Question

part c - thinking (2 marks)

1. a cereal box is shaped like a rectangular prism. you want to reduce the amount of cardboard used without changing the volume.

explain one way you could reduce the surface area of the box and why it works.
(2 marks - explanation and reasoning)

Explanation:

For a rectangular prism (cereal box), the volume formula is \(V = l\times w\times h\) (length \(l\), width \(w\), height \(h\)) and the surface - area formula is \(SA=2(lw + lh+wh)\). If we make the shape of the prism closer to a cube (i.e., make \(l\), \(w\), and \(h\) as close in value as possible while keeping \(V = l\times w\times h\) constant), the surface area will decrease. This is because for a given volume, a cube has the minimum surface area among all rectangular prisms. Mathematically, using the AM - GM (Arithmetic - Mean Geometric - Mean) inequality, for positive numbers \(a\), \(b\), \(c\) (representing \(l\), \(w\), \(h\)), \(\frac{a + b + c}{3}\geq\sqrt[3]{abc}\), and the equality holds when \(a = b = c\). When \(a\), \(b\), \(c\) are closer to each other, the sum \(ab+bc + ca\) (which is part of the surface - area formula \(SA = 2(ab + bc+ca)\)) is minimized for a fixed \(abc\) (volume).

Answer:

Make the length, width, and height of the cereal box (rectangular prism) as close in value as possible. This works because for a given volume, a cube (a special case of a rectangular prism where \(l = w=h\)) has the minimum surface area among all rectangular prisms. By the AM - GM inequality, when the three dimensions are closer to each other, the sum of the products of the pairs of dimensions (which is related to the surface area formula \(SA = 2(lw+lh + wh)\)) is minimized while keeping the volume \(V=lwh\) constant.