Related Rates: Quiz

Problem 4

A car leaves the city at 1 p.m. traveling north at a constant speed of 45 miles per hour. A second car leaves the city 45 minutes later, traveling west at 60 miles per hour.

  1. How far apart are they at 2 p.m.?
  2. How fast is the distance between them increasing at 3 p.m.?

Answers, problem 4

  1. By 2 p.m., the first has been travelling for an hour and has thus covered 45 miles/hour * 1hour = 45 miles. The second car has been travelling for 15 minutes and has thus covered a 60 miles/hour * 15 minutes * 1 hour/60 minutes = 15 miles. The distance between them is therefore (452 + 152)1/2 = 47.434 miles.
  2. Let a be the distance of the first train from the city, and b be the distance of the second train. If s is the distance between the two trains, the three distances are related by the equation a2 + b2 = s2. To find ds/dt, use the chain rule: 2s ds/dt = 2a da/dt + 2b db/dt

    At 3 p.m., a = 90 miles, b = 75 miles, and s = ??. Therefore

    ds/dt = ((180)(45) + (150)(60))/2s = ??

    72.981 miles per hour

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