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?
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 = 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
- 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.