A model consists of a solid hemisphere attached to a solid cylinder as shown in diagram. The height of the cylinder is 26 cm and the base area is 198 cm^2.

1. Calculate the volume of the cylinder using its base area and height.

$\text{Volume of cylinder}=\text{base area}\times\text{height}$
$=198\times 26=5148\text{ cm}^3$

1. Find the radius of the hemisphere using the base area of the cylinder.

Since the hemisphere has the same radius as the cylinder base,
$\pi r^2=198$
$r^2=\frac{198}{\pi}$
$r=\sqrt{\frac{198}{\pi}}$

2. Use the formula for the volume of a hemisphere.

Volume of hemisphere
$=\frac{2}{3}\pi r^3$
$=\frac{2}{3}\pi\left(\sqrt{\frac{198}{\pi}}\right)^3$

3. Calculate and round the volume to the nearest whole number.

$\frac{2}{3}\pi\left(\sqrt{\frac{198}{\pi}}\right)^3=1056.9\ldots$
To the nearest whole number,
$1056.9\ldots\approx 1057$

1. Substitute the cone volume and base area into the cone volume formula, then solve for the height.

For a cone,
$V=\frac{1}{3}\times \text{base area}\times h$
Given $V=630$ and base area $=198$,
$630=\frac{1}{3}(198)h$
$h=\frac{630\times 3}{198}=9.545\ldots$
So the height of the cone is $9.55\text{ cm}$ to 3 significant figures.

1. Find the common radius of the cylinder, hemisphere and cone.

Since the base area of the cylinder is $198\text{ cm}^2$,
$\pi r^2=198$
$r=\sqrt{\frac{198}{\pi}}=7.9377\ldots\text{ cm}$

2. Find the slant height of the cone removed.

From part (b)(i), the height of the cone is
$h=\frac{3(630)}{198}=9.5454\ldots\text{ cm}$
Using Pythagoras' theorem,
$l=\sqrt{r^2+h^2}$
$l=\sqrt{7.9377\ldots^2+9.5454\ldots^2}=12.4153\ldots\text{ cm}$

3. Write the total surface area as the sum of all exposed surfaces.

Exposed surfaces are:

• curved surface area of cylinder: $2\pi rh$
• curved surface area of hemisphere: $2\pi r^2$
• circular base of cylinder: $\pi r^2$
• curved surface area of conical hole: $\pi rl$

So
$\text{TSA}=2\pi r(26)+2\pi r^2+\pi r^2+\pi rl.$

4. Calculate the total surface area and round to the nearest whole number.

$\text{TSA}=2\pi(7.9377\ldots)(26)+2(198)+198+\pi(7.9377\ldots)(12.4153\ldots)$
$=1296.79\ldots+396+198+309.65\ldots$
$=2200.44\ldots$
To the nearest whole number,
$\text{TSA}=2200\text{ cm}^2.$