Question

a carpenter has been asked to build an open box with a sqaure base. the sides of the box will cost $3 per m^2 and the base will cost $4 per m^2. What are the dimensions of the box of greatest volume that can be constructed for $48

Answer 1

Dimension of box is 2 m x 2 m x 1.33 m

Step-by-step explanation:

Let a be base side and h be the height.

Volume of box, V = a²h

The sides of the box will cost $3 per m² and the base will cost $4 per m². Cost for making is $48.

That is

                 4a² + 3 x 4 x a x h = 48

                 4a² + 12 a x h = 48

                 a² + 3 ah = 12  

                 $h=\frac{12-a^2}{3a}$

So volume is

                    $V=a^2\times \frac{12-a^2}{3a}=4a-\frac{a^3}{3}$

At maximum volume we have derivative is zero,

                    $dV=4da-3\times \frac{a^2}{3}da\\\\0=4-a^2\\\\a=\pm 2$

Negative side is not possible, hence side of square base is 2m.

Substituting in  a² + 3 ah = 12  

                           2² + 3 x  2 x h = 12  

                             h = 1.33 m

Dimension of box is 2 m x 2 m x 1.33 m