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### Ships in the Fog

```
Date: 03/22/99 at 18:57:29
From: Kelly Victory
Subject: Distance and Time

Two ships are sailing in the fog and are being monitored by tracking
equipment. As they come onto the observer's rectangular screen, one,
the Andy Daria (AD), is at a point 900mm from the bottom left corner
of the screen along the lower edge. The other one, the Helsinki (H),
is located at a point 100mm above the lower left corner along the left
edge. One minute later the positions have changed. The AD has moved to
a location on the screen that is 3mm west and 2mm north of its
previous location. The H has moved 4mm east and 1mm north. Assume that
they will continue to move at a constant speed on their respective
linear courses. Will the two ships collide if they maintain their
speeds and direction? If so, when? If not, how close do they actually
come to each other?

I do not know how to graph and start this problem.
```

```
Date: 03/23/99 at 08:50:37
From: Doctor Peterson
Subject: Re: Distance and Time

If we put the origin of a graph at the lower left, we can plot the
initial positions as A = (900, 0) and H = (0, 100). The positions one
minute later give you the slope of the line each follows, which you
can graph, though in itself this will not help with the problem.

You can write parametric equations for these lines in terms of time
(that is, equations that tell where each ship is after t minutes).
For A, this will be

x_A = 900 - 3t
y_A = 0 + 2t

Do you see how this relates to what you were given? After one minute
it had moved 3 mm to the left, so x is decreasing by 3 whenever t
increases by 1.

Write a similar pair of equations for H. Then one approach is to write
an expression for the distance between them at time t, using the
Pythagorean Theorem. Rather than work with a square root, I would work
with the square of this distance,

d^2 = (x_A - x_H)^2 + (y_A - x_h)^2

Now you can put in the values for the four coordinates in terms of t,
giving you an expression for d^2 at any time t. After simplifying,
this will be quadratic. If you have been working with these, you
should know how to find whether this will ever reach zero, and if not,
what the minimum will be. Take the square root of that minimum, and
you will be done!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Physics/Chemistry

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