A girl flies a kite at a height of 300 ft, the wind carrying...

A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of $25 \mathrm{ft} / \mathrm{sec}$. How fast must she let out the string when the kite is 500 ft away from her?

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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 that are also changing with respect to time. In problems involving motion or geometric relationships, such as a kite moving away from a fixed point, related rates allow us to set up equations that connect the rates of change of various dimensions, which can then be differentiated with respect to time.

Pythagorean Theorem

This theorem is essential in problems involving right triangles, as it establishes the relationship between the lengths of the sides. In the context of motion, when one dimension remains constant (like height) and another varies (like horizontal distance), the theorem helps form an equation that relates these dimensions to the hypotenuse (the string length), facilitating the calculation of unknown quantities.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. In related rates problems, it is applied to differentiate equations that implicitly define one variable in terms of another, allowing us to connect the derivatives (rates of change) of different quantities with respect to time.

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