To determine the correct equation that models the height [tex]\( h \)[/tex] of the tip of the hour hand as a function of time [tex]\( t \)[/tex], we need to analyze the given conditions and how they influence the mathematical model.
1. Maximum and Minimum Heights:
The height varies between 9 feet and 10 feet.
- Maximum height, [tex]\( h_{\text{max}} = 10 \)[/tex] feet.
- Minimum height, [tex]\( h_{\text{min}} = 9 \)[/tex] feet.
2. Amplitude and Midline:
- The amplitude [tex]\( A \)[/tex] is calculated as half the range of the height changes:
[tex]\[
A = \frac{h_{\text{max}} - h_{\text{min}}}{2} = \frac{10 - 9}{2} = 0.5 \ \text{feet}
\][/tex]
- The midline [tex]\( C \)[/tex] (average height) is:
[tex]\[
C = \frac{h_{\text{max}} + h_{\text{min}}}{2} = \frac{10 + 9}{2} = 9.5 \ \text{feet}
\][/tex]
3. Period of the Function:
- The period of the height function in hours is 24 hours, as the hour hand returns to the same position every 24 hours.
- For cosine functions, the period [tex]\( T \)[/tex] is related to the coefficient [tex]\( B \)[/tex] inside the cosine argument by the equation:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
- Given [tex]\( T = 24 \)[/tex] hours, we solve for [tex]\( B \)[/tex]:
[tex]\[
24 = \frac{2\pi}{B} \implies B = \frac{2\pi}{24} = \frac{\pi}{12}
\][/tex]
4. Final Equation:
- Combine the amplitude, midline, and the coefficient [tex]\( B \)[/tex] into the cosine function:
[tex]\[
h = A \cos(Bt) + C \Rightarrow h = 0.5 \cos \left( \frac{\pi}{12} t \right) + 9.5
\][/tex]
The correct equation that models the height [tex]\( h \)[/tex] of the tip of the hour hand as a function of time [tex]\( t \)[/tex] is:
[tex]\[
h = 0.5 \cos \left( \frac{\pi}{12} t \right) + 9.5
\][/tex]
Therefore, the correct answer to the problem is:
[tex]\[
h = 0.5 \cos \left( \frac{\pi}{12} t \right) + 9.5
\][/tex]
