(i) The area of base of a right circular cone is `60 cm^(2)` and its volume is `200 cm^(3)`. Find the height of the cone.

Answer

To find the height of the right circular cone given the area of the base and the volume, we can follow these steps:

### Step 1: Understand the formulas
The area of the base of a cone (which is a circle) is given by the formula:
\[ \text{Area} = \pi r^2 \]
The volume of a cone is given by the formula:
\[ \text{Volume} = \frac{1}{3} \pi r^2 h \]

### Step 2: Use the area to find \( r^2 \)
We know the area of the base is \( 60 \, \text{cm}^2 \):
\[ \pi r^2 = 60 \]
To find \( r^2 \), we can rearrange this equation:
\[ r^2 = \frac{60}{\pi} \]

### Step 3: Substitute \( r^2 \) into the volume formula
We know the volume of the cone is \( 200 \, \text{cm}^3 \):
\[ \frac{1}{3} \pi r^2 h = 200 \]
Substituting \( r^2 \) from Step 2 into the volume formula:
\[ \frac{1}{3} \pi \left(\frac{60}{\pi}\right) h = 200 \]

### Step 4: Simplify the equation
Now, simplify the left side:
\[ \frac{1}{3} \cdot 60 \cdot h = 200 \]
This simplifies to:
\[ 20h = 200 \]

### Step 5: Solve for \( h \)
Now, divide both sides by \( 20 \):
\[ h = \frac{200}{20} = 10 \]

### Conclusion
The height of the cone is:
\[ h = 10 \, \text{cm} \]

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