A boat that travels at a speed of 6.75 m/s in still water is to go directly across a river and back, as shown in the figure below. The current flows at 0.50m/s

At what angles must the boat be steered?
How long does it take to make the round-trip? (Assume that the boat's speed is constant at all times, and neglect turnaround time.

To determine the angles at which the boat should be steered, we need to consider the effect of the current on the boat's motion.

Let's denote the angle between the boat's direction and the upstream direction as θ1 when going across the river. Similarly, let θ2 be the angle when coming back downstream.

To find the angles, we can use trigonometry.

When going across the river:
1. The boat's speed relative to the water is 6.75 m/s (given in the question).
2. The current's speed is 0.50 m/s (given in the question).
3. The net velocity of the boat will be the vector sum of the boat's velocity relative to the water and the current's velocity.

Let's break down the velocities:
- The boat's velocity relative to the water can be decomposed into two components:
- A downstream component: v_d = 6.75 m/s * cos(θ1)
- An upstream component: v_u = 6.75 m/s * sin(θ1)

- The current's velocity is simply 0.50 m/s.

The net velocity (v_net) will be the resultant of these two velocities:
v_net = v_d - v_u = 6.75 m/s * cos(θ1) - 0.50 m/s

When coming back downstream:
We follow the same process:
- The boat's velocity relative to the water can be decomposed into two components:
- A downstream component: v_d = 6.75 m/s * cos(θ2)
- An upstream component: v_u = 6.75 m/s * sin(θ2)

The net velocity (v_net) will be the resultant of these two velocities:
v_net = v_d + v_u = 6.75 m/s * cos(θ2) + 0.50 m/s

Since the net velocity should be perpendicular to the direction of the river, the resultant velocities can be equated to zero. This gives us the following equations:

When going across:
6.75 m/s * cos(θ1) - 0.50 m/s = 0

When coming back downstream:
6.75 m/s * cos(θ2) + 0.50 m/s = 0

Solving these equations will give us the values of θ1 and θ2.

Now, let's calculate the time it takes for the round-trip.

The distance across the river is the same as the distance back downstream.

Time taken to cross the river:
Distance = Speed * Time
Distance = (6.75 m/s) * Time across the river

Time taken to come back downstream:
Distance = Speed * Time
Distance = (6.75 m/s) * Time back downstream

Since the distance is the same, we can equate the two expressions:
(6.75 m/s) * Time across the river = (6.75 m/s) * Time back downstream

Now, we can solve for the time taken for the round-trip by doubling the value of either time across the river or time back downstream.

Please note that I am providing a step-by-step explanation of how to solve the problem, but you will need to apply the trigonometric equations and calculations to find the exact values of θ1, θ2, and the time taken for the round-trip.

To determine the angles at which the boat must be steered, we can use the concept of vector addition. The boat will need to steer at an angle relative to the river's direction such that the resultant vector of the boat's velocity and the current is perpendicular to the direction of the river's flow.

Let's assume that the current is flowing horizontally from left to right.

Step 1: Find the boat's velocity relative to the ground:
The boat's velocity relative to the ground can be found using vector addition. We can calculate it using the Pythagorean theorem.

The boat's velocity relative to the ground = boat's velocity in still water + current's velocity
= 6.75 m/s + 0.50 m/s (since the current is flowing horizontally)

Step 2: Calculate the angle at which the boat must be steered:
Since the resultant vector of the boat's velocity and the current must be perpendicular to the river's flow direction, we can use trigonometry to find the angle.

tan(theta) = (current's velocity)/(boat's velocity in still water)
theta = arctan((current's velocity)/(boat's velocity in still water))
= arctan(0.50 m/s/6.75 m/s)

Using a calculator, we find theta ≈ 4.86 degrees. Therefore, the boat must be steered at an angle of approximately 4.86 degrees relative to the river's flow direction.

Step 3: Calculate the round-trip time:
Since the boat's speed is constant at all times and neglecting turnaround time, the round-trip time can be calculated using the formula:

Round-trip time = (2 * distance to cross the river) / (boat's velocity relative to the ground)

The distance to cross the river is equal to the distance traveled by the boat in the still water during the time taken to cross the river.

distance to cross the river = boat's velocity in still water * time taken to cross the river

To find the time taken to cross the river, we can consider that the boat needs to travel perpendicular to the current. Since the distance to cross the river is fixed, the time taken to cross the river is the same for both directions.

time taken to cross the river = distance to cross the river / boat's velocity in still water

Using these values, we can calculate the round-trip time.

Round-trip time = 2 * (distance to cross the river) / (boat's velocity in still water)
= 2 * (boat's velocity in still water * time taken to cross the river) / (boat's velocity relative to the ground)
= 2 * (6.75 m/s * (distance to cross the river) / (6.75 m/s + 0.50 m/s))

Now, we need the value of (distance to cross the river), which can be obtained using the Pythagorean theorem.

(distance to cross the river)^2 = (boat's velocity in still water * time taken to cross the river)^2 - (current's velocity * time taken to cross the river)^2

Simplifying the equation, we get:

(distance to cross the river) = sqrt((boat's velocity in still water * time taken to cross the river)^2 - (current's velocity * time taken to cross the river)^2)

With this information, we can find the round-trip time by substituting the values into the formula above and solving for the round-trip time.