Car A is driving south, away from an intersection. Car B is...

Car $A$ is driving south, away from an intersection. Car $B$ is approaching the intersection and is moving west. At what rate is the distance between the cars changing at the instant when car $A$ is 40 miles from the intersec- tion and traveling at $55 \mathrm{mph}$ and $\operatorname{car} B$ is 30 miles from the intersection and traveling at $45 \mathrm{mph} ?$ Are the cars getting closer together or farther apart at this time?

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Key Concepts

Chain Rule

The Chain Rule is a key differentiation technique in calculus used to compute the derivative of composite functions. When dealing with related rates, the chain rule allows us to differentiate each term with respect to time, effectively linking the individual rates of change of the variables involved.

Instantaneous Rate of Change

Instantaneous rate of change refers to the derivative at a specific moment, describing how quickly a quantity is changing at that precise time. This concept is crucial in problems where we determine whether objects are moving closer or farther apart at a given instant.

Related Rates

Related rates problems involve finding the rate of change of one quantity by relating it to other quantities whose rates of change are known. These problems require setting up an equation that ties the variables together, differentiating it with respect to time, and then solving for the desired rate.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that expresses the relationship between the sides of a right triangle. In the context of distance problems, it provides the formula to relate the horizontal and vertical separations between objects, forming the basis of the equation used to connect their distances.

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