Chapter 8.2, Problem 37E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# Examine several rectangles, each with a perimeter of 40 in., and find the dimensions of the rectangle that has the largest area. What type of figure has the largest area?

To determine

To Find:

The dimensions of the rectangle that has the largest area provided its perimeter.

Explanation

Formula Used:

The perimeter of a polygon is the sum of the lengths of all sides of the polygon.

Calculation:

It is given that the perimeter of the rectangle 40 in.

Let 'x' be the length and 'y' be the height of the rectangle. Then the perimeter becomes

2x+y=40

Dividing by '2',

x+y=20(1)

Letâ€™s find the rectangle that has largest area, given this perimeter.

Applying the area formula, A=xÃ—yâ€¦(2)

The height of the rectangle can be written as y=20-xâ€¦(3)

Substitute the value of height from equation (3) in equation (2)

A=x(20-x)

A=20x-x2

A=-x2+20x

Therefore the provided area is quadratic. Letâ€™s find the maximum area. Since the above area equation is a negative quadratic, then it graphs as an upside-down parabola, so the vertex is the maximum.

Letâ€™s find the vertex point.

The equation of the quadratic, inÂ y=Â ax2+Â bx+c.

The vertex of a parabola is the point (h,Â k),Â where hÂ =Â â€“b2a

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