The height in inches about the ground of the tip of the hour hand is a function, h(t), of the time (t), in hours. The hour hand is 6 inches long and the middle of the clock face is 68 inches about the ground. Identify the period, midline, and amplitude of the function.
since the center of the clock is 68" above the ground, you know the function will be
h(t) = 6*cos(kt) + 68
why? because at t=0, it's 12:00, and the hour hand is at its max height, 6" above center.
Now, we know that at t=12, it is back on top, so the period is 12:
h(t) = 6cos(2pi/12 t) + 68
To identify the period, midline, and amplitude of the function h(t), we need to understand the characteristics of the function based on the description provided.
1. Period: The period of a function represents the length of one complete cycle or revolution. In this case, the period of the function h(t) relates to the time it takes for the hour hand to complete a full rotation, which is 12 hours.
2. Midline: The midline of a function represents the average or central value of the function. In this case, the midline of the function h(t) corresponds to the height of the center of the clock face, which is 68 inches above the ground.
3. Amplitude: The amplitude of a function represents the maximum distance between the function's midline and its maximum/minimum values. Here, since the description mentions that the hour hand is 6 inches long, the amplitude of the function h(t) is equivalent to this distance, which is 6 inches.
Therefore, we have:
- Period = 12 hours
- Midline = 68 inches
- Amplitude = 6 inches
The function h(t) can now be described as:
h(t) = A * sin(Bt - C) + D
Where:
A = Amplitude = 6 inches
B = 2π / Period = 2π / 12 = π/6
C = Phase Shift (Not given in the description)
D = Midline = 68 inches
Please note that to provide a complete equation for the function h(t), we would need additional information, such as the starting position of the hour hand (phase shift). Without the phase shift, we cannot fully determine the equation.