After the car has traveled 15 inches, the pebble wedged in the tire tread is approximately 21 inches away from the pavement. This is determined by using the tire's diameter to find the circumference, which serves as the period of the sinusoidal function describing the pebble's motion.
The question is asking for the distance of the pebble from the pavement after the car has traveled 15 inches when the diameter of the tire is 24 inches. The sinusoidal variation of this distance means the maximum distance is achieved when the pebble is at the top of the wheel, which is the radius of the wheel above the pavement (12 inches). As the wheel rotates, the pebble's height above the pavement follows a sinusoidal pattern based on the circumference of the wheel, which can be calculated using the formula C = πd where d is the diameter.
The circumference of the tire, which is the period of the sinusoidal function, is C = π × 24 inches = 75.4 inches. When the car has moved 15 inches, it has completed 15/75.4 of a full rotation. We can find the height using the cosine of the proportional angle, which is cos(2π × 15/75.4). The final height would be the radius plus the cosine of the angle times the radius: 12 inches + 12 inches × cos(2π × 15/75.4). This gives us the pebble's distance from the pavement, which computes to approximately 21 inches.