Shape and Space 7
4 marks · HardTwo solid spheres are similar. The surface area of the smaller sphere is 48 cm² and the surface area of the larger sphere is 192 cm².
Explain how you would find the ratio of the radii of the two spheres.
Using your answer to (a), work out the ratio of the volumes of the smaller sphere to the larger sphere.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) Ratio of radii = 1:2 (use square root of surface area ratio)
(b) Ratio of volumes = 1:8
Common mistakes: using area ratio directly as volume ratio
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · States or implies that for similar solids, (radius ratio)² = (area ratio).
A1 (a) · 1 mark · Finds correct ratio: √(48/192) = 1/2, so ratio 1:2.
M2 (b) · 1 mark · States that volumes ratio is (r₁/r₂)³ or uses (area ratio)^(3/2). (follow-through)
A2 (b) · 1 mark · Gets correct ratio of volumes: (1/2)³ = 1/8. (follow-through)
Worked solution
a) For similar solids, the ratio of surface areas = (ratio of radii)².
So the ratio of radii is 1:2.
b) The ratio of volumes is (radius ratio)³ = (1/2)³ = 1/8.