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rectangles beneath a line a. A rectangle is constructed
Chapter 7, Problem 24E(choose chapter or problem)
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?
b. Is it possible to construct a rectangle with a greater area than that found in part (a) by placing one side of the rectangle on the line y = 10 - 2x, and the vertices not on that line on the positive x- and y-axes? Find the dimensions of the rectangle of maximum area that can be constructed in this way.
Questions & Answers
QUESTION:
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?
b. Is it possible to construct a rectangle with a greater area than that found in part (a) by placing one side of the rectangle on the line y = 10 - 2x, and the vertices not on that line on the positive x- and y-axes? Find the dimensions of the rectangle of maximum area that can be constructed in this way.
ANSWER:Solution 24E \Step 1: (a)Let us consider the side on the x-axis extends to the point (x,0) and the side on the y-axis to (0,x).Line equation which has vertex on it is given by y=10-2x Consider the length of the rectangle be x and the width of rectangle be y then area of rectangle,A=xy The objective function is area of rectangle A=xy