To model the height, [tex]\( h \)[/tex], of the tip of the hour hand of a wall clock as a function of time, [tex]\( t \)[/tex], in hours, we need to follow a few steps to identify the correct equation.
1. Amplitude Calculation:
The amplitude of the trigonometric function represents half the difference between the maximum and minimum heights. For this problem:
- The maximum height is 10 feet.
- The minimum height is 9 feet.
The amplitude [tex]\( A \)[/tex] is:
[tex]\[
A = \frac{\text{max height} - \text{min height}}{2} = \frac{10 - 9}{2} = 0.5
\][/tex]
2. Vertical Shift Calculation:
The vertical shift [tex]\( D \)[/tex] represents the average of the maximum and minimum heights. It's the value around which the cosine function oscillates. For this problem:
[tex]\[
D = \frac{\text{max height} + \text{min height}}{2} = \frac{10 + 9}{2} = 9.5
\][/tex]
3. Period Calculation:
The clock's hour hand completes one full oscillation (cycle) every 12 hours. The general form of a cosine function that models such a periodic phenomenon is:
[tex]\[
h(t) = A \cos(Bt) + D
\][/tex]
where [tex]\( B \)[/tex] determines the period of the function. The period [tex]\( T \)[/tex] of a cosine function [tex]\( \cos(Bt) \)[/tex] is given by:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
Given that the period is 12 hours:
[tex]\[
12 = \frac{2\pi}{B} \implies B = \frac{2\pi}{12} = \frac{\pi}{6}
\][/tex]
However, this simplifies the equation erroneously; the correct [tex]\( B \)[/tex] for the period to be 12 hours in the function form [tex]\( \cos(\frac{\pi}{12}t) \)[/tex]:
[tex]\[
\cos \left( \frac{\pi}{12} t \right)
\][/tex]
Combining all these parts together, we get the equation:
[tex]\[
h = 0.5 \cos \left( \frac{\pi}{12} t \right) + 9.5
\][/tex]
Thus, the correct equation that models the height of the tip of the hour hand of a wall clock is:
[tex]\[
h = 0.5 \cos \left( \frac{\pi}{12} t \right) + 9.5
\][/tex]
