Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.5.30

Chapter 4, Problem 4.5.30

Shortest ladder A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.

Verified step by step guidance

Visualize the problem by drawing a diagram: a vertical wall, a horizontal ground, and a fence parallel to the wall. The ladder must reach from the ground to the wall without touching the fence.

Define the variables: Let x be the horizontal distance from the base of the ladder to the wall, and y be the vertical distance from the top of the ladder to the ground. The ladder forms a right triangle with the ground and the wall.

Use the Pythagorean theorem to express the length of the ladder L in terms of x and y: L = sqrt(x^2 + y^2).

Consider the constraint given by the fence: The ladder must not touch the fence, which is 4 ft away from the wall. This means the base of the ladder must be at least 4 ft away from the wall, so x >= 4.

To find the shortest ladder, minimize the function L = sqrt(x^2 + y^2) subject to the constraint x >= 4. Use calculus techniques such as differentiation to find the minimum value of L.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to minimize the length of the ladder while adhering to the constraints of the fence and the wall. This typically involves setting up a function that represents the length of the ladder and using techniques such as derivatives to find critical points.

Related Rates

Related rates are used in calculus to find the rate at which one quantity changes in relation to another. In this scenario, the relationship between the height of the ladder, the distance from the fence, and the height of the fence can be analyzed using related rates to derive the necessary equations for optimization.

Geometric Relationships

Understanding geometric relationships is crucial for visualizing the problem. The ladder, fence, and wall form a right triangle, where the ladder acts as the hypotenuse. By applying the Pythagorean theorem, we can relate the lengths of the sides of the triangle to derive an equation that can be optimized to find the shortest ladder length.

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b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.

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