To determine the number (from 1 to 12) that the minute hand is pointing to at [tex]\( t = 0 \)[/tex] minutes, we need to understand how the function [tex]\( h=0.75 \cos \left(\frac{\pi}{30}(t-15)\right)+8 \)[/tex] models the height of the tip of the minute hand over time.
1. Evaluate the function at [tex]\( t = 0 \)[/tex]:
[tex]\[
h = 0.75 \cos \left(\frac{\pi}{30}(0-15)\right) + 8
\][/tex]
Simplify inside the cosine term:
[tex]\[
h = 0.75 \cos \left(\frac{\pi}{30} \cdot (-15)\right) + 8
\][/tex]
[tex]\[
h = 0.75 \cos \left(-\frac{15\pi}{30}\right) + 8
\][/tex]
[tex]\[
h = 0.75 \cos \left(-\frac{\pi}{2}\right) + 8
\][/tex]
2. Evaluate the cosine:
[tex]\[
h = 0.75 \cos \left(-\frac{\pi}{2}\right) + 8
\][/tex]
Since [tex]\( \cos \left(-\frac{\pi}{2}\right) = 0 \)[/tex]:
[tex]\[
h = 0.75 \cdot 0 + 8
\][/tex]
[tex]\[
h = 8
\][/tex]
3. Interpreting the height [tex]\( h \)[/tex] in terms of the minute hand position:
- The overall height of the minute hand ranges from 8 (uppermost) to [tex]\( 8 - 0.75 = 7.25 \)[/tex] (lowermost).
- At [tex]\( h = 8 \)[/tex], the minute hand is at the topmost position.
4. Identifying the topmost position on a clock:
- On a standard 12-hour clock, the minute hand points to 12 at the topmost position.
Therefore, at [tex]\( t = 0 \)[/tex] minutes, the minute hand is pointing to:
[tex]\[ \boxed{12} \][/tex]
