A open topped box is to be constructed with an equilateral triangular base of side a. The sides of the box are to be rectangles (a by h). Find the dimensions of a box which has the maximum volume and surface area of 300 square centimeters.
I know that the area of the triangle is the square root of 3 divided by 4 multiplied by a square.
I assumed that h is the height of the sides. Then we would have the volume given by the following expression: $$V = \frac{a^2\sqrt3}{4}*h$$ Also, the box is opened on top, so the area is given by: $$A =\frac{a^2\sqrt3}{4}+3ah=300$$ Isolating h, we have: $$h = \frac{300 - \frac{a^2\sqrt3}{4}}{3a} $$ Substituting h in the volume expression, we get: $$V = \frac{a^2\sqrt3}{4}*\frac{300 - \frac{a^2\sqrt3}{4}}{3a}$$ Deriving V in respect to a, and then equalizing to 0, in order to find a local maximum, we have: $$a = \frac{20}{\sqrt[4]{3}}$$ Applying that value of a to V: $$V_{max} = \frac{1000}{(\sqrt[4]{3})^3}$$