A hot water tank is made by joining a hemisphere of radius 30 cm to a cylinder of radius 30 cm and height 70 cm.

1. Identify the external surfaces to include

The tank consists of a hemisphere joined to a cylinder.

External surface area includes:

$\text{curved area of hemisphere} + \text{curved area of cylinder} + \text{bottom circular base}$

The circular face where the hemisphere joins the cylinder is not exposed.

2. Substitute the radius and height into the surface area formulae

Radius $r=30$ cm, cylinder height $h=70$ cm.

Curved surface area of hemisphere:
$2\pi r^2=2\pi(30)^2=1800\pi$

Curved surface area of cylinder:
$2\pi rh=2\pi(30)(70)=4200\pi$

Area of bottom circular base:
$\pi r^2=\pi(30)^2=900\pi$

3. Add the required surface areas

$\text{Total surface area}=1800\pi+4200\pi+900\pi$

$=6900\pi\text{ cm}^2$

Using a calculator,
$6900\pi=21676.989\ldots$

$\approx 21\,700\text{ cm}^2 \text{ correct to 3 significant figures}$

1. Find the volume of the cylindrical part

For the cylinder, $r=30$ cm and $h=70$ cm.

$V_{\text{cylinder}}=\pi r^2h$

$=\pi(30)^2(70)=63000\pi\text{ cm}^3$

2. Find the volume of the hemispherical part and the total volume

Volume of a hemisphere:

$V_{\text{hemisphere}}=\frac{1}{2}\left(\frac{4}{3}\pi r^3\right)=\frac{2}{3}\pi r^3$

$V_{\text{hemisphere}}=\frac{2}{3}\pi(30)^3=18000\pi\text{ cm}^3$

Total volume:

$63000\pi+18000\pi=81000\pi\text{ cm}^3$

3. Convert the volume from cubic centimetres to litres

Since $1000\text{ cm}^3=1$ litre,

$81000\pi\div1000=81\pi$

$81\pi=254.469\ldots$

So the number of litres is approximately $254$ litres.

1. Find the time taken in seconds

From part (b)(i), the tank contains $84.8$ litres of water.

At $3$ litres per second,
$\text{time}=\frac{84.8}{3}=28.266\ldots\text{ seconds}$

2. Convert the time into minutes and seconds

$28.266\ldots\text{ seconds}=0\text{ minutes }28.266\ldots\text{ seconds}$

To the nearest second, this is $28$ seconds.

1. Convert the tank volume from litres to cubic metres

From part (b)(i), the volume of water is $84.8$ litres.

Since $1000$ litres $=1\text{ m}^3$,

$84.8\text{ litres}=\frac{84.8}{1000}=0.0848\text{ m}^3.$

2. Find the area of the trapezium cross-section of the bath

Area of trapezium

$=\frac{1}{2}(0.4+0.6)(0.3)$

$=\frac{1}{2}(1.0)(0.3)=0.15\text{ m}^2.$

3. Use the prism volume formula to find the length of the bath

For a prism,

$\text{Volume}=\text{cross-sectional area}\times \text{length}.$

So

$0.0848=0.15\times \text{length}$

$\text{length}=\frac{0.0848}{0.15}=0.565333\ldots$

$\text{length}=0.565\text{ m} \quad \text{to 3 significant figures}.$