Two triangular ends and three rectangles between them: \(5\).

Three round each triangle, and three joining them: \(3 + 3 + 3 = 9\).

Three corners on each triangle: \(6\).

The cross-section is a triangle with base \(8\) cm and height \(4\) cm:

\(\frac{1}{2} \times 8 \times 4 = 16\) cm²

A prism's volume is its cross-section times its length:

\(16 \times 11 = 176\) cm³

A cuboid is a prism with a rectangular cross-section, so multiply the three dimensions together.

\(20 \times 40 \times 30 = 24000\ \text{cm}^3\)

Divide the volume by \(1000\), because that is how many cubic centimetres fit in one litre.

\(24000 \div 1000 = 24\) litres

A cube has six identical square faces, so one face has area

\(96 \div 6 = 16\ \text{cm}^2\)

The edge is the side of that square.

\(\sqrt{ 16 } = 4\ \text{cm}\)

The volume is the edge cubed.

\(4^3 = 4 \times 4 \times 4 = 64\ \text{cm}^3\)

The front elevation shows the width and the height of the cuboid, but not its depth. The volume is what carries the depth.

The front elevation is \(7\) cm by \(3\) cm, so the area of the end of the cuboid is

\(7 \times 3 = 21\ \text{cm}^2\)

Volume is this area multiplied by the depth, so

\(\text{depth} = 126 \div 21 = 6\) cm

The side elevation keeps the height of \(3\) cm and shows the depth of \(6\) cm, so it is a rectangle \(6\) cm across and \(3\) cm up.

A cylinder is a prism whose cross-section is a circle, so its volume is the area of that circle times the height.

The formula uses the radius, and what is given is the diameter, so halve it first.

\(8 \div 2 = 4\) cm

Area of the circle: \(\pi r^2 = \pi \times 4^2 = 16\pi\ \text{cm}^2\)

Volume: \(16\pi \times 4 = 64\pi\ \text{cm}^3\)

Leaving the \(\pi\) in place keeps the answer exact, so nothing is lost to rounding.

The cross-section is a trapezium. Add the two parallel sides, halve the total, and multiply by the perpendicular height between them.

\(16 + 12 = 28\)

\(\frac{1}{2} \times 28 \times 8 = 112\ \text{cm}^2\)

The volume of any prism is the area of its cross-section times its length.

\(112 \times 17 = 1904\ \text{cm}^3\)

The cross-section is a right-angled triangle, so its area is half the base times the height.

\(\frac{1}{2} \times 4 \times 5 = 10\ \text{cm}^2\)

The volume of a prism is the area of its cross-section times its length, so here

\(10 \times \text{length} = 70\)

Divide to undo the multiplication.

\(\text{length} = 70 \div 10 = 7\) cm

Dividing the volumes only gives the number of boxes if they fill the crate with no gaps, so count whole boxes along each edge instead. Work in cm.

length: \(160 \div 40 = 4\), so \(4\) whole boxes

width: \(120 \div 40 = 3\), so \(3\) whole boxes

height: \(80 \div 40 = 2\), so \(2\) whole boxes

\(4 \times 3 \times 2 = 24\) boxes

So the claim of \(24\) boxes is valid. The method only worked because the box side divides every edge exactly, so the boxes leave no gaps.

Volume of the bath, using the cuboid:

\(160 \times 60 \times 42 = 403200\ \text{cm}^3\)

Change to litres by dividing by \(1000\):

\(403200 \div 1000 = 403.2\) litres

Water used by the shower:

\(6.5 \times 7 = 45.5\) litres

Water saved:

\(403.2 - 45.5 = 357.7\) litres

Both cuboids have the same length and width, so compare the depths.

The real water is \(53\) cm deep and the assumed cuboid is \(42\) cm deep, so the water is deeper than Yusuf assumed.

The bath holds more water than part (a) used, so Yusuf saves more than \(357.7\) litres.

Tank A: take the waste-water compartment from the total.

\(477 - 275 = 202\) litres

Tank B: the cross-section is a triangle, so its area is half base times height.

\(\frac{1}{2} \times 0.9 \times 0.5 = 0.225\ \text{m}^2\)

The volume of a prism is the cross-section area times the length.

\(0.225 \times 1 = 0.225\ \text{m}^3\)

Change to litres so both tanks are in the same unit.

\(0.225 \times 1000 = 225\) litres

\(225\) litres is greater than \(202\) litres, so Joel buys tank B.