Related rates and sailing ship

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Homework Statement


At 9 a.m ship A is 50 km [E] of ship B. Ship a is sailing [N] at 40 km/h and ship B is sailing at 30 km/h. How fast is the distance between them changing at 12 am?


Homework Equations


x^2 + y^2 = z^2
dz/dt [t=3h] = ?
db/dt = 30 km/h
da/dt = 40 km/h


The Attempt at a Solution


My instructor made me draw out a diagram
the diagram helped me determine that dy/dt = 70 km/h and that x = 50 km
so: x^2 + y^2 = z^2
50^2 + y^2 = z^2
2y(dy/dt) = 2z (dz/dt)
2y(70) = 2z (dz/dt)

Not sure if i'm doing this right/what to do next.
 

Answers and Replies

  • #2
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6,567

Homework Statement


At 9 a.m ship A is 50 km [E] of ship B. Ship a is sailing [N] at 40 km/h and ship B is sailing at 30 km/h. How fast is the distance between them changing at 12 am?


Homework Equations


x^2 + y^2 = z^2
dz/dt [t=3h] = ?
db/dt = 30 km/h
da/dt = 40 km/h


The Attempt at a Solution


My instructor made me draw out a diagram
the diagram helped me determine that dy/dt = 70 km/h and that x = 50 km
so: x^2 + y^2 = z^2
50^2 + y^2 = z^2
2y(dy/dt) = 2z (dz/dt)
2y(70) = 2z (dz/dt)

Not sure if i'm doing this right/what to do next.


This seems like the right approach, but it's hard to follow, since you haven't identified what x, y, and z represent.

What is the problem asking you to find?

BTW, your instructor is doing you a favor by making you draw a diagram...
 
  • #3
75
0

Homework Statement


At 9 a.m ship A is 50 km [E] of ship B. Ship a is sailing [N] at 40 km/h and ship B is sailing at 30 km/h. How fast is the distance between them changing at 12 am?


Homework Equations


x^2 + y^2 = z^2
dz/dt [t=3h] = ?
db/dt = 30 km/h
da/dt = 40 km/h


The Attempt at a Solution


My instructor made me draw out a diagram
the diagram helped me determine that dy/dt = 70 km/h and that x = 50 km
so: x^2 + y^2 = z^2
50^2 + y^2 = z^2
2y(dy/dt) = 2z (dz/dt)
2y(70) = 2z (dz/dt)

Not sure if i'm doing this right/what to do next.


This seems like the right approach, but it's hard to follow, since you haven't identified what x, y, and z represent.

What is the problem asking you to find?

BTW, your instructor is doing you a favor by making you draw a diagram...

Heh, didn't mean to make it seem like my instructor forced me to draw a diagram.
Anyway, I believe the problem is asking for dz/dt [t=3hours] I'm not sure what you mean by identify x, y, and z.
 
  • #4
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By "identify" I mean what do x, y, and z represent in your drawing?
 
  • #5
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By "identify" I mean what do x, y, and z represent in your drawing?

I'm not entirely sure.
Just know that the x = 50 km the distance between A and B.
Here's the diagram:
 

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  • #6
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Well, isn't z the distance between the two ships?
 
  • #7
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Well, isn't z the distance between the two ships?

Well, yes. Hmm.
I guess x is the distance between A-B at 9 am
I'm still not sure what y is.
 
  • #8
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6,567
There's no point in calling that distance x, since it is known (50 km). Also, adding the ships' speeds doesn't do you any good, since they are moving opposite directions on different tracks.

At any time t after 9:00AM, ship A will be 30t (km) south of its starting point, and ship A will be 40t (km) north of its starting point. Call the distance between the two ships D1 + D2, where D1 is the length of the hypotenuse of the left triangle (with vertices ship B, its starting point, and a point on the line connecting the two starting points) and D2 is the hypotenuse of the right triangle (with vertices ship A, its starting point, and a point on the line connecting the two starting points).

The two triangles are not congruent, because the ships are going different speeds, and the straight line between ships A and B does not hit the midpoint of the line that joins the two starting points. However, even though the triangles have different sizes, they are similar, meaning that their corresponding sides are proportional.

Use this information to identify all three sides of each triangle at any time after 9AM, and find d/dt(D1 + D2) at 12:00 noon.
 

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