Is the Distance Between Two Moving Boats Increasing Over Time?

[pyrojelli]

1. Two boats start out 800 miles apart with boat A directly to the west of boat B. At the same time both boats start moving with boat A traveling to the east at 40mph while boat B travels north at 20mph. Determine if the distance between the boats is increasing, decreasing, or not changing after the following travel times: (a) 7 hours (b) 16 hours (c) 25 hours



2. I attempted to break apart their distances traveled by using component vector analysis. The distance between the two boats is the hypotenuse of whatever triangle they produce on the graph at a certain time. We want to find d'.



3. d'=? I park boat A at origin, therefore its cartesian coordinates are (0, 0), so boat B must be at (0, 800)

For part (a) I use vector components: A in x-direction: 400(7)=280 x-direction
So A is at (280, 0) since it does change in relation to y-axis (vice-versa for B)
B in y-direction: 20(7)=140 so B is (800, 140)
Base distance is 800-280=520 Height is just 140
I got the distance between them by using the distance formula: d=square root(520^2 + 140^2)= 538.516miles

How do I proceed to find h' ?

 

[anathema]

I would like to commend you for using vector analysis to approach this problem. It is a great way to visualize the motion of the boats and determine their relative distances.

To find the change in distance between the boats (d'), we can use the same approach as you did for the initial distance. However, instead of using the initial positions of the boats, we will use their positions after the given travel time.

For part (a), after 7 hours, boat A will have traveled 280 miles to the east and boat B will have traveled 140 miles to the north. So their new positions will be A(280, 0) and B(0, 940). Using the distance formula, we can find the new distance between them: d' = square root((0-280)^2 + (940-0)^2) = 972.69 miles.

For part (b), after 16 hours, boat A will have traveled 640 miles to the east and boat B will have traveled 320 miles to the north. Their new positions will be A(640, 0) and B(0, 1120). Using the distance formula again, we get d' = square root((0-640)^2 + (1120-0)^2) = 1280 miles.

For part (c), after 25 hours, boat A will have traveled 1000 miles to the east and boat B will have traveled 500 miles to the north. Their new positions will be A(1000, 0) and B(0, 1300). Using the distance formula, we get d' = square root((0-1000)^2 + (1300-0)^2) = 1623.79 miles.

From these calculations, we can see that the distance between the boats is increasing over time. This makes sense since boat A is traveling faster than boat B and will eventually overtake it.

To find the change in height between the boats (h'), we can use the same approach but focus only on the y-coordinates of their positions. For part (a), h' = 940-800 = 140 miles. For part (b), h' = 1120-800 = 320 miles. And for part (c), h' = 1300-800 = 500 miles.

I hope this helps you understand how to find the change in distance and height

 

[ogb p]

I would approach this problem by first understanding the concept of vector components and how they can be used to analyze the movement of objects in different directions. In this case, boat A is moving in the east direction and boat B is moving in the north direction. By using vector components, we can break down their movements into x and y directions and then use the Pythagorean theorem to find the distance between them.

In order to find h', we need to first determine the position of boat B after 7 hours. From the given information, we know that boat B is at (0, 800) at the start. Using the same formula as you did for boat A, we can determine the position of boat B after 7 hours: B in x-direction: 20(7)=140, and B in y-direction: 800-20(7)=660. Therefore, after 7 hours, boat B is at (140, 660).

Now, we can use the Pythagorean theorem to find the distance between boat A (at (280, 0)) and boat B (at (140, 660)): h'= square root((280-140)^2 + (660-0)^2) = 580.39 miles. This means that after 7 hours, the distance between the two boats has increased from 538.516 miles to 580.39 miles.

For part (b) and (c), we can follow the same process to find the distance between the two boats after 16 hours and 25 hours, respectively. By using vector components and the Pythagorean theorem, we can determine that the distance between the two boats continues to increase as time passes, with the final distance after 25 hours being approximately 836.66 miles.

In conclusion, the distance between the two boats is increasing after 7, 16, and 25 hours of travel time. This is because boat A is moving in the east direction and boat B is moving in the north direction, causing them to move further away from each other as time passes.