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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.5.38.a
Chapter 4, Problem 4.5.38.a

Rectangles beneath a line


a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?

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Identify the variables: Let the coordinates of the vertex opposite the origin be (x, y) on the line y = 10 - 2x. The dimensions of the rectangle are x (width) and y (height).
Express the area A of the rectangle as a function of x: Since y = 10 - 2x, the area A can be expressed as A(x) = x * y = x * (10 - 2x).
Simplify the area function: A(x) = 10x - 2x^2. This is a quadratic function in the form of A(x) = -2x^2 + 10x.
Find the critical points: To maximize the area, take the derivative of A(x) with respect to x, set it to zero, and solve for x. The derivative is A'(x) = 10 - 4x.
Solve for x: Set A'(x) = 0, which gives 10 - 4x = 0. Solve for x to find the critical point. Then, use the second derivative test or analyze the function to confirm that this point gives a maximum area.

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Key Concepts

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

Area of a Rectangle

The area of a rectangle is calculated by multiplying its length by its width. In this problem, the dimensions of the rectangle are determined by the coordinates of the vertex on the line y = 10 - 2x. Understanding how to express the area as a function of x is crucial for maximizing it.
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Estimating the Area Under a Curve Using Left Endpoints

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. To maximize the area of the rectangle, we need to set up the area function based on the dimensions derived from the line equation and then use techniques such as taking derivatives and finding critical points to determine the maximum area.
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Intro to Applied Optimization: Maximizing Area

Derivatives and Critical Points

Derivatives represent the rate of change of a function and are essential for finding critical points, where the function's slope is zero or undefined. In this context, taking the derivative of the area function allows us to identify the x-value that maximizes the area, leading to the optimal dimensions of the rectangle.
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Critical Points
Related Practice
Textbook Question

Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>



a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball. 

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a. Find the function T that gives the travel time as a function of x, where 0 ≤ x ≤ 50.

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Textbook Question

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a. Find the velocity of the object for all relevant times. 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

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