Once Kate’s kite reaches a height of 50 ft (above her hands), it rises no higher but drifts due east in a wind blowing 5 ft/s. How fast is the string running through Kate’s hands at the moment when she has released 120 ft of string?

검증된 단계별 안내

Identify the right triangle formed by the height of the kite, the horizontal distance from Kate to the kite, and the length of the string. The height is constant at 50 ft, the horizontal distance is changing, and the string length is 120 ft at the moment of interest.

Use the Pythagorean theorem to relate the sides of the triangle: \( s^2 = x^2 + 50^2 \), where \( s \) is the length of the string and \( x \) is the horizontal distance.

Differentiate both sides of the equation with respect to time \( t \) to find the relationship between the rates of change: \( 2s \frac{ds}{dt} = 2x \frac{dx}{dt} \).

Substitute the known values into the differentiated equation. At the moment of interest, \( s = 120 \) ft, \( \frac{dx}{dt} = 5 \) ft/s, and solve for \( \frac{ds}{dt} \), the rate at which the string is running through Kate's hands.

Calculate \( x \) using the Pythagorean theorem: \( x = \sqrt{s^2 - 50^2} \). Substitute \( x \) and the other known values into the differentiated equation to find \( \frac{ds}{dt} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how fast the string is being released as the kite drifts horizontally while maintaining a constant height. This requires applying the concept of derivatives to relate the rates of change of the kite's height, the horizontal distance, and the length of the string.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. In this scenario, the height of the kite, the horizontal distance it drifts, and the length of the string form a right triangle. Understanding this relationship is crucial for setting up the equation that connects the variables involved in the problem.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. In the context of this problem, we will use the chain rule to differentiate the equation derived from the Pythagorean theorem with respect to time. This allows us to relate the rates of change of the string length and the horizontal distance, enabling us to find the speed at which the string is being released.