the question is "Joe rode his bike over a piece of gum. Joe continued riding his bike at a constant rate. At time t=1.25 seconds the gum was at a maximum height above the ground and 1 second later the gum was at a minimum. If the wheel diameter is 68cm find a trigonometric equation that will find the height in centimeters at any time t.

assuming that at t = 0 the height was zero (at ground level)

Let's use a sine curve.
The amplitude is 34 and the period is 2.5 seconds
2.5 = 2π/k
k = π/1.25

so our basic curve is
h = 34sin(πt/1.25)

We need to raise this 34 cm so it doesn't go below road level
h = 34sin((π/1.25)t) + 34

This does not give us a height of 0 when t = 0, we have to move the curve 1/4 period to the right

h = 34sin((π/1.25)(t - .625)) + 34

testing:
when t = 0 , h = 0 , good!
when t = 1.25, h = 68 , good!
My equation is correct.

To find a trigonometric equation that relates the height of the gum (above the ground) to time, we can use the concept of periodic motion.

Let's start by understanding the motion of the gum on the wheel. When the gum is at its maximum height, it will be located on the topmost point of the wheel, and when the gum is at its minimum height, it will be located on the bottommost point of the wheel.

Since the gum's height varies sinusoidally over time, we can use a sine function to describe its motion:

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

In this equation,
- A represents the amplitude of the motion (the maximum height - the minimum height)/2.
- B represents the frequency of the motion, and it is related to the time period by B = (2π) / T, where T is the time period.
- C represents the phase shift of the motion, and it can be calculated using the time at maximum height.
- D represents the vertical shift of the motion, and it can be calculated using the minimum height.

To determine the amplitude (A), we will need the maximum height and the minimum height. However, the problem statement does not provide this information. If you can provide the maximum and minimum heights, I can help you find the trigonometric equation.

To find a trigonometric equation that relates the height of the gum to time, we can use the fact that the gum goes through maximum and minimum heights during its motion. This suggests that the height can be modeled using a sinusoidal function, such as the sine or cosine function.

Let's assume that the gum's height, h, in centimeters, is given as a function of time, t, in seconds. Since the gum goes through a maximum height at t = 1.25 seconds, we can adjust the sine function to have a phase shift of -1.25 seconds.

A general form for the height as a function of time can be represented as:

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

where A, B, C, and D are constants to be determined.

Using the given information, we can proceed with the following steps to find the trigonometric equation:

Step 1: Find the amplitude, A:
Since the gum's height has a maximum and a minimum, the amplitude is half the difference between these two extremes. Given that the wheel diameter is 68 cm, the radius is half of this, which is 34 cm. Therefore, the amplitude is 34 cm.

A = 34

Step 2: Find the period, T:
The period is the time it takes for the height to complete one full cycle. Given that the gum reaches its minimum height 1 second after the maximum, the total time for one full cycle is 1 + 1 = 2 seconds.

T = 2

Step 3: Find the angular frequency, ω:
The angular frequency, which is equal to 2π divided by the period, T, can be used to adjust the speed of oscillation.

ω = 2π / T

Substituting the value of T:
ω = 2π / 2 = π

Step 4: Find the phase shift, C:
The phase shift, C, determines the starting point of the height function. Since the maximum height occurs at t = 1.25 seconds, we need to shift the function to the left by 1.25 seconds.

C = -1.25

Step 5: Final equation:
Replacing A, B, and C in the general form equation with the discovered values, the trigonometric equation becomes:

h(t) = 34 * sin(π * (t + 1.25)) + D

The vertical shift, D, is not given in the problem, so we don't have enough information to determine its value. If you have the value for D, you can substitute it into the equation to complete it.

Therefore, the trigonometric equation that represents the height in centimeters, h, as a function of time, t, is:

h(t) = 34 * sin(π * (t + 1.25)) + D