A manufacturer sells drinks in bottles of two sizes that are geometrically similar. Some specifications for the bottles are shown below.

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.

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