A cylindrical container whose diameter is 12 cm and height is 15 cm, is filled with ice cream. The whole ice-cream is distributed to 10 children in equal cones having hemispherical tops. If the height of the conical portion is twice the diameter of its base, find the diameter of the ice-cream cone?

Options

A. A)6 cm
B. B)12 cm
C. C)3 cm
D. D)18 cm

Answer

To solve the problem step by step, we will first calculate the volume of the cylindrical container and then distribute that volume among the cones with hemispherical tops.

### Step 1: Calculate the volume of the cylindrical container
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Diameter of the cylinder = 12 cm, so the radius \( r = \frac{12}{2} = 6 \) cm.
- Height \( h = 15 \) cm.

Substituting the values into the formula:
\[ V = \pi (6)^2 (15) = \pi (36)(15) = 540\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of ice cream distributed to each child
The total volume of ice cream is distributed equally among 10 children. Therefore, the volume of ice cream per child is:
\[ \text{Volume per child} = \frac{540\pi}{10} = 54\pi \text{ cm}^3 \]

### Step 3: Set up the equation for the volume of the cone with a hemispherical top
Let the diameter of the cone be \( d \). The radius of the base of the cone is \( \frac{d}{2} \).

The height of the conical portion is given as twice the diameter of its base:
\[ \text{Height of cone} = 2d \]

The volume \( V_c \) of the cone is given by:
\[ V_c = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \left(\frac{d}{2}\right)^2 (2d) = \frac{1}{3} \pi \left(\frac{d^2}{4}\right) (2d) = \frac{1}{3} \pi \frac{d^3}{2} = \frac{\pi d^3}{6} \]

The volume \( V_h \) of the hemispherical top is given by:
\[ V_h = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{2}{3} \pi \frac{d^3}{8} = \frac{\pi d^3}{12} \]

### Step 4: Total volume of the cone with hemispherical top
The total volume \( V_{total} \) for one child is:
\[ V_{total} = V_c + V_h = \frac{\pi d^3}{6} + \frac{\pi d^3}{12} \]

To combine these fractions:
\[ V_{total} = \frac{2\pi d^3}{12} + \frac{\pi d^3}{12} = \frac{3\pi d^3}{12} = \frac{\pi d^3}{4} \]

### Step 5: Equate the volumes and solve for \( d \)
We know that the volume per child is \( 54\pi \):
\[ \frac{\pi d^3}{4} = 54\pi \]

Dividing both sides by \( \pi \):
\[ \frac{d^3}{4} = 54 \]

Multiplying both sides by 4:
\[ d^3 = 216 \]

Taking the cube root of both sides:
\[ d = \sqrt[3]{216} = 6 \text{ cm} \]

### Conclusion
The diameter of the ice-cream cone is \( 6 \) cm.

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