The diagram shows a cylindrical container used to dispense coffee in a hotel. Calculate the volume of the cylinder. Answer ... cm^3 25 litres of coffee are poured into the empty container. Work out the height, $h$, of the empty space in the container. Answer $h$ = ... cm Cups in the shape of a hemisphere of radius 3.5 cm are filled with coffee from the container. Work out the maximum number of these cups that can be completely filled from the 25 litres of coffee in the container. A solid shape consists of a cube with a pyramid on top has a total height of 10x cm.
The pyramid sits perfectly on one surface of the cube. Each side of the cube is 6x cm.
Find an expression, in terms of x, for the surface area of the solid.
Give your answer in its simplest form.

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
$=180x^2+60x^2=240x^2.$