Question

Generate a general formula, in simplest form, that will give the maximum volume of a rectangular based prism made from a rectangular piece of paper. The dimensions of the rectangular page are w ×2w with a square of dimensions x×x cut from each corner.

Answer 1

As per the question,

the length will be $2w-2x,$

width will be $w-2x,$

height will be $x$

of the rectangular prism.

The required volume will be $x(2w-2x)(w-2x)$

So,

$V=x(2w-2x)(w-2x)$

$V=2w^2x-6wx^2+4x^3$

For V to be maximum,$\frac{dV}{dx}=0$

$$\frac{dV}{dx}=2w^2-12wx+12x^2=0$$

Solving this equation,

we get,

$6x^2-6wx+w^2=0$

By using quadratic formula

$$x=\frac{-b+\sqrt{b^2-4ac}}{2a}$$

and

$$x=\frac{-b-\sqrt{b^2-4ac}}{2a}$$

Discriminant(D)=$\sqrt{b^2-4ac}$

So

$$x=\frac{6w-\sqrt{36w^2-24w^2}}{12}$$

(Neglected positive discriminant because $w-2x$ would be negative for that case, which is not possible)

$$x=\frac{(3-\sqrt{3})w}{6}$$

which can be further written as

$$x=\frac{w}{3+\sqrt{3}}$$

So by using value of $\sqrt{3}=1.732$

we can solve and get,

$$x=0.211w$$