两个空纸杯在几何上是相似的。
所用纸张的厚度忽略不计。
大杯子的容量是小杯子的 8 倍。
小杯子的高度为 5 cm，大杯子的底面积为 26 cm^2。

1. Relate the cup capacities to the volume ratio and find the linear scale factor.

The big cup can be filled by 8 small cups, so
$\frac{\text{volume of big cup}}{\text{volume of small cup}}=8.$
For geometrically similar cups,
$\text{volume ratio}=(\text{linear scale factor})^3.$
Hence
$\text{linear scale factor}=\sqrt[3]{8}=2.$

2. Use the linear scale factor to find the height of the big cup.

$\text{height of big cup}=2\times 5=10\text{ cm}.$

1. Use the square of the linear scale factor to find the small cup's base area.

From part (a), the linear scale factor from the small cup to the big cup is $2$.

For similar figures,
$\frac{\text{base area of big cup}}{\text{base area of small cup}}=2^2=4.$

So,
$\text{base area of small cup}=\frac{26}{4}=6.5\text{ cm}^2.$

1. Explain why the mass ratio follows the area ratio, not the volume ratio.

The big cup can be filled by $8$ small cups, so the volume ratio is $8:1$ and the linear scale factor is $2:1$.

Since the cups are made of paper with negligible thickness, the mass depends on the surface area of paper used, not the volume capacity.

Area ratio $=2^2:1^2=4:1$.

So the mass ratio is $4:1$, not $8:1$.