Multiple choice

The capacities of two hemispherical vessels are $6.4$litres and $21.6$ litres. The areas of inner curved surfaces of the vessels will be in the ratio of __________.

  1. $\sqrt{2}:\sqrt{3}$
  2. $2:3$
  3. $4:9$
  4. $16:81$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The volume of a hemisphere is proportional to the cube of the radius (V proportional to r^3). The ratio of volumes is 6.4 : 21.6 = 64 : 216 = 8 : 27. Thus, the ratio of radii is the cube root of 8 : 27, which is 2 : 3. The surface area is proportional to the square of the radius (r^2), so the ratio of areas is 2^2 : 3^2 = 4 : 9.

AI explanation

The capacity of a hemisphere is its volume (2/3)πr^3, so the volumes are in the ratio 6.4 to 21.6, which simplifies to 64 to 216 or 4 to 9. Since volume is proportional to the cube of the radius, the ratio of their radii is the cube root of this ratio, giving 2 to 3. The inner curved surface area of a hemisphere is 2πr^2, meaning area is proportional to the square of the radius. Squaring the radius ratio gives the surface area ratio as 4 to 9.

Keep practicing — related questions