Small cones of same size are formed by melting a big cone, whose radius and height are two times and 400% more than the smaller cones respectively. Find the total number of small cones formed? (Use ∏= 3)

Given:

- The radius of the big cone is two times the radius of the small cone.
- The height of the big cone is 400% more than the height of the small cone.

Let \( r \) and \( h \) be the radius and height of the small cone, respectively.

### Step 1: Define the dimensions of the big cone
- The radius of the big cone, \( R \), is \( 2r \).
- The height of the big cone, \( H \), is 400% more than \( h \), which means:
  \[
  H = h + 4h = 5h
  \]

### Step 2: Calculate the volume of the big cone and small cone
The volume of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]

- **Volume of the big cone:**
  \[
  V_{\text{big}} = \frac{1}{3} \pi R^2 H = \frac{1}{3} \pi (2r)^2 (5h)
  \]
  \[
  V_{\text{big}} = \frac{1}{3} \pi \times 4r^2 \times 5h = \frac{20}{3} \pi r^2 h
  \]

- **Volume of one small cone:**
  \[
  V_{\text{small}} = \frac{1}{3} \pi r^2 h
  \]

### Step 3: Find the total number of small cones formed
The total number of small cones formed is the ratio of the volume of the big cone to the volume of one small cone:
\[
\text{Number of small cones} = \frac{V_{\text{big}}}{V_{\text{small}}} = \frac{\frac{20}{3} \pi r^2 h}{\frac{1}{3} \pi r^2 h} = 20
\]

Thus, the total number of small cones formed is **20**.

---

Do you want more details or have any questions? Here are some related problems to consider:

1. What if the height of the big cone was 300% more than the height of the small cone?
2. How would the result change if the radius of the big cone was 3 times that of the small cone?
3. What if the big cone's height was double that of the small cone instead of 400% more?
4. How does the volume relationship change if the cone dimensions are in different units?
5. What is the ratio of the surface area of the big cone to the small cone?
6. How would the total number of small cones change if the radius and height both doubled?
7. What if the big cone was hollow? How would that affect the result?
8. Can you generalize this problem for cones of different shapes?

**Tip:** Always check the percentage increase carefully when applying to different geometric dimensions.

Learn how to calculate the total number of small cones formed from a big cone using geometry and the volume formula of cones. Detailed step-by-step solution provided.

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