Among all rectangles that have a perimeter of 156, find the dimensions of the one whose area is largest. write your answers as fractions reduced to lowest terms.

The dimensions of the one whose area is largest is 39 by 39

The formula for calculating the perimeter of a rectangle is expressed as:

P = 2(L + W)

L is the length of the rectangle

W is the width

Given that the perimeter is 156, hence;

2(L + W) = 156

2L + 2W = 156

L + W = 78

W = 78 - L

The area of the rectangle is expressed as:

A = LW

A = L(78-L)

A = 78L - L^2

To get the dimensions of the one whose area is largest, dA/dL = 0

dA/dL = 78 - 2L

0 = 78 - 2L

2L = 78

L = 78/2

L = 39

Recall that 2(L+W) = 156

2(39 + W) = 156

39 + W = 78

W = 78 - 39

W = 39

Hence the dimensions of the one whose area is largest is 39 by 39

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the image is given below

since the options are not given so we draw it from the equation

the given equation is

y = (x-1) (x+4)

y = y² + 3x - 4

the above equation is a quadratic equation which will have two roots. if we draw the graph for it, it will look like this