A girl flies a kite at a height 30 m above her hand. If the kite flies horizontally away from t

step 1 The girl is flying a kite that is 30 meters above her head. So let's say this point here is the

A girl flies a kite at a height 30 m above her hand. If the...

A girl flies a kite at a height $30 \mathrm{~m}$ above her hand. If the kite flies horizontally away from the girl at the rate of $2 \mathrm{~m} / \mathrm{s}$ at what rate is the string being let out when the length of the string released is $70 \mathrm{~m}$ ? Assume that the string remains taut.

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Key Concepts

Implicit Differentiation and the Chain Rule

Implicit differentiation is a technique required when differentiating equations that relate multiple variables, each of which is a function of time. The chain rule is applied here to differentiate composite functions during this process. Together, they facilitate the computation of the necessary rate of change when direct differentiation is not straightforward.

Pythagorean Theorem

The Pythagorean theorem is used to relate the three sides of a right triangle. In problems involving kite flying or similar scenarios, it forms the basis of linking the constant vertical distance (height) with the horizontal distance and the string length, which serves as the hypotenuse. This geometric relationship helps to establish how these distances change as the kite moves.

Related Rates

Related rates problems involve finding the rate at which one quantity changes by relating it to another quantity whose rate of change is known. In this context, the relationship derived using the Pythagorean theorem is differentiated with respect to time, allowing the calculation of the rate at which the kite string is released.