Question

A tire is 22 inches in diameter and rests on a platform that is 4 inches above the ground. The six o’clock position on the tire is level with the platform. A piece of gum stuck to the three o’clock position of the tire completed 1 full revolution in 12 seconds. The function h(t) gives the height of the piece of gum in inches above the ground t seconds after the tire begins to turn. Find a formula for the height function h(t)

Answer 1

h(t) = 11 * sin(π/6 * t) + 15

Explanation:

To find the height function h(t) for the piece of gum, we'll use a sinusoidal function since the height of the gum changes in a periodic manner as the tire rotates. The general formula for a sinusoidal function is:

h(t) = A * sin(B * (t - C)) + D

where A is the amplitude, B is the angular frequency, C is the horizontal shift, and D is the vertical shift.

1. Amplitude (A): The amplitude is half the distance between the highest and lowest points of the gum. Since the diameter of the tire is 22 inches, the radius is 11 inches. Thus, the amplitude is 11 inches.

2. Angular frequency (B): Since the gum completes 1 full revolution in 12 seconds, the period of the function is 12 seconds. The angular frequency, B, is found by dividing 2π by the period: B = 2π / 12 = π/6.

3. Horizontal shift (C): Since the gum starts at the three o'clock position, and we want the height to be at its maximum at t = 0, there is no horizontal shift. So, C = 0.

4. Vertical shift (D): The platform is 4 inches above the ground, and the radius of the tire is 11 inches. Therefore, the midpoint of the gum's motion is 4 + 11 = 15 inches above the ground. So, D = 15.

Now we can write the height function h(t):

h(t) = 11 * sin(π/6 * t) + 15