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:

Pay close attention to the final ratio requested.
The calculation gives \(h = r\), which means the ratio of height to radius is \(1 : 1\).
However, the question asks for the ratio of \(h\) to \(2r\).
Substituting \(h = r\) directly gives \(r : 2r = 1 : 2\). Always double-check what is being asked before choosing your option!

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 a right circular ice-cream cone of radius \(r\) and height \(h\).
The cone is completely filled by two identical spherical scoops of ice-cream.
This means the total volume of the cone is exactly equal to the combined volume of the two spherical scoops.
We need to find the ratio of the height \(h\) to twice the radius \(2r\) of the cone.

Step 2: Key Formula or Approach:
- Volume of a right circular cone of radius \(r\) and height \(h\):
\[ V_{\text{cone}} = \frac{1}{3}\pi r^2 h \] - Volume of a sphere of radius \(R\):
\[ V_{\text{sphere}} = \frac{4}{3}\pi R^3 \] - Since the radius of each spherical scoop is given as \(R = \frac{r}{2}\), the volume of one scoop is:
\[ V_{\text{scoop}} = \frac{4}{3}\pi \left(\frac{r}{2}\right)^3 \] - Equate the volume of the cone to the volume of the two scoops:
\[ V_{\text{cone}} = 2 \times V_{\text{scoop}} \]

Step 3: Detailed Explanation:

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

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

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

• Set the volume of the cone equal to the total volume of the two scoops:
\[ V_{\text{cone}} = \text{Total Volume} \] \[ \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi r^3 \]

• Simplify the equation by dividing both sides by \(\frac{1}{3}\pi r^2\):
\[ h = r \]

• 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).