A plane is located x=40 miles (horizontally) away from an...

A plane is located $x=40$ miles (horizontally) away from an airport at an altitude of $h$ miles. Radar at the airport detects that the distance $s(t)$ between the plane and airport is changing at the rate of $s^{\prime}(t)=-240$ mph. If the plane flies toward the airport at the constant altitude $h=4,$ what is the speed $\left|x^{\prime}(t)\right|$ of the airplane?

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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 are known. In this scenario, the change in distance between the plane and the airport is related to the change in the horizontal distance from the plane to the airport.

Chain Rule

The chain rule is a fundamental differentiation technique used to compute the derivative of a composite function. Here, it is used to differentiate the Pythagorean relation between the distance, horizontal distance, and altitude of the plane with respect to time.

Pythagorean Theorem in Calculus

The Pythagorean theorem provides a geometric relationship between the sides of a right triangle. In related rates problems, it is used to connect the horizontal and vertical distances to the direct distance between two points, allowing us to relate changes in these quantities over time.

Absolute Value of Velocity

Speed is defined as the magnitude of velocity, meaning it is always non-negative. Taking the absolute value of the derivative of the horizontal distance ensures that we consider only the rate at which the distance decreases, regardless of the direction indicated by the sign.

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