Let's analyze each of the given equations to determine which one best models the height, [tex]\( h \)[/tex], of the tip of the hour hand of a wall clock, varying from 9 feet to 10 feet over a period of time.
### Equations to Consider:
1. [tex]\( h = 0.5 \cos \left(\frac{\pi}{12} t\right) + 9.5 \)[/tex]
2. [tex]\( h = 0.5 \cos \left(\frac{\pi}{6} t\right) + 9.5 \)[/tex]
3. [tex]\( h = \cos \left(\frac{\pi}{12} t\right) + 9 \)[/tex]
4. [tex]\( h = \cos \left(\frac{\pi}{6} t\right) + 9 \)[/tex]
### Analysis
1. Equation [tex]\( h = 0.5 \cos \left(\frac{\pi}{12} t\right) + 9.5 \)[/tex]
- The cosine function, [tex]\( \cos(\theta) \)[/tex], varies from -1 to 1.
- Thus, [tex]\( \cos \left(\frac{\pi}{12} t \right) \)[/tex] varies from -1 to 1.
- When multiplied by 0.5, this becomes [tex]\( 0.5 \cos \left(\frac{\pi}{12} t \right) \)[/tex], which varies from -0.5 to 0.5.
- Adding 9.5 shifts the range to [tex]\( 9.5 - 0.5 \)[/tex] to [tex]\( 9.5 + 0.5 \)[/tex], or [tex]\( 9 \)[/tex] to [tex]\( 10 \)[/tex].
Therefore, this equation correctly models the height from 9 feet to 10 feet.
2. Equation [tex]\( h = 0.5 \cos \left(\frac{\pi}{6} t\right) + 9.5 \)[/tex]
- Similarly, [tex]\( \cos \left(\frac{\pi}{6} t\right) \)[/tex] varies from -1 to 1.
- [tex]\( 0.5 \cos \left(\frac{\pi}{6} t \right) \)[/tex] varies from -0.5 to 0.5.
- Adding 9.5 results in a range of [tex]\( 9.5 - 0.5 \)[/tex] to [tex]\( 9.5 + 0.5 \)[/tex], or [tex]\( 9 \)[/tex] to [tex]\( 10 \)[/tex].
However, the period of this function is not consistent with the height variation over 12 hours.
3. Equation [tex]\( h = \cos \left(\frac{\pi}{12} t\right) + 9 \)[/tex]
- Here, [tex]\( \cos \left(\frac{\pi}{12} t\right) \)[/tex] varies from -1 to 1.
- Adding 9 results in a range of [tex]\( 9 - 1 \)[/tex] to [tex]\( 9 + 1 \)[/tex], or [tex]\( 8 \)[/tex] to [tex]\( 10 \)[/tex].
This does not correctly model the height from 9 feet to 10 feet.
4. Equation [tex]\( h = \cos \left(\frac{\pi}{6} t\right) + 9 \)[/tex]
- Similarly, [tex]\( \cos \left(\frac{\pi}{6} t\right) \)[/tex] varies from -1 to 1.
- Adding 9 results in a range of [tex]\( 9 - 1 \)[/tex] to [tex]\( 9 + 1 \)[/tex], or [tex]\( 8 \)[/tex] to [tex]\( 10 \)[/tex].
This also does not correctly model the height from 9 feet to 10 feet.
### Conclusion
Considering the detailed analysis, the correct equation that models the height, [tex]\( h \)[/tex], varying from 9 feet to 10 feet over 12 hours is:
[tex]\[ h = 0.5 \cos \left(\frac{\pi}{12} t\right) + 9.5 \][/tex]
Thus, the suitable equation is:
[tex]\[ \boxed{h=0.5 \cos \left(\frac{\pi}{12} t\right)+9.5} \][/tex]
