A 1250 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the frictional force fk between boat and water is proportional to the speed v of the boat. Thus, fk = 80v, where v is in meters per second and fk (the magnitude of the frictional force) is in newtons. Find the time required for the boat to slow down to 45 km/h.

To find the time required for the boat to slow down to 45 km/h, we need to determine the acceleration of the boat first. Once we have the acceleration, we can use kinematic equations to find the time.

1. Convert the initial speed of the boat to meters per second:
Initial speed = 90 km/h
Conversion factor: 1 km/h = 1/3.6 m/s
Initial speed = 90 km/h * (1/3.6) m/s = 25 m/s

2. Convert the final speed of the boat to meters per second:
Final speed = 45 km/h
Final speed = 45 km/h * (1/3.6) m/s = 12.5 m/s

3. The frictional force, fk, is given by the equation fk = 80v, where v is the speed of the boat in meters per second.

4. Apply Newton's second law of motion:
Sum of forces = mass * acceleration
The only force acting on the boat is the frictional force, fk, which opposes the motion. So, the equation becomes:
fk = mass * acceleration

5. Substitute the given values into the equation:
80v = 1250 kg * acceleration

6. Rearrange the equation to solve for the acceleration:
acceleration = (80v) / 1250

7. Substitute the final speed of the boat into the equation to find the acceleration:
acceleration = (80 * 12.5) / 1250
= 0.8 m/s^2

8. Use the kinematic equation to find the time required to slow down:
Final speed = Initial speed + (acceleration * time)
Substitute the values:
12.5 m/s = 25 m/s + (0.8 m/s^2 * time)

9. Rearrange the equation to solve for time:
0.8 m/s^2 * time = 12.5 m/s - 25 m/s
0.8 m/s^2 * time = -12.5 m/s
time = (-12.5 m/s) / (0.8 m/s^2) ≈ -15.63 s

Note: The negative sign indicates that the boat is slowing down. However, time cannot be negative in this context. Therefore, the boat does not slow down to 45 km/h in this situation.

To find a valid solution, double-check the problem statement and verify the given data and equations.