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:

Be careful with the ratio required. The question asks for \(h : 2r\), not \(h : r\). Always simplify expressions before plugging in values to save time.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The problem states that the volume of the cone is exactly equal to the combined volume of two spherical scoops.
Step 2: Key Formula or Approach:
Volume of Cone = \(\frac{1}{3} \pi r^2 h\)
Volume of Sphere = \(\frac{4}{3} \pi R^3\)
Given: Radius of cone is \(r\). Radius of scoop \(R = \frac{r}{2}\).
Step 3: Detailed Explanation:
According to the question:
\[ \text{Volume of cone} = 2 \times \text{Volume of one scoop} \]
\[ \frac{1}{3} \pi r^2 h = 2 \times \left( \frac{4}{3} \pi \left( \frac{r}{2} \right)^3 \right) \]
Canceling \(\frac{1}{3} \pi\) from both sides:
\[ r^2 h = 2 \times 4 \times \frac{r^3}{8} \]
\[ r^2 h = 8 \times \frac{r^3}{8} \]
\[ r^2 h = r^3 \]
Dividing both sides by \(r^2\):
\[ h = r \]
We need to find the ratio \(h : 2r\).
\[ \frac{h}{2r} = \frac{r}{2r} = \frac{1}{2} \]
Step 4: Final Answer:
The ratio \(h : 2r\) is \(1 : 2\).