An ice cream cone is filled with ice cream and more ice 
cream in the form of a hemisphere is equal to the diameter 
of the cone. If the hemispherical surface is equal to lateral 
area of the cone, find the total volume of ice cream if the 
radius of the hemisphere is 2 cm.

Let's break down the problem step-by-step:

1. **Given Information:**
   - The radius of the hemisphere, \( r = 2 \) cm.
   - The diameter of the hemisphere is equal to the diameter of the cone, hence the radius of the cone's base is also \( r = 2 \) cm.
   - The hemispherical surface area is equal to the lateral surface area of the cone.

2. **Formulas Required:**
   - **Volume of a hemisphere:**
     \[
     V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
     \]
   - **Lateral surface area of a cone:**
     \[
     A_{\text{cone}} = \pi r l
     \]
     where \( l \) is the slant height of the cone.
   - **Surface area of a hemisphere:**
     \[
     A_{\text{hemisphere}} = 2 \pi r^2
     \]

   Since the hemispherical surface area is equal to the lateral area of the cone:
   \[
   2 \pi r^2 = \pi r l
   \]

   Simplifying this:
   \[
   2 r = l \implies l = 2 \times 2 = 4 \text{ cm}
   \]

3. **Height of the Cone:**
   - Use the Pythagorean theorem for the right triangle formed by the height \( h \), the radius \( r \), and the slant height \( l \) of the cone:
   \[
   l^2 = r^2 + h^2
   \]
   Substituting \( l = 4 \) cm and \( r = 2 \) cm:
   \[
   4^2 = 2^2 + h^2
   \]
   \[
   16 = 4 + h^2
   \]
   \[
   h^2 = 12
   \]
   \[
   h = \sqrt{12} = 2\sqrt{3} \text{ cm}
   \]

4. **Volume of the Cone:**
   \[
   V_{\text{cone}} = \frac{1}{3} \pi r^2 h
   \]
   Substituting \( r = 2 \) cm and \( h = 2\sqrt{3} \) cm:
   \[
   V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (2\sqrt{3})
   \]
   \[
   V_{\text{cone}} = \frac{1}{3} \pi \times 4 \times 2\sqrt{3}
   \]
   \[
   V_{\text{cone}} = \frac{8}{3} \pi \sqrt{3} \text{ cm}^3
   \]

5. **Volume of the Hemisphere:**
   \[
   V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
   \]
   Substituting \( r = 2 \) cm:
   \[
   V_{\text{hemisphere}} = \frac{2}{3} \pi (2)^3
   \]
   \[
   V_{\text{hemisphere}} = \frac{2}{3} \pi \times 8
   \]
   \[
   V_{\text{hemisphere}} = \frac{16}{3} \pi \text{ cm}^3
   \]

6. **Total Volume of Ice Cream:**
   The total volume is the sum of the volume of the cone and the volume of the hemisphere:
   \[
   V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}}
   \]
   \[
   V_{\text{total}} = \frac{8}{3} \pi \sqrt{3} + \frac{16}{3} \pi
   \]

Therefore, the total volume of ice cream is:
\[
\frac{8}{3} \pi \sqrt{3} + \frac{16}{3} \pi \text{ cm}^3
\]

If you have any questions or need further details, feel free to ask!

Here are 5 related questions to explore further:
1. What would be the volume if the radius of the hemisphere was doubled?
2. How would the volume of the ice cream change if the cone was filled up to half its height?
3. What is the relationship between the surface area of the hemisphere and its volume?
4. How would the total volume change if the radius of the cone was increased while keeping the height the same?
5. Can you derive the formula for the surface area of a hemisphere?

**Tip:** Always double-check your unit conversions and dimensions to ensure consistency when solving geometric problems.

An ice cream cone is filled with ice cream and more ice cream in the form of a hemisphere is equal to the diameter of the cone. If the hemispherical surface is equal to lateral area of the cone, find the total volume of ice cream if the radius of the hemisphere is 2 cm.

Calculate the total volume of ice cream in a cone and hemisphere, where the hemispherical surface area equals the cone's lateral area. Explore geometric concepts and use formulas for volume, surface area, and the Pythagorean theorem. Suitable for high school students (Grades 10-12).

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