The diagram shows two geometrically similar cup and saucer sets. The diameter of the smaller saucer is 5 cm. The diameter of the larger saucer is 6.5 cm. A coffee shop sells the smaller cup of coffee at $1 and the larger cup at $2. Calculate which is a better buy. Explain your answer. Answer ...

1. Find the linear scale factor from the smaller set to the larger set.

$\text{Linear scale factor}=\frac{6.5}{5}=1.3$

2. Find the volume scale factor for the coffee cups.

Since the cup-and-saucer sets are geometrically similar, volumes scale by the cube of the linear scale factor.

$\text{Volume scale factor}=1.3^3=2.197$

So the larger cup holds $2.197$ times as much coffee as the smaller cup.

3. Compare the price scale factor with the volume scale factor.

$\text{Price scale factor}=\frac{\$2}{\$1}=2$

The larger cup costs $2$ times as much, but gives $2.197$ times as much coffee.

4. Decide which cup gives better value for money and explain.

Since $2.197>2$, the increase in coffee is greater than the increase in price.

Therefore, the larger cup gives more coffee per dollar.