be provided without assumptions.