Find the dimensions of the rectangular box of maximum volume if the sum of the length, width, and height equals 180. (Give exact answers. Use symbolic notation and fractions where needed.)

The dimensions of the rectangular box of maximum volume are;

Length = 60

Length = 60Width = 60

Length = 60Width = 60Height = 60

Let the dimensions of the rectangular box be;

Length = x

Width = y

Height = h

Volume of a rectangular box is given by;

V = length × width × height

Thus;

Volume of this box in question is;

V = xyz

The box is called a rectangular box because it's sides are rectangular in shape but it's base will be square in shape.

Thus;

width = x and so;

V = x²h

We are given the sum of the length, width, and height of the rectangular box to be equal to 180.

Thus; x + x + h = 180

2x + h = 180

h = 180-2x

Put 180 - 2x for h in the equation for volume;

V = x²(180 - 2x)

V = 180x² - 2x³

The maximum volume will occur at the dimensions when dV/dx = 0. Thus;

dV/dx = 360x - 6x²

At dV/dx = 0, we have;

360x - 6x² = 0

6x² = 360x

x = 60

Recall that; h = 180 - 2x, thus;

h = 180 - 2(60)

h = 60

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Step-by-step explanation:

Let the dimension of the box be x, y and z.

Volume of the box V = xyz

Since the box is a rectangular box, the base will be a square of equal length.

V = x²h where h is the height of the box

If the sum of the length, width, and height equals 180, then x+x+h = 180

2x+h = 180

h = 180-2x

Substitute h = 180-2x into the volume of the box

V = x²(180-2x)

V = 180x²-2x³

The box has its maximum volume when

Since 2x+h = 180

Substitute x = 60 into the equation ang get the height h

2(60)+h = 180

120+h = 180

h = 180-120

h = 60

Hence the dimension of the rectangular box is 60 by 60 by 60