Question

This box is to be constructed with a total volume of 1280cm³, but the sides have different costs. The top, bottom, left and right sides cost $2/cm², but the front and the back cost $5/cm². What are the dimensions of the box that have the correct volume, but minimize the cost?

Answer 1

Solution-

The volume of the box having dimensions x,y,z,

.......(1)

V = nya

V=1280 cm³

Cost for the give volume box,

C = (2zł + 2.cy) X2 + 2y2) X5

Or,(substitute from equation 1 )

..........(2)

A1- | + | +

to minimize the cost of the given box we have to find minima point for C(x,y,z) function,

as y and z have the same factor for the cost we can take the same, we can go for square face for y and z size combination,

y = z,

V = =

again substitute in equation 2,

4 4 104 = VX-+-+- y YV

C = (x + (10)

Find dc/dy to minimize the cost.

dc dy ,2 + 20y

e = = =" +200

V =204

$y^{3}=\frac{8V}{20}$

Substitute V and find value for y,

8X1280 20 -=512

y = 8 cm

= 8 cm

V = =

1280 = 82X.

T= 20

Find total cost use equation 2

C = (2x8X20 + 2X20X8) X2 + (2x8X8) X5

C = (320 + 320) X2 + (128) X5

C= 1280 + 640 = 1920

Dimensions for the minimum cost of box are x,y,z,= 20 cm, 8 cm, 8 cm