Ships in the FogDate: 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/ |
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