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You are required to construct an open box from a paper by cutting four squared from the corners of the paper. The rectangular piece of paper is 5cm long and 3cm wide. What should be the length of the sides of such four (identical) squares that will create a box of largest volume? What will be the largest volume?

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Let $s$ be the side length of the square which is cut out.

The Volume of the box will be given by $$V = (3-2s)(5-2s)s \space\space\space\space\space\space\text{ (do you see why? Draw a picture to help) }$$ $$\implies V = 4s^3 - 16s^2 + 15s $$

To find extrema, take a derivative and set equal to $0$.

$$\frac{dV}{ds} = 12s^2 - 32s + 15 = 0$$

Use the quadratic formula to find solutions to this equation, and note that $0 < s < 1.5$ (otherwise, you would cut off nothing at all or two full rectangles from the sheet).

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