一家制造商销售两种尺寸、几何相似的瓶装饮料。瓶子的一些规格如下所示。

1. Form the volume ratio using the bottle capacities.

Regular capacity : Large capacity
$330 : 1500 = 11 : 50$

2. Take the cube root of the volume ratio to find the corresponding height ratio, then express it as $1:n$.

For similar bottles,
$(\text{height ratio})^3 = \text{volume ratio}$
So
$\text{Regular height : Large height} = \sqrt[3]{11} : \sqrt[3]{50}$
Divide both parts by $\sqrt[3]{11}$:
$1 : \sqrt[3]{\frac{50}{11}} = 1 : 1.66\text{ (3 s.f.)}$

1. Use the linear scale factor from part (i) to form the material ratio.

From part (i),
$\text{regular height} : \text{large height} = 1 : 1.66.$
Since the thicknesses are the same, the amount of material needed is proportional to the surface area.

For similar solids,
$\text{surface area ratio} = (\text{linear scale factor})^2.$

2. Calculate the material needed for the large bottle.

$\frac{x}{24} = 1.66^2$
$x = 24(1.66)^2$
$x = 66.1344$
$x \approx 66.1$