The diagram shows a solid, made from a hemisphere and a cone. The hemisphere has radius r cm. The cone has radius r cm and height h cm. The total volume of the solid is 500π cm^3. The total surface area of the solid is A cm^2. Show that A = 500π/r + πr^2.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1 for (2/3)πr^3 or (1/3)πr^2h (correct formula for volume of hemisphere or volume of cone)
2. M1 (dep) for (2/3)πr^3 + (1/3)πr^2h = 500π (oe)
3. A1 for h = (1500 – 2r^3)/r^2 or r^2h = 1500 – 2r^3 (oe) (for making h the subject)
4. M1 for A = 2πr^2 + πrl (where l is slant height)
5. C1 (for final step) for A = 2πr^2 + πr * sqrt(r^2 + ((1500 – 2r^3)/r^2)^2) leading to A = 500π/r + πr^2 (oe, for showing the result)