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Mathematics — Standard (041) — CBSE Class 10 board question

Q1. [1] § 12.2 Surface Area of a Combination of Solids
An ice-cream cone of radius $r$ and height $h$ is completely filled by two spherical scoops of ice-cream. If radius of each spherical scoop is $\dfrac{r}{2}$, then $h : 2r$ equals
  1. A $1 : 8$
  2. B $1 : 2$
  3. C $1 : 1$
  4. D $2 : 1$
Previously asked in CBSE board exam
2026 30/4/1 Q16
Generated by claude-sonnet-4-6 · 2026-06-15 10:32 · grounding rag
Model Answer

Volume of cone = Volume of two spherical scoops

$$\frac{1}{3}\pi r^2 h = 2 \times \frac{4}{3}\pi \left(\frac{r}{2}\right)^3 = 2 \times \frac{4}{3}\pi \cdot \frac{r^3}{8} = \frac{\pi r^3}{3}$$

$$\frac{1}{3}\pi r^2 h = \frac{\pi r^3}{3} \implies h = r \implies h : 2r = 1 : 2$$

Answer: B) 1 : 2

Explanation

Set volume of cone equal to total volume of the two scoops. Each scoop has radius $r/2$, so its volume is $\frac{4}{3}\pi(r/2)^3 = \frac{\pi r^3}{6}$; two scoops give $\frac{\pi r^3}{3}$. Equate with $\frac{1}{3}\pi r^2 h$ to get $h = r$, so $h:2r = r:2r = 1:2$.

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