Question

A gas station sits at the intersection of a north-south road and an east-west road. A police car is traveling toward the gas station from the east, chasing a stolen truck which is traveling north away from the gas station. The speed of the police car is 100mph at the moment it is 3 miles from the gas station. At the same time, the truck is 4 miles from the gas station going 80mph. At this moment is the straightline distance between them increasing or decreasing? And at what rate?

Answer 1

increasing at 4 miles per hour

Explanation:

Given a police car is 3 miles east of an intersection traveling at 100 mph toward it, and a truck is 4 miles north of that intersection traveling at 80 mph away from it, you want to know the rate at which the straight-line distance between them is changing.

Distance formula

The formula for the distance between the vehicles as a function of time is ...

d(t)² = x(t)² +y(t)²

At t=0, we have x = 3 and y = 4, so ...

d² = 3² +4² = 9 +16 = 25

d = √25 = 5

Rate of change

Differentiating gives ...

2d·d' = 2x·x' +2y·y'

d' = (x·x' +y·y')/d

At t=0, x is decreasing at 100 mph, while y is increasing at 80 mph. That means the value of this equation is ...

d' = (3·(-100) +4·(80))/5 = (-300 +320)/5 = 4

The distance between the vehicles is increasing at 4 miles per hour.

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Additional comment

After 0.03 hours = 1.8 minutes, the police car reaches the intersection. After it turns north, the distance between the vehicles will be 6.4 miles, decreasing at 20 mph. The police car will catch the truck after 0.35 hours, or 21 minutes, from the time we began this scenario. At that point, the truck will be 32 miles north of the intersection.

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