该图展示了一个用于在酒店分发咖啡的圆柱形容器。 计算该圆柱体的体积。 答案 ... cm^3 将 25 升咖啡倒入该空容器中。计算该容器中空白空间的高度 $h$。 答案 $h$ = ... cm 用该容器中的咖啡装满半径为 3.5 cm 的半球形杯子。 求用容器中的 25 升咖啡最多可以完全装满多少个这样的杯子。 一个立体图形由一个立方体和其顶部的一个棱锥组成，总高度为 10x cm。
该棱锥完美地放置在立方体的一个面上。 立方体的每条边长为 6x cm。
求该立体图形表面积关于 x 的表达式。
给出你答案的最简形式。

1. Calculate the volume of the cylinder using its radius and height.

$V=\pi r^2h$
$=\pi(18)^2(50)$
$=16200\pi$
$=50\,894.\ldots$

So the volume is $50\,894\text{ cm}^3$ to the nearest whole number.

1. Convert 25 litres to cubic centimetres.

$25\text{ litres}=25\times1000=25\,000\text{ cm}^3$

2. Find the height of coffee in the cylinder.

Volume of coffee $=\pi r^2(\text{filled height})$

$25\,000=\pi(18)^2(\text{filled height})$

$\text{filled height}=\frac{25\,000}{\pi(18)^2}=24.56\ldots\text{ cm}$

3. Subtract the filled height from the total height to find the empty height.

$h=50-24.56\ldots=25.43\ldots\text{ cm}$

So, to 3 significant figures,

$h=25.4\text{ cm}$

1. Calculate the volume of one hemispherical cup.

Volume of a hemisphere
$=\frac{1}{2}\left(\frac{4}{3}\pi r^3\right)=\frac{2}{3}\pi r^3$
For $r=3.5$ cm,
$\text{Volume of 1 cup}=\frac{2}{3}\pi(3.5)^3=89.797\ldots\text{ cm}^3$

2. Divide the total coffee volume by the volume of one cup and take the maximum whole number.

$25$ litres $=25\,000\text{ cm}^3$.

Number of complete cups
$=\frac{25\,000}{89.797\ldots}=278.40\ldots$
So the maximum number of cups that can be completely filled is
$278.$

1. Find the vertical height of the pyramid.

Total height of solid $=10x$ cm.

Height of cube $=6x$ cm.

So height of pyramid
$=10x-6x=4x\text{ cm}.$

2. Find the slant height of one triangular face of the pyramid.

The pyramid has square base side $6x$, so half the base side is
$\frac{6x}{2}=3x.$

Using Pythagoras' theorem,
$l^2=(4x)^2+(3x)^2$
$l^2=16x^2+9x^2=25x^2$
$l=5x.$

3. Find the total exposed area of the cube.

The top face of the cube is covered by the pyramid, so only $5$ square faces of the cube are exposed.

Area of exposed cube faces
$=5(6x)^2$
$=5(36x^2)=180x^2.$

4. Find the area of the four triangular faces and add to the cube area.

Area of one triangular face
$=\frac12(6x)(5x)=15x^2.$

Area of $4$ triangular faces
$=4(15x^2)=60x^2.$

Total surface area
$=180x^2+60x^2=240x^2.$