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: Understanding the Question:
The topic of this question is Surface Areas and Volumes.
We are given an ice-cream cone of radius \(r\) and height \(h\).
The cone is completely filled by exactly two identical spherical scoops of ice-cream.
This means the total volume of the cone is equal to the combined volume of the two spherical scoops.
The radius of each of these spherical scoops is \(\frac{r}{2}\).
We need to find the ratio of the height \(h\) to twice the radius \(2r\) of the cone, which is written as \(h : 2r\).

Step 2: Key Formula or Approach:
- The volume of a right circular cone of radius \(r\) and height \(h\) is:
\[ V_{\text{cone}} = \frac{1}{3}\pi r^2 h \] - The volume of a sphere of radius \(R\) is:
\[ V_{\text{sphere}} = \frac{4}{3}\pi R^3 \] - Since there are two identical spherical scoops, their total volume is:
\[ V_{\text{total}} = 2 \times \left(\frac{4}{3}\pi R^3\right) \] where \(R = \frac{r}{2}\).
- Since the cone is completely filled by the ice-cream, we will equate the volume of the cone to the total volume of the two scoops and solve for the relation between \(h\) and \(r\).

Step 3: Detailed Explanation:

• Write down the volume formula for the cone:
\[ V_{\text{cone}} = \frac{1}{3}\pi r^2 h \]

• Calculate the volume of one spherical scoop with radius \(R = \frac{r}{2}\):
\[ V_{\text{sphere}} = \frac{4}{3}\pi \left(\frac{r}{2}\right)^3 \] \[ V_{\text{sphere}} = \frac{4}{3}\pi \left(\frac{r^3}{8}\right) = \frac{1}{6}\pi r^3 \]

• Calculate the total volume of the two identical scoops:
\[ V_{\text{total}} = 2 \times V_{\text{sphere}} = 2 \times \left(\frac{1}{6}\pi r^3\right) = \frac{1}{3}\pi r^3 \]

• Equate the volume of the cone to the total volume of the two scoops:
\[ \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi r^3 \]

• Divide both sides of the equation by the common term \(\frac{1}{3}\pi r^2\):
\[ h = r \]

• Find the required ratio \(h : 2r\):
Substitute \(h = r\) into the ratio expression:
\[ \frac{h}{2r} = \frac{r}{2r} = \frac{1}{2} \] This gives the ratio as \(1 : 2\).


Step 4: Final Answer:
The ratio \(h : 2r\) is equal to 1 : 2.
Therefore, the correct option is (B).