Question

An open box (i.e. without a lid) is built in the shape of a cuboid to minimise the total

Answer 1

The lenght of the base is four times the width of the base, so the volume must be:

$V=l\cdot w\cdot h$$400cm^3=4x\cdot x\cdot h$$h=\frac{400}{4x^2}$$h=\frac{100}{x^2}$

For the surface area we have, the area of the base:

$A_b=4x\cdot x$$A_b=4x^2^{}$

the area of 2 sides must be:

$2A_s=2\cdot w\cdot h$$2A_s=2\cdot x\cdot\frac{100}{x^2}$$2A_s=\frac{200}{x}$

and for the other 2:

$2A_{s2}=2\cdot l\cdot h$$2A_{s2}=2\cdot4x\cdot\frac{100}{x^2}$$2A_{s2}=\frac{800}{x}$

And adding all we have:

$A=A_b+2A_s+2A_{s2}$$A=4x^2+\frac{200}{x}+\frac{800}{x}$$A=4x^2+\frac{1000}{x}$