Question

A sail boat is crossing a river of width 42.5 meters, sailing at 17.4 m/s vertically across the river. If a cross wind of 5.9 m/s directed 13.2 degrees off of vertical pushes the sailboat, then how far downstream (horizontally) will the boat be when it reaches the other side of the river? (please provide your answer to 1 decimal place)

Answer 1

Answer: 0.0 meters

To solve this problem, we will need to use trigonometry. We can use the formula `distance = speed x time` to find how far downstream the boat will be when it reaches the other side of the river. We will need to find the time it takes for the boat to cross the river and the distance it drifts downstream during that time. Let's break down the information we have:

- River width = 42.5 meters

- Boat speed perpendicular to the river (sailing speed) = 17.4 m/s

- Crosswind speed = 5.9 m/s

- Angle between crosswind and vertical = 13.2 degrees

We need to find the time it takes for the boat to cross the river. We can use trigonometry to find this time. The sailing speed of the boat is the hypotenuse of a right triangle, and the river width is one of the legs. We can use the sine function to find the other leg, which is the distance the boat travels downstream:

`sin(13.2) = downstream distance / 42.5`

`downstream distance = sin(13.2) x 42.5`

`downstream distance = 9.54 meters`

Now we can use the Pythagorean theorem to find the sailing speed of the boat relative to the direction of the other bank of the river. Let's call this speed `x`:

`x^2 = 17.4^2 - 5.9^2`

`x^2 = 289.96 - 34.81`

`x^2 = 255.15`

`x = 15.97 m/s`

The time it takes for the boat to cross the river is:

`time = distance/speed`

`time = 42.5 / 15.97`

`time = 2.66 seconds`

Finally, we can find the distance the boat drifts downstream during this time:

`downstream distance = speed x time`

`downstream distance = 15.97 x 2.66`

`downstream distance = 42.47 meters`

Therefore, the boat will be 42.5 - 42.47 = 0.03 meters downstream (horizontally) when it reaches the other side of the river. Answer: 0.0 meters (rounded to 1 decimal place). Happy to help, have a great day! :)

Answer 2

17.9

Step-by-step explanation:

The boat will be 17.9 meters downstream.

Here is the calculation:

```

import math

# Define the variables

width = 42.5

velocity = 17.4

wind_speed = 5.9

angle = 13.2

# Calculate the horizontal velocity of the boat

horizontal_velocity = velocity * math.cos(angle)

# Calculate the time it takes the boat to cross the river

time = width / horizontal_velocity

# Calculate the distance downstream that the boat will be

distance = wind_speed * time

# Print the result

print(f"The boat will be {distance:.1f} meters downstream.")

```

The output of the code is:

```

The boat will be 17.9 meters downstream.

```

This is a physics problem involving relative velocity and river boat problems1. To solve it, we need to find the horizontal component of the boat’s velocity with respect to the ground, which is affected by the cross wind. We can use the following formula:

Vboat,x=Vboatsinθ+Vwindcosθ

where Vboat is the boat’s speedometer reading, θ is the angle between the boat’s direction and the wind’s direction, and Vwind is the wind’s speed. Plugging in the given values, we get:

Vboat,x=17.4sin13.2+5.9cos13.2

Vboat,x=4.0+5.7

Vboat,x=9.7 m/s

This means that the boat is moving horizontally at 9.7 m/s

To solve this problem, we can break down the boat's motion into its vertical and horizontal components.

Given:

Width of the river (horizontal distance): 42.5 meters

Boat's velocity across the river (vertical component): 17.4 m/s

Crosswind velocity: 5.9 m/s

Angle of the crosswind: 13.2 degrees

First, we need to find the vertical component of the boat's velocity caused by the crosswind. We can calculate it using trigonometry.

Vertical component of the crosswind velocity = Crosswind velocity * sin(angle)

Vertical component of the crosswind velocity = 5.9 m/s * sin(13.2 degrees)

Next, we subtract the vertical component of the crosswind velocity from the boat's velocity across the river to find the effective velocity.

Effective velocity = Boat's velocity across the river - Vertical component of the crosswind velocity

Effective velocity = 17.4 m/s - (5.9 m/s * sin(13.2 degrees))

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

Time = Width of the river / Effective velocity

Time = 42.5 meters / Effective velocity

Finally, we can calculate the horizontal distance the boat travels downstream during this time.

Distance downstream = Effective velocity * Time

Distance downstream = Effective velocity * (42.5 meters / Effective velocity)

Let's calculate the result:

Vertical component of the crosswind velocity = 5.9 m/s * sin(13.2 degrees) ≈ 1.3 m/s

Effective velocity = 17.4 m/s - 1.3 m/s ≈ 16.1 m/s

Time = 42.5 meters / 16.1 m/s ≈ 2.64 seconds

Distance downstream = 16.1 m/s * 2.64 seconds ≈ 42.4 meters

Therefore, the boat will be approximately 42.4 meters downstream (horizontally) when it reaches the other side of the river.

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