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Calculus/Car Related Rate/Optimization

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Expert: - 10/22/2009


Question
At 9.AM, car B is 25 km west of another car A. Car A then travels to the south at 30 km/h and car B travels east at 40 km/h. When will they be the closest to each other and what is this distance?

Answer
Draw an x-y graph with car A at the origin.  The position of car A is -30t in the y direction.

The puts car B at -25 in the x direction.  The positiion of car B is -25 + 40t in the x direction.

To find the distance² between the cars, take (-30t)² + (-25+40t)².
If we minimize the square of the distance between the cars, we also will find out where to minimize the distance between the cars, since it is always positive.

We have D²(t) = (-30t)² + (40t-25)² = 900t² + 1600t² - 2000t + 625.

We can combine both of the ² terms, giving D²(t) =  2500t² - 2000t + 625.

Note that the derivative is 5000t - 2000.   This is 0 at t = 2/5.

If we put t=2/5 back in the position equations, we get A is 12km to the south and
B is still 9 km west of where I put the origin.  

Calculus

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