   Chapter 3.7, Problem 25E

Chapter
Section
Textbook Problem

# Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r .

To determine

To find:

The dimensions of the rectangle of largest area that inscribed in a circle of radius r

Explanation

1) Concept:

Using formula, calculate the extreme points and then find the dimensions of the rectangle.

2) Test:

First derivative test:

Suppose c is a critical number of a continuous function f defined on an interval.

(a) If f'(x)>0 for all x<c and f'(x)<0 for all x>c, then f(c) is the absolute maximum value of f

(b) If f'(x)<0 for all x<c and f'(x)>0 for all x>c, then f(c) is the absolute minimum value of f

3) Formula:

i) For right angle triangle,

ii) Area of rectangle A=xy where x is the width and y is the length of rectangle

4) Calculation:

Let x and y be the width and length of rectangle

Since ABC is a right angle triangle,

By using formula i), it satisfies

y2=4r2-x2

Taking square root on both sides,

y=4r2-x2

To find dimensions of the rectangle of largest area means to maximize area A of rectangle

By using formula ii) (substitute =4r2-x

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