# Maximize volume of a rectangular box

From a rectangular sheet of metal measuring 120mm by 75mm, equal squares of side x are cut from each of the four corners. The remaining flaps are then folded upwards to form an open box.

a) Draw a neat and simple diagram of the rectangular sheet of metal and show the dimensions (not drawn to scale) given including the squares of side x.

b) Show that the volume of the box is given by:

V= 9000x - 390x^2 + 4x^3

c) Find the value of x such that the volume is a maximum.

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#### Solution Preview

a) Draw a neat and simple diagram of the rectangular sheet of metal and show the dimensions (not drawn to scale) given including the squares of side x.

b) Show that the volume of the box is given by:

V= 9000x - ...

#### Solution Summary

We had to maximize the volume of a rectangular box. Since the volume is expressed as a quadratic function, we had to solve for x, setting the first derivative of a quadratic function to zero.