Determine the dimensions of the box (in metres) that will minimize the cost of production. What is the minimum cost? Given:A woman wants to construct a box whose base length is twice the base...

Determine the dimensions of the box (in metres) that will minimize the cost of production. What is the minimum cost? Given:

A woman wants to construct a box whose base length is twice the base width. The woman wants the box to have a volume of 70m^3

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should come up with the following substitution:x=width, 2x=length.

You need to evaluate the areas of top and bottom of the box such that:

`A_base= x*2x =gt A_base=2x^2`

You need to evaluate the area of one side of the box such that:

`A = x*h`

You need to evaluate the area of next side of the box such that:

`A = 2x*h`

h expresses the height of box

You should remember that there are 4 sides, the area of two of them is A=x*h and the area of the other two is `A = 2x*h.`

You need to determine the height using the volume of the box:

`V = A_base*h `

`70 = 2x^2*h =gt `

`h = 35/x^2`

Hence, the needed amount of material is given by the formula:

`A_base + A_top + 2*x*h + 2*2x*h`

Substituting `35/x^2`  for h yields:

`A_base + A_top + 70/x + 140/x`

You need to notice that area of base is equal to area of top, hence:

`A(x) = 2*2x^2 + 210/x =gt A(x) = 4x^2 + 210/x`

You have formed a function of x that expresses the total area of the box. You need to find what dimensions of the box minimize the cost, hence you need to differentiate the function `A(x) ` with respect to x such that:

`A'(x) = 8x - 210/x^2`

You need to solve for x the equation `A'(x) = 0`  such that:

`8x - 210/x^2 = 0 =gt 8x^3 - 210 = 0`

You need to expand the difference of cubes:

`(2x - root(3)210)(4x^2 + 2*root(3)210*x + root(3)(210^2))`

Hence, the real root of equation is `x = (root(3)210)/2`

`x~~2.971m`

Hence, the dimensions of the box that minimize the cost of production are width `~~`  3 m and length `~~`  6m.

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