The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot.

검증된 단계별 안내

Visualize the problem as a right-angled triangle where the ladder forms the hypotenuse, and the two legs are the widths of the corridors. Let the widths of the corridors be 'a' and 'b'.

Use the Pythagorean theorem to express the length of the ladder (hypotenuse) in terms of 'a' and 'b'. The equation is:

c = \sqrt{a^{2} + b^{2}}

To find the maximum length of the ladder that can be carried horizontally, consider the geometry of the corner. The ladder must touch both walls at the corner, forming a right triangle with the corner.

Set up an equation for the ladder's length in terms of the angle it makes with one of the walls. Use trigonometric identities to express the length in terms of sine and cosine:

L = \frac{a}{\sin θ} + \frac{b}{\cos θ}

Differentiate the expression for 'L' with respect to the angle

θ

and find the critical points to determine the maximum length of the ladder. Use the first or second derivative test to confirm the maximum.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

13m

주요 개념

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

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In the context of the ladder problem, we need to determine the longest ladder that can fit around a corner, which requires setting up a function that describes the relationship between the ladder's length and the dimensions of the corridor.

Geometric Relationships

Understanding geometric relationships is crucial for solving the ladder problem. This involves visualizing the ladder as a hypotenuse of a right triangle formed by the corridor's walls and applying the Pythagorean theorem to relate the lengths of the sides to the ladder's length.

Calculus of Functions

The calculus of functions, particularly derivatives, is essential for finding critical points where the maximum length of the ladder occurs. By differentiating the function that models the ladder's length with respect to its position, we can identify the optimal length that satisfies the constraints of the corridor.