如图所示为一个高为 16 cm 的实心棱台，它是由一个正四棱锥截得的。水平顶面的面积为 5 cm^2，而底面面积为 125 cm^2。截去的部分高为 $h$ cm。 证明 $h = 4$。 答 计算该棱台的体积。 答：... cm^3 求截去部分占原正四棱锥的百分比。 答：... % 两个实心圆锥的高度相同，但圆锥 A 的半径是圆锥 B 的 1.5 倍。已知圆锥 B 的体积为 240 cm^3。 求圆锥 A 的体积。

1. Set up the area ratio of the similar pyramids

The removed small pyramid and the original pyramid are similar.

Area of top square $=5\text{ cm}^2$ and area of base square $=125\text{ cm}^2$.

$\frac{\text{area of top square}}{\text{area of base square}}=\frac{5}{125}=\frac{1}{25}$

2. Convert area ratio to height ratio and solve for $h$

For similar pyramids,

$\frac{\text{small height}}{\text{original height}}=\sqrt{\frac{1}{25}}=\frac{1}{5}$

The small height is $h$ and the original height is $h+16$.

$\frac{h}{h+16}=\frac{1}{5}$

$5h=h+16$

$4h=16$

$h=4$

1. Find the height of the original right pyramid

From part (a)(i), the removed small pyramid has height $h=4$ cm.

Height of original pyramid
$=4+16=20\text{ cm}$

2. Calculate the volumes of the original pyramid and the removed small pyramid

Volume of a pyramid $=\dfrac{1}{3}\times \text{base area}\times \text{height}$.

Original pyramid:
$V_{\text{large}}=\frac{1}{3}\times 125\times 20=\frac{2500}{3}$

Removed pyramid:
$V_{\text{small}}=\frac{1}{3}\times 5\times 4=\frac{20}{3}$

3. Subtract to find the volume of the frustum

Volume of frustum
$=V_{\text{large}}-V_{\text{small}}$
$=\frac{2500}{3}-\frac{20}{3}$
$=\frac{2480}{3}=826.666\ldots$
$\approx 827\text{ cm}^3$

1. Identify the relevant volumes from earlier work

From 7(a)(ii),

height of original pyramid $= h+16=4+16=20\text{ cm}$.

Volume of original pyramid:
$\frac13 \times 125 \times 20=\frac{2500}{3}\text{ cm}^3$

Volume of removed small pyramid:
$\frac13 \times 5 \times 4=\frac{20}{3}\text{ cm}^3$

2. Calculate the percentage of the original pyramid that was removed

Percentage removed
$=\frac{\text{volume removed}}{\text{volume of original pyramid}}\times 100\%$

$=\frac{\frac{20}{3}}{\frac{2500}{3}}\times 100\%$

$=\frac{20}{2500}\times 100\%=0.8\%$

1. Relate the volumes using the radius scale factor

For a cone,
$V=\frac{1}{3}\pi r^2h.$
The cones have the same height, so
$\frac{V_A}{V_B}=\left(\frac{r_A}{r_B}\right)^2.$
Given $r_A=1.5r_B$,
$\frac{V_A}{V_B}=1.5^2=2.25.$

2. Calculate the volume of cone A

$V_A=2.25\times 240=540.$