To determine the position of the minute hand at [tex]\(t = 0\)[/tex] given the height equation [tex]\(h = 0.75 \cos \left(\frac{\pi}{30}(t - 15)\right) + 8\)[/tex], let's follow these steps:
1. Substitute [tex]\(t = 0\)[/tex] into the equation:
[tex]\[
h = 0.75 \cos \left(\frac{\pi}{30}(0 - 15)\right) + 8
\][/tex]
2. Simplify the argument of the cosine function:
[tex]\[
\frac{\pi}{30} (0 - 15) = \frac{\pi}{30} \cdot (-15) = -\frac{15\pi}{30} = -\frac{\pi}{2}
\][/tex]
3. Evaluate the cosine function:
[tex]\[
\cos \left(-\frac{\pi}{2}\right) = 0
\][/tex]
4. Calculate the height [tex]\(h\)[/tex] at [tex]\(t = 0\)[/tex]:
[tex]\[
h = 0.75 \cdot 0 + 8 = 8
\][/tex]
Thus, the height of the tip of the minute hand at [tex]\(t = 0\)[/tex] is 8 feet.
Next, we need to determine which number the minute hand is pointing to when its height is 8 feet.
- The cosine function in the height equation oscillates between -1 and 1, causing the height [tex]\(h\)[/tex] to oscillate between [tex]\(8 - 0.75\)[/tex] and [tex]\(8 + 0.75\)[/tex], which means the height [tex]\(h\)[/tex] ranges from 7.25 feet to 8.75 feet.
- The height of 8 feet corresponds to the horizontal position of the minute hand, which occurs when the minute hand points directly to the numbers on the clock (3, 6, 9, or 12).
Since the [tex]\( \cos \left( \frac{\pi}{30} (t - 15) \right) \)[/tex] term is zero at [tex]\(t = 0\)[/tex], the cosine function is at a point where it crosses the horizontal axis. When the hour is at 12:00, the minute hand is at the topmost position and at the highest vertical displacement (which we determined is at a height of 8.75 feet).
At [tex]\(t = 0\)[/tex] (which corresponds to the topmost point of the minute hand), the minute hand must be pointing to the number 3 to be at a horizontal position.
Therefore, the minute hand is pointing to:
[tex]\[
\boxed{9}
\][/tex]
