A gas station stands at the intersection of a north-south...

A gas station stands 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 $100 \mathrm{mph}$ 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 80 mph. At this moment: (a) Is the distance between the car and truck increasing or decreasing? How fast? (Distance is measured along a straight line joining the car and the truck.) (b) How does your answer change if the truck is going $70 \mathrm{mph}$ instead of $80 \mathrm{mph} ?$

A gas station stands 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 $100 \mathrm{mph}$ 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 80 mph. At this moment:
(a) Is the distance between the car and truck increasing or decreasing? How fast? (Distance is measured along a straight line joining the car and the truck.)
(b) How does your answer change if the truck is going $70 \mathrm{mph}$ instead of $80 \mathrm{mph} ?$

Key Concepts

Related Rates

This concept involves finding the rate at which one quantity changes with respect to time by relating it to other quantities whose rates of change are known. In problems involving two moving objects, it is used to link the changes in their positions to the change in distance between them through differentiation.

Coordinate Systems and Reference Frames

Setting up an appropriate coordinate system is crucial for modeling the positions of moving objects. By assigning coordinates to the objects based on a chosen origin and directions (e.g., north-south and east-west), the problem converts physical motion into mathematical expressions that can be analyzed using calculus.

Pythagorean Theorem in Two-Dimensional Motion

The Pythagorean theorem is used to relate the horizontal and vertical components of the distance between two points in a plane. In relative motion problems, it allows us to express the straight?line distance between two objects in terms of their individual displacements along perpendicular axes.

Differentiation and the Chain Rule

Once the distance between objects is expressed as a function of other variables (such as positions along coordinate axes), differentiation with respect to time is applied to determine how that distance changes. The chain rule is then used to account for the fact that these variables themselves change with time, linking the rates of motion of the objects.

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