A company needs to create a box with a square base and no top, The volume of the box must be `500 cm^3` . What should be the dimensions of the box to minimize the amount fo material that will be used to make it?

**Answer:**

Dimensions should = 5 x 10 x 10

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The square box that has to be created by the company should have a volume of 500 cm^3 and a square base. Let the height of the box be h. If the length of each side of the square base is L, L^2*h=500

=> h = 500/L^2

The surface area of the open top box is

SA =L^2 + 4*L*h

= L^2 + 4*(500/L^2)*L

= L^2 + 2000/L

To minimize the surface area, solve `(dSA)/(DL)=0` for L

`(dSA)/(DL) = 2*L - 2000/L^2`

2*L - 2000/L^2 =0

=> 2*L = 2000/L^2

=> L^3 = 1000

=> L = 10

If the length of each side of the base is 10, the height of the box is 5.

**The required dimensions of the box to minimize the material used for making the box is 5*10*10.**

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