A particular power boat travelling at V0 km/h when its power is turned off will have its speed V given by V(t) = V0 × 10-0.18t where t is the time in seconds since the engine was turned off. If this boat was travelling at 50km/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?
The O after the V is in subscript form.
so you have
V(t) = 50 x 10^(-.18t)
t=0 , V(0) = 50 (10^0) = 50
t=1 , V(1) = 50 (10^-.18) = 33.03
t = 2 , V(2) = 50 (10^-.36) = 21.83
you do the rest of the button-pushing
mathematically the boat would never come to a stop
since 10^?? can never become zero
Ah thanks that helps explain it much easier for me.
To construct a table showing the speed of the boat at different time intervals, we can substitute the given values of V0 and t into the equation V(t) = V0 × 10^(-0.18t).
Given that the boat was traveling at 50 km/h before powering down (V0 = 50), we can calculate the speed using the formula for different values of t.
Using the equation V(t) = V0 × 10^(-0.18t):
For t = 0 seconds:
V(0) = 50 × 10^(-0.18 × 0) = 50 km/h
For t = 1 second:
V(1) = 50 × 10^(-0.18 × 1) = 44.97 km/h (rounded to two decimal places)
For t = 2 seconds:
V(2) = 50 × 10^(-0.18 × 2) = 40.11 km/h (rounded to two decimal places)
For t = 3 seconds:
V(3) = 50 × 10^(-0.18 × 3) = 35.68 km/h (rounded to two decimal places)
For t = 4 seconds:
V(4) = 50 × 10^(-0.18 × 4) = 31.64 km/h (rounded to two decimal places)
For t = 5 seconds:
V(5) = 50 × 10^(-0.18 × 5) = 27.95 km/h (rounded to two decimal places)
So, the table showing the speed of the boat at different time intervals would look like:
| t (seconds) | V(t) (km/h) |
|------------|-------------|
| 0 | 50 |
| 1 | 44.97 |
| 2 | 40.11 |
| 3 | 35.68 |
| 4 | 31.64 |
| 5 | 27.95 |
Now, let's discuss the limitations of using this exponential function to describe the speed of the boat:
1. The model assumes a continuous and uninterrupted decay of speed. In reality, there are various factors that can affect the boat's speed, such as wind, waves, and other external forces. These factors are not accounted for in the exponential function.
2. The model assumes a constant rate of decay. In reality, the rate at which the boat slows down may vary due to factors mentioned earlier or other influences, such as friction or mechanical resistance. These variations are not captured in the exponential function.
3. The model assumes that the boat will continue to slow down indefinitely. However, in reality, a boat may eventually reach a minimum speed (e.g., its terminal velocity) or come to a complete stop due to other external factors or the boat's inherent design limitations. This behavior is not represented in the exponential function.
4. The model assumes a linear relationship between time and speed decay, which may not always hold true. In some cases, the speed decay may follow a different pattern, such as logarithmic or polynomial decay. Again, these variations are not accounted for in the exponential function.
Therefore, while the exponential function provides a simple and mathematical representation of the boat's speed decay, it has limitations when it comes to accurately describing the real-world behavior of the boat in all situations.