A right circular cone is 5.4cm high and radius of its base is 2cm. It is melted and recast into another right circular one with radius of base as 1.5 cm. find the height of new cone formed.

Answer

To find the height of the new cone formed after melting and recasting the original cone, we can follow these steps:

### Step 1: Calculate the volume of the original cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
For the original cone:
- Height \( h_1 = 5.4 \) cm
- Radius \( r_1 = 2 \) cm

Substituting the values:
\[ V_1 = \frac{1}{3} \pi (2)^2 (5.4) \]
\[ V_1 = \frac{1}{3} \pi (4)(5.4) \]
\[ V_1 = \frac{1}{3} \pi (21.6) \]
\[ V_1 = 7.2 \pi \text{ cm}^3 \]

### Step 2: Set up the equation for the volume of the new cone.
Let the height of the new cone be \( h_2 \) and the radius be \( r_2 = 1.5 \) cm. The volume of the new cone is:
\[ V_2 = \frac{1}{3} \pi (r_2)^2 h_2 \]
Substituting the value of \( r_2 \):
\[ V_2 = \frac{1}{3} \pi (1.5)^2 h_2 \]
\[ V_2 = \frac{1}{3} \pi (2.25) h_2 \]
\[ V_2 = 0.75 \pi h_2 \text{ cm}^3 \]

### Step 3: Equate the volumes of the original and new cones.
Since the volume of the melted cone remains the same, we have:
\[ V_1 = V_2 \]
\[ 7.2 \pi = 0.75 \pi h_2 \]

### Step 4: Cancel \( \pi \) from both sides.
\[ 7.2 = 0.75 h_2 \]

### Step 5: Solve for \( h_2 \).
To find \( h_2 \), divide both sides by \( 0.75 \):
\[ h_2 = \frac{7.2}{0.75} \]
Calculating the division:
\[ h_2 = 9.6 \text{ cm} \]

### Final Answer:
The height of the new cone formed is \( 9.6 \) cm.

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