The volume of a cylinder is the area of its circular end multiplied by its height:

\(V = \pi r^2 h\)

Square the radius first, and leave \(\pi\) alone throughout:

\(V = \pi \times 3^2 \times 6 = \pi \times 9 \times 6 = 54\pi\)

\(54\pi\) cm³

When the open cylinder is cut along its height and unrolled, one side of the rectangle is the height of the cylinder, \(7.5\) cm. The length \(l\) of the rectangle is the distance round the circular end, the circumference.

\(l = 2\pi r = 2 \times \pi \times 1.9\) cm

Keeping the full value of \(l\) on the calculator,

\(\text{area} = l \times 7.5 = 2 \times \pi \times 1.9 \times 7.5 \approx 89.5\) cm²

The net still wraps round the same cylinder: the base and the perpendicular height are the same as the rectangle, so base times perpendicular height gives the same area.

So the area of the rectangle is equal to the area of the parallelogram.

The net still wraps round the same cylinder: a sloping side is the longest side of a right-angled triangle whose other side is the perpendicular height, so it is longer than the height of the rectangle.

So the length of a sloping side of the parallelogram is more than the height of the rectangle.

The formula uses the radius, and the drum has been measured across, so halve the diameter first. Squaring the diameter by mistake is the commonest way this mark is lost.

\(r = 18 \div 2 = 9\) cm

\(V = \pi r^2 h = \pi \times 9^2 \times 12 = \pi \times 81 \times 12\)

Using the \(\pi\) key on a calculator, \(V = 3053.6\) cm³ to 1 decimal place.

Divide the volume in cm³ by \(1000\):

\(3053.6 \div 1000 = 3.05\)

The drum holds about \(3.05\) litres.

Unroll the curved surface and it is a rectangle. Its height is the height of the cylinder, and its width is the distance all the way round the circular end — the circumference.

\(\text{curved surface area} = 2\pi r h = 2 \times \pi \times 2 \times 8 = 32\pi\)

\(32\pi\) cm²

The total surface area is the curved part and the two circular ends. Leaving the ends out is the mark most often dropped here.

Each circular end has area \(\pi r^2 = \pi \times 2^2 = 4\pi\) cm², and there are two of them:

\(2 \times 4\pi = 8\pi\)

Add that to the curved surface area from part (a):

\(32\pi + 8\pi = 40\pi\)

\(40\pi\) cm²

Put what you know into the formula for the volume of a cylinder:

\(\pi r^2 h = 425\pi\)

\(\pi \times 5^2 \times h = 425\pi\)

\(25\pi h = 425\pi\)

Divide both sides by \(\pi\) — this is why the volume was given in terms of \(\pi\), so nothing has to be rounded:

\(25h = 425\)

\(h = 425 \div 25 = 17\)

The height is \(17\) cm.

Put what you know into the formula for the volume of a cylinder:

\(\pi r^2 h = 500\pi\)

\(\pi \times r^2 \times 20 = 500\pi\)

Divide both sides by \(\pi\):

\(20r^2 = 500\)

\(r^2 = 500 \div 20 = 25\)

The last step is the square root, and it is the one that is forgotten:

\(r = \sqrt{ 25 } = 5\)

The radius is \(5\) cm.

First find the total volume that must be filled.

Radius: \(r = 80 \div 2 = 40\) cm

Volume of one tank: \(\pi r^2 h = \pi \times 40^2 \times 140 \approx 703716.8\) cm³

Volume of all 5 tanks: \(5 \times 703716.8 \approx 3518583.8\) cm³

The ratio of fertiliser to water is \(1 : 100\), so the mixture has \(1 + 100 = 101\) parts in total, and fertiliser is \(\frac{1}{101}\) of it by volume — not \(\frac{1}{100}\).

Fertiliser needed: \(3518583.8 \div 101 \approx 34837.5\) cm³

Convert to litres: \(34837.5 \div 1000 \approx 34.8\) litres

Ben has \(35.1\) litres, which is more than \(34.8\) litres, so Ben has enough fertiliser for the 5 tanks.

Yes

Each tank has a top and a bottom, so the total surface area is the curved side and both circular ends.

Radius: \(r = 3 \div 2 = 1.5\) m

Area of both ends of one tank: \(2\pi r^2 = 2 \times \pi \times (1.5)^2 = 4.5\pi \approx 14.1\) m²

Curved surface area of one tank: \(\pi d h = \pi \times 3 \times 4 = 12\pi \approx 37.7\) m²

Total surface area of one tank: \(2\pi r^2 + \pi d h = 4.5\pi + 12\pi = 16.5\pi \approx 51.8\) m²

For 4 tanks: \(4 \times 16.5\pi = 66\pi \approx 207.35\) m²

60 tins of paint cover: \(60 \times 3 = 180\) m²

180 m² is less than 207.35 m², so Amara does not have enough paint to cover 4 tanks.

No

The volume of a cylinder is the area of its circular end multiplied by its height:

\(V = \pi r^2 h\)

Square the radius first, and leave \(\pi\) alone throughout:

\(V = \pi \times 8^2 \times 9 = \pi \times 64 \times 9 = 576\pi\)

\(576\pi\) cm³

When the open cylinder is cut along its height and unrolled, one side of the rectangle is the height of the cylinder, \(12.4\) cm. The length \(l\) of the rectangle is the distance round the circular end, the circumference.

\(l = 2\pi r = 2 \times \pi \times 1.1\) cm

Keeping the full value of \(l\) on the calculator,

\(\text{area} = l \times 12.4 = 2 \times \pi \times 1.1 \times 12.4 \approx 85.7\) cm²

The net still wraps round the same cylinder: a sloping side is the longest side of a right-angled triangle whose other side is the perpendicular height, so it is longer than the height of the rectangle.

So the length of a sloping side of the parallelogram is more than the height of the rectangle.

The net still wraps round the same cylinder: the cylinder has not changed, so its height is still the perpendicular distance between the top and bottom edges of the net.

So the perpendicular height of the parallelogram is equal to the height of the rectangle.