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
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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
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