A boat can travel at 1.7 m/s in still water. The boat must cross a river with a current of 0.7 m/s. The captain of the boat is trying to reach a point on shore that is 110 m upstream from where they are starting. In order to do this, the captain must aim the boat at a 45 degree angle upstream. How wide is the river?

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To solve this problem, we can break it down into horizontal and vertical components.

Let's consider the horizontal component first:
The boat's speed relative to the ground in the horizontal direction can be calculated using the cosine function, given that the angle formed by the boat's path and the river is 45 degrees.
Horizontal velocity of the boat = Boat's speed in still water * cos(angle)
= 1.7 m/s * cos(45 degrees)
= 1.7 m/s * 0.707 (rounded to 3 decimal places)
≈ 1.202 m/s

Next, let's consider the vertical component:
The current of the river is acting against the boat's motion, so the effective speed of the boat in the vertical direction will be its speed in still water minus the current speed.
Vertical velocity of the boat = Boat's speed in still water - River current
= 1.7 m/s - 0.7 m/s
= 1.0 m/s

Now, we can find the time it takes for the boat to travel upstream to the point on the shore. We can use the formula:
time = distance / velocity

time = upstream distance / horizontal velocity
= 110 m / 1.202 m/s
≈ 91.606 seconds (rounded to 3 decimal places)

Since the boat is traveling at an angle of 45 degrees upstream, the distance crossed in the vertical direction is the same as the horizontal distance.
Therefore, the width of the river can be calculated using the formula:
width of the river = time * vertical velocity
= 91.606 s * 1.0 m/s
≈ 91.606 m

So, the width of the river is approximately 91.606 meters.

To find the width of the river, we can use the concept of vector addition. Let's break down the motion of the boat into its components.

Let's assume that the width of the river is represented by the distance perpendicular to the boat's starting point.

1. The boat's velocity in still water is 1.7 m/s. We can break it down into horizontal and vertical components:
- Horizontal component (x-direction): 1.7 m/s * cos(45°)
- Vertical component (y-direction): 1.7 m/s * sin(45°)

2. The river's current is 0.7 m/s, and it acts in the horizontal direction (x-direction).

3. The boat aims at a 45 degree angle upstream, which means it opposes the current. This results in the boat's actual velocity in the x-direction being the difference between the boat's horizontal velocity component and the current:
- Actual velocity in the x-direction: (1.7 m/s * cos(45°)) - 0.7 m/s

Now we can calculate the time it takes for the boat to cross the river:

4. The time taken to cross the river is determined by the width of the river divided by the boat's actual velocity in the x-direction:
- Time taken: Width / (1.7 m/s * cos(45°) - 0.7 m/s)

Since we know that Time = Distance / Speed, we can rearrange the equation to find the width of the river:

5. Width = Time * (1.7 m/s * cos(45°) - 0.7 m/s)

Now we can substitute the values and calculate the width:

Width = (110 m) * (1.7 m/s * cos(45°) - 0.7 m/s)

By evaluating the equation, we can determine the width of the river.