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

To solve this problem, we need to analyze the volumes involved and their dimensions. We have an ice cream cone and two spherical scoops of ice cream that completely fill the cone. Here's the step-by-step solution:

1. Volume of the cone is given by the formula: \(V_{cone} = \frac{1}{3} \pi r^2 h\).
2. The two spheres have a radius of \(\frac{r}{2}\), which means the volume of one sphere is: \(V_{sphere} = \frac{4}{3} \pi \left(\frac{r}{2}\right)^3 = \frac{4}{3} \pi \frac{r^3}{8} = \frac{\pi r^3}{6}\).
3. For two spheres, the total volume is: \(V_{2 \, spheres} = 2 \times \frac{\pi r^3}{6} = \frac{\pi r^3}{3}\).
4. Since the cone is completely filled by these two spheres, the volume of the cone equals the volume of the two spheres: \(\frac{1}{3} \pi r^2 h = \frac{\pi r^3}{3}\).
5. By cancelling \(\pi\) and multiplying both sides by 3, we get: \(r^2 h = r^3\).
6. Dividing both sides by \(r^2\), we find: \(h = r\).
7. We are asked to find the ratio of \(h : 2r\):
   • \(h = r\)
   • \(2r = 2r\)

The ratio is \(r : 2r = 1 : 2\). Therefore, the correct answer is \((1 : 2)\).