Question

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 \(\frac{r}{2}\), then h : 2r equals

• A. 1 : 8
• B. 1 : 2
• C. 1 : 1
• D. 2 : 1

Hint:

A common mistake in exams is calculating the ratio \(h : r\) instead of \(h : 2r\).
Our derivation shows \(h = r\), which represents a ratio of \(1 : 1\).
However, the question asks for the ratio of the height to the diameter \(2r\), which is \(r : 2r = 1 : 2\).
Always double-check what is being requested in the final ratio to avoid this trap!

Answer 1

The Correct Option is B

Step 1: Write both volumes in terms of $r$.
The cone has volume $\frac{1}{3}\pi r^2 h$. Each scoop is a sphere of radius $\frac{r}{2}$, so one scoop has volume $\frac{4}{3}\pi\left(\frac{r}{2}\right)^3 = \frac{\pi r^3}{6}$.
Step 2: Equate the cone to two scoops and simplify in one go.
$\frac{1}{3}\pi r^2 h = 2\times\frac{\pi r^3}{6} = \frac{\pi r^3}{3}$. Dividing both sides by $\frac{\pi r^3}{3}$ gives $\frac{h}{r}=1$, so $h=r$.
Step 3: Convert straight to the ratio that's actually asked.
We need $h:2r$, not $h:r$. Since $h=r$, $\frac{h}{2r}=\frac{r}{2r}=\frac{1}{2}$, so $h:2r = 1:2$, matching option (B).
\[ \boxed{h:2r = 1:2} \]