The curved surface area of a cone is `12320 cm^(2).` If the radius of its base is 56 cm, find its height.

Answer

To find the height of the cone given its curved surface area and the radius of its base, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Curved Surface Area (CSA) of the cone, \( CSA = 12320 \, cm^2 \)
- Radius of the base, \( r = 56 \, cm \)

2. **Use the Formula for Curved Surface Area of a Cone**:
The formula for the curved surface area of a cone is given by:
\[ CSA = \pi r l \]
where \( l \) is the slant height of the cone.

3. **Substitute the Known Values into the Formula**:
We can substitute the values of CSA and \( r \) into the formula:
\[ 12320 = \frac{22}{7} \times 56 \times l \]

4. **Calculate \( \frac{22}{7} \times 56 \)**:
First, calculate \( \frac{22}{7} \times 56 \):
\[ \frac{22 \times 56}{7} = \frac{1232}{7} = 176 \, cm \]
Therefore, we can rewrite the equation as:
\[ 12320 = 176l \]

5. **Solve for \( l \)**:
To find \( l \), divide both sides by 176:
\[ l = \frac{12320}{176} = 70 \, cm \]

6. **Use the Pythagorean Theorem to Find the Height**:
The relationship between the height \( h \), radius \( r \), and slant height \( l \) is given by:
\[ l^2 = h^2 + r^2 \]
Rearranging gives:
\[ h^2 = l^2 - r^2 \]

7. **Substitute the Values of \( l \) and \( r \)**:
Substitute \( l = 70 \, cm \) and \( r = 56 \, cm \):
\[ h^2 = 70^2 - 56^2 \]
Calculate \( 70^2 \) and \( 56^2 \):
\[ 70^2 = 4900, \quad 56^2 = 3136 \]
Thus:
\[ h^2 = 4900 - 3136 = 1764 \]

8. **Find the Height \( h \)**:
Taking the square root gives:
\[ h = \sqrt{1764} = 42 \, cm \]

### Final Answer:
The height of the cone is \( 42 \, cm \).

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