图中显示了一个倒置的空心纸圆锥，其底面半径为 $r$ cm，高度为 $h$ cm。圆锥中装有水（由阴影区域表示），水深为 $\frac{1}{3}h$ cm。圆锥的内表面积为 106 cm^2。求圆锥内部与水接触的表面积制造。

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完整解法

1. Identify the linear scale factor of the smaller cone to the whole cone.

The water forms a smaller cone similar to the whole cone.

Since its height is $\dfrac{1}{3}h$ and the whole cone has height $h$ ,
$\text{linear scale factor}=\frac{\frac{1}{3}h}{h}=\frac{1}{3}.$

2. Use the area scale factor to find the curved surface area in contact with water.

For similar figures, surface areas are in the ratio of the squares of corresponding lengths.

$\text{area scale factor}=\left(\frac{1}{3}\right)^2=\frac{1}{9}.$

So the internal surface area in contact with the water is
$106\times \frac{1}{9}=\frac{106}{9}\text{ cm}^2.$

答案

$\boxed{\dfrac{106}{9}\text{ cm}^2}$

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