Question

The function h(t) =0.015P-3t + 160 models the height h (in feet) of an amusement park ride t seconds after it starts.

What is the minimum height of the ride?

____feet

How long does it take the ride to reach the minimum height?

_____seconds

Answer 1

The function given is \( h(t) = -3t + 0.015P + 160 \). This is a linear function in terms of \( t \), where the coefficient of \( t \) is negative (-3), indicating that the height decreases as time increases. Therefore, there will not be a minimum height in the traditional sense for \( t \geq 0 \).

However, if we are looking for specific values for \( P \) or a certain context — for instance, if \( P \) is adjusted or if the ride operates within a bounded time interval — the results can change.

### Analyzing the equation:
1. **Height Function**: The height \( h(t) \) decreases over time indefinitely.
2. **Initial Height**: At \( t = 0 \):
\[
h(0) = -3(0) + 0.015P + 160 = 0.015P + 160
\]
3. **Behavior as \( t \) Approaches Infinity**: As \( t \) increases, \( h(t) \) keeps decreasing, indicating that there is no minimum within the defined function unless bounded by a valid operational time frame or an external edge condition (ground level, for example).

### Contextual Interpretation:
- If we consider real-world applications, the height cannot be negative because negative heights do not make practical sense. Thus, you would need to know conditions where \( h(t) = 0 \) to find bounds.
- Solve \( 0 = -3t + 0.015P + 160\):
\[
3t = 0.015P + 160 \\
t = \frac{0.015P + 160}{3}
\]

### Conclusions:
- **Minimum Height**: In practical scenarios, the minimum height can be considered as the threshold where the ride reaches the ground. The minimum height would be \( 0 \) feet, related to how long it takes to reach that point.
- **Time to Minimum Height**: The ride takes \( t = \frac{0.015P + 160}{3} \) seconds to reach that point.

Without specific values for \( P \), we cannot provide numerical answers directly. If \( P \) were provided, we could compute those accordingly.