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?
Dimensions should = 5 x 10 x 10
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.