12. A particular power boat travelling at V0 km/h when its power is turned off it will have its speed V given by V(t) = V0 x 10^-0.18t where t is the time in seconds since the engine was turned off. If this boat was travelling at 50k/h before powering down, construct a table showing its speed at t = 0, 1, 2… 5. What are the limitations in using this exponential function to describe the speed of the boat?
To construct a table showing the boat's speed at various time intervals, we can simply substitute the given values of time into the given equation V(t) = V0 x 10^(-0.18t), where V0 is the initial speed of the boat.
Given that the boat is traveling at 50 km/h before powering down, we can substitute V0 = 50 into the equation.
Let's calculate the boat's speed at t = 0, 1, 2, 3, 4, and 5 seconds:
For t = 0 seconds:
V(0) = V0 x 10^(-0.18(0))
V(0) = 50 x 10^0
V(0) = 50 km/h
For t = 1 second:
V(1) = V0 x 10^(-0.18(1))
V(1) = 50 x 10^(-0.18)
V(1) ≈ 48.4509 km/h
For t = 2 seconds:
V(2) = V0 x 10^(-0.18(2))
V(2) = 50 x 10^(-0.36)
V(2) ≈ 46.9945 km/h
For t = 3 seconds:
V(3) = V0 x 10^(-0.18(3))
V(3) = 50 x 10^(-0.54)
V(3) ≈ 45.6234 km/h
For t = 4 seconds:
V(4) = V0 x 10^(-0.18(4))
V(4) = 50 x 10^(-0.72)
V(4) ≈ 44.3301 km/h
For t = 5 seconds:
V(5) = V0 x 10^(-0.18(5))
V(5) = 50 x 10^(-0.9)
V(5) ≈ 43.1086 km/h
The table would look like this:
| t | V(t) |
|----------|----------|
| 0 | 50 |
| 1 | 48.4509 |
| 2 | 46.9945 |
| 3 | 45.6234 |
| 4 | 44.3301 |
| 5 | 43.1086 |
As for the limitations of using this exponential function to describe the speed of the boat, here are a few considerations:
1. Simplified Model: The equation assumes a simple exponential decay model, neglecting other factors like air resistance, water turbulence, and changing sea conditions. In reality, these factors can significantly affect the boat's speed, leading to deviations from the model's predictions.
2. Simplified Power Down Process: The equation assumes an instantaneous power down, meaning the boat's engine goes from full power to completely off in an instant. In actual situations, the power may decrease gradually, resulting in a more complex speed-time relationship.
3. Linear Time Interval: The equation uses time in seconds as the input parameter. However, in practical scenarios, the boat's power-down process may take longer intervals, making the time intervals used in the table unrealistic.
4. Sensitivity to Initial Speed: The equation heavily depends on the initial speed of the boat (V0). Even a slight inaccuracy in measuring the initial speed can lead to large discrepancies in the predicted speeds at different time intervals.
Therefore, while the exponential function can provide an approximate estimation of the boat's speed after powering down, it may not accurately represent real-world conditions due to the limitations mentioned above.