A lifeguard who can swim at 1.2 m/s in still water wants to reach a dock positioned perpendicularly directly across a 550 m wide river.

If the current in the river is 0.80 m/s, how long will it take the lifeguard to reach the dock?
If instead she had decided to swim in such a way that will allow her to cross the river in a minimum amount of time, where would she land relative to the dock?

V = 1.2 - 0.80i = 1.44m/s[-33.7o] = 1.44m/s[33.7o] S. of E. = 326.3o CCW.

a. Cos326.3 = 550/d.
d1 = 550/Cos326.3 = 661 m. across due to current.
V*T1 = 661.
1.44 * T1 = 661,
T1 = 459 s. = 7.7 min. to cross river.

Tan326.3 = d2/550.
d2 = 550*Tan326.3 = 367 m. Downstream from dock.
d2 = V2 * T2 = 367.
(1.2-0.8) * T2 = 367,
T2 = 917 s. = 15.3 min. from downstream to dock.
T1 + T2 = 7.7 + 15.3 = 23 min. = Total time to reach dock.

b. Directly in front of the dock.

To solve this problem, we can use the concept of relative velocity. Let's break it down step-by-step:

Step 1: Determine the current's effect on the lifeguard's swimming speed.
The current is moving at a speed of 0.80 m/s. Since the lifeguard is swimming against the current, the effective speed will be reduced. This can be calculated by subtracting the current speed from the lifeguard's swimming speed:
Effective speed = 1.2 m/s - 0.80 m/s = 0.40 m/s

Step 2: Calculate the time it takes the lifeguard to cross the river.
The lifeguard needs to swim across a 550 m wide river. Since her effective speed is 0.40 m/s, we can use the formula: time = distance / speed
Time = 550 m / 0.40 m/s = 1375 seconds

So, it will take the lifeguard 1375 seconds to reach the dock.

Step 3: Determine where the lifeguard would land if she chose the minimum time path.
To minimize the time it takes to cross the river, the lifeguard should swim perpendicular to the current. In this case, the lifeguard would be pushed downstream while swimming sideways. The distance the lifeguard would travel downstream can be calculated by multiplying the current speed by the time taken to cross the river:
Distance downstream = current speed * time taken to cross the river
Distance downstream = 0.80 m/s * 1375 seconds
Distance downstream = 1100 meters

Therefore, the lifeguard would land 1100 meters downstream from the dock if she chose the path that minimizes the crossing time.

To find how long it will take for the lifeguard to reach the dock, we need to consider the speed of the lifeguard and the current of the river. Let's break down the problem step by step:

1. Calculate the effective speed of the lifeguard:
The effective speed is the combination of the lifeguard's swimming speed and the speed of the current. In this case, the lifeguard's swimming speed is 1.2 m/s, and the current speed is 0.80 m/s.
So, the effective speed is the sum of these two: 1.2 m/s + 0.80 m/s = 2 m/s.

2. Calculate the time taken to cross the river:
To find the time taken, we can use the formula: time = distance / speed.
The distance the lifeguard needs to cross is 550 m (width of the river).
The speed of the lifeguard, which is the effective speed calculated earlier, is 2 m/s.
Therefore, the time it will take for the lifeguard to reach the dock is: time = 550 m / 2 m/s = 275 seconds.

Now, let's consider the second part of the question. If the lifeguard wants to cross the river in a minimum amount of time:

To minimize the time taken, the lifeguard should swim at an angle such that the component of the swimming speed perpendicular to the river current is maximized. This will allow the lifeguard to reach the other side quickly without being affected by the current.

If the lifeguard swims at an angle such that the net velocity vector is perpendicular to the river current, she will land directly across the dock. This means she will land at the same point as the dock or very close to it.

To summarize:
- It will take the lifeguard 275 seconds to reach the dock if she swims directly across the river.
- If she swims in a way that allows her to cross the river in minimum time, she will land at the same point or very close to the dock.