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:
- Height Function: The height \( h(t) \) decreases over time indefinitely.
- Initial Height: At \( t = 0 \): \[ h(0) = -3(0) + 0.015P + 160 = 0.015P + 160 \]
- 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.