a large bin for holding heavy material must be in the shape of a box with an open top and square base.the base will cost $7 a square foot and sides cost $9 a foot
if the volume must be 170 cubic feet. find the dimesions that will minimize the cost of the box's contruction
Let us assume the base dimension as xft and height of the box as yft.
Volume of the box `(V) = x*x*y`
`170 = x*x*y`
`y = 170/x^2`
We have one square base and four side faces in the box.
If the cost of construction is A;
`A = x^2*7+4*x*y*9`
`A = 7x^2+36*x*170/x^2`
`A = 7x^2+6120/x`
For cost of construction to be maximum or minimum` (dA)/dx = 0`
`(dA)/dx = 14x-6120/x^2`
When `(dA)/dx = 0` ;
`14x-6120/x^2 = 0`
`x^3 = 6120/14`
`x = 7.56`
If the cost is minimum at x = 7.56 then `((d^2A)/(dx^2))_(x=7.56)>0`
`(d^2A)/(dx^2) = 14+12240/x^3`
Since x>0 then `14+12240/x^3>0` always.
`(d^2A)/(dx^2)>0` always. This means A has a minimum.
So the cost is minimum at x = 7.56
`y = 170/7.56^2 = 2.97`
base = 7.56ft
side = 2.97ft