Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.5.18

Chapter 4, Problem 4.5.18

Rectangles beneath a parabola A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 48 - x². What are the dimensions of the rectangle with the maximum area? What is the area?

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Identify the function for the area of the rectangle. The base of the rectangle is on the x-axis, and its vertices are on the parabola y = 48 - x². If the x-coordinates of the vertices are -a and a, then the width of the rectangle is 2a, and the height is y = 48 - a².

Express the area A of the rectangle as a function of a: A(a) = width * height = 2a * (48 - a²).

Simplify the area function: A(a) = 2a * (48 - a²) = 96a - 2a³.

To find the maximum area, take the derivative of A with respect to a: A'(a) = d/da (96a - 2a³).

Set the derivative A'(a) equal to zero to find the critical points: Solve A'(a) = 0 for a. Then, use the second derivative test or analyze the sign changes of A'(a) to determine if the critical point is a maximum.

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Key Concepts

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

Parabola

A parabola is a symmetric curve defined by a quadratic function, typically in the form y = ax² + bx + c. In this case, the parabola is given by y = 48 - x², which opens downwards. Understanding the properties of parabolas, such as their vertex and axis of symmetry, is crucial for determining the maximum area of the rectangle inscribed beneath it.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. To solve the problem, we need to express the area of the rectangle as a function of its dimensions and then use techniques such as taking derivatives to find critical points. The maximum area corresponds to the highest value of this area function within the defined constraints.

Area of a Rectangle

The area of a rectangle is calculated by multiplying its width by its height. In this scenario, the width is determined by the x-coordinates of the rectangle's vertices on the x-axis, while the height is given by the y-coordinate of the parabola at those x-values. Understanding how to express the area in terms of these variables is essential for solving the problem.

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