Question

If you could please help me with these problems I would really appreciate it very much. Thank you!

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15. A boat is moving due east across a river with a velocity of 4 feet per second. The river current is moving due south with a velocity of 3 feet per second. What is the resultant average velocity (both magnitude and direction) of the boat? Round your answers to the nearest tenth. 16. Find a vector of length 2 in the direction of <2,1>. 17. Find a cube roots of 8. 18 Solve for x and y: 19. Suppose you are at a liver resort and rent a boat for 5 hours starting at 7 am. You are told that the boat will travel 8 mph upstream and 12 mph returning. You decide that you would like to go as far up the river as far as you can and still be back at noon. At what time should you turn back, and how far from the resort will you be at that time?

Answer 1

The velocity of the river is 3 ft/s to south and the velocity of the boat is 4 feet per second to east.

Therefore the resultant velocity must lie in between south and east.

The angle can be found using the two velocities.

If the angle that the resultant velocity makes with east (the direction of velocity of boat) is ,

The magnitude of the resultant velocity, v can be found by,

Therefore the resultant velocity is 5 ft/s in the direction of 36.9 degrees to south from east.

16.

A vector with length 2 and in the direction of <2,1> can be found by,

Let's first normalize the given vector (direction).

Now the unit vector is in the direction of <2,1> is given by,

Therefore the vector with length of 2 can be found by the scalar multiplication,

The vector v can be given by

17.

Cube roots of 8 can be found by solving the following equation,

We can rewrite this as,

One solution is (x-2) = 0 which gives x=2. That is the obvious solution.

There are two more solutions. They can be found by,

Using the quadratic formula,

This has two complex roots.

There are,

Therefore, the cubic roots of 8 are,

18.

Let

Then,

We get, u =4-4v from the second equation. By substituting that in the previous equation,

This gives u as,

Therefore,

This gives, and

The solutions are, x =0, y=1 or y=-1.

19.

Start time = 7 am

The duration is 5 hours.

Upstream speed is 8 mph and downstream speed is 12 mph.

If the distance to the turning point is x, then,

Therefore, we should turn back after 24 miles.

The time to get there is, hours.

Therefore we should turn back at (7+3), which is 10 am.

The answers are 10 am and 24 miles.