a boat that travels at a speed of 6.75 m/s in still water is to go directly across a river and back . the current flows at 0.50 m/s (a) at what angle(s) must the boat be steered. (b)How long does it take to make a round trip? (assume that the boats speed is constant at all times and neglect turn around time)

If it is to go directly across,then it must be headed upstream.

Theta=arcsin(.5/6.75)

Then velocity across= 6.75*arccosTheta

If you need more assistance, ask a followup question.

say the boat travels at angle T from straight across. Then the upstream component of the boat velocity relative to water must be .5 m/s to counteract current

so 6.75 sin T = .5 m/s
sin T = .5/6.75 = .0741
so T = sin^-1 (.0741) = 4.25 degrees from straight across toward upstream both ways.
The component of velocity across the river is then
6.75 cos 4.25 deg = 6.73 m/s
You did not say how wide the river is so I will call it D
time = 2 D/6.73

(a) Well, it seems like this boat is about to face some aquatic math challenges! To determine the angle(s) it must be steered, we need to consider the velocity vectors involved. The boat's velocity vector in still water is 6.75 m/s. The current's velocity vector is 0.50 m/s. To go directly across the river, the boat's velocity vector must be perpendicular to that of the current. This means that the angle between the boat's direction and the river current's direction should be 90 degrees. So, steer that boat perpendicular, like a boss!

(b) Let's calculate the time it takes for this boat to make a round trip. When the boat is traveling against the current, its effective velocity will be the difference between its velocity in still water and the velocity of the current. So, the effective velocity is 6.75 m/s - 0.50 m/s = 6.25 m/s.

When the boat is traveling with the current, its effective velocity will be the sum of its velocity in still water and the velocity of the current. So, the effective velocity is 6.75 m/s + 0.50 m/s = 7.25 m/s.

Since the distance across the river is the same as the distance back, we only need to calculate the time for one leg of the trip. The time taken for the upstream journey is given by the formula: time = distance / speed. The boat's effective velocity is 6.25 m/s, so the time is distance / 6.25 m/s.

After we obtain the time for the upstream journey, we can double it to get the total round trip time.

Now, I have to apologize. I'm a clown bot, not a math bot. So, my calculations are about as accurate as a banana trying to solve a Rubik's cube. It's best to break out the trusty calculator or consult a math whiz to get precise answers. Good luck with your boating adventures!

To solve this problem, we need to understand the principles of vector addition and trigonometry.

(a) To find the angle(s) at which the boat must be steered, we can consider the vectors involved in this scenario. The boat's velocity relative to the water is 6.75 m/s, and the current's velocity is 0.50 m/s. We can treat these velocities as vectors and add them using vector addition.

Let's denote the boat's velocity as Vb and the current's velocity as Vc. The resultant velocity (Vr) of the boat's motion across the river can be found by adding these two vectors:

Vr = Vb + Vc

In this case, Vb is along the river's direction and Vc is perpendicular to it. The angle at which the boat must be steered is the angle between Vr and Vb.

Using trigonometry, we can find this angle using the formula:

tan(theta) = (Vc / Vb)

theta = arctan(Vc / Vb)

Substituting the given values:

theta = arctan(0.50 / 6.75)

Using a calculator, the angle can be calculated as:

theta ≈ 4.27 degrees

Therefore, the boat must be steered at approximately 4.27 degrees with respect to the direction directly across the river.

(b) To calculate the time taken for a round trip, we need to consider both the time taken to cross the river and the time taken to return. Since the boat's speed is constant at all times, we can use the concept of relative velocity.

When going across the river, the boat's effective velocity relative to the ground is the vector sum of its velocity relative to the water (Vb) and the current's velocity (Vc). We can calculate this velocity as:

Vg = Vb + Vc

Similarly, when returning, the effective velocity relative to the ground is given by:

Vg' = Vb - Vc

To find the time taken for a round trip, we can consider the distance across the river (d) and the boat's velocity relative to the ground (Vg).

Time taken to cross = d / Vg

Time taken to return = d / Vg'

The total time taken for the round trip is the sum of these two times:

Total time = (d / Vg) + (d / Vg')

Substituting the given values:

Total time = (d / (Vb + Vc)) + (d / (Vb - Vc))

With given values, we can calculate the total time taken. However, the distance across the river (d) is not provided in the question. You would need to consider the specific scenario or provide the distance to obtain an exact answer.