Question

g a box has a square base of side x and height h find the dimensions x, h for which the volume is 14 and the surface area is as small as possible

Answer 1

Explanation:

Base area =
`x^2`

side area=
`h*x`

Volume=
`x^2*h`=14

Surface area =
`4hx+2x^2`

upon substituting
`h=14/x^2`

surface area =
`4*(14)/(x^2)*x+2x^2`

upon differentiating and solving we get f"(
`\sqrt[3]{14\\}`) >0 proves that this point is of minima

hence, dimensions are

`x=\sqrt[3]{14}` ,h=
`\sqrt[3]{14}`