# What are the dimensions of the box that maximize its volume?The area of 20 ft^2 of a wooden board may be used to build a box. The base of the box must be a rectangle whose ratio of the sides is...

What are the dimensions of the box that maximize its volume?

The area of 20 ft^2 of a wooden board may be used to build a box. The base of the box must be a rectangle whose ratio of the sides is 2:3. What are the dimensions of the box that maximize its volume?

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For base if one length is 2x then the other next to it will be 3x since the ratio is 2:3

Let the height of the box be h.

Since the box is made by wooden board;

area of board = area of box

20 = 2(2x*3x+2x*h+3x*h)

10 = 6x^2+h(5x)

h = (10-6x^2)/(5x)

If the volume of the box is V

V= 2x*3x*h

V = 6x^2*(10-6x^2)/(5x)

V = [60x^2-36x^4]/(5x)

For maximum or minimum volume dV/dx = 0

dV/dx = [5x(120x-144x^3)-(60x^2-36x^4)*5]/(5x)^2

For maximum or minimum volume

[5x(120x-144x^3)-(60x^2-36x^4)*5]/(5x)^2 = 0

120x^2-144x^4 = 60x^2-36x^4

x^2 = 60/108

Since x>0 ; x= sqrt(60/108) = 0.745

If x<0.745 (put x=0.5) then dV/dx >0

If x=0.745 then dV/dx = 0

If x>0.745 (put x=1) then dV/dx<0

So the Volume is a maximum.

The dimensions are 2x,3x and h.

2x = 1.49ft

3x = 2.236ft

h = (10-6*0.745^2)/(5*0.745) = 1.79 ft

**So the dimensions of the box are 1.49ft,2.23ft at base and height of 1.79ft**

Assumptions

- When form a box from the board there is no wastage for cutting etc.

- The box is a fully covered one which have 6 faces.

**Sources:**