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

Answer

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

### Step 1: Write down the formula for the curved surface area of a cone.
The formula for the curved surface area (CSA) of a right circular cone is given by:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height of the cone.

### Step 2: Substitute the given values into the formula.
Given:
- Curved Surface Area (CSA) = 12320 cm²
- Radius (r) = 56 cm

Substituting these values into the formula:
\[ 12320 = \pi \times 56 \times l \]

### Step 3: Solve for the slant height (l).
We can rearrange the equation to solve for \( l \):
\[ l = \frac{12320}{\pi \times 56} \]
Using \( \pi \approx 3.14 \):
\[ l = \frac{12320}{3.14 \times 56} \]
Calculating the denominator:
\[ 3.14 \times 56 = 1758.4 \]
Now substituting back:
\[ l = \frac{12320}{1758.4} \approx 7.0 \text{ cm} \]

### Step 4: Use the Pythagorean theorem to find the height (h).
The relationship between the height (h), radius (r), and slant height (l) is given by:
\[ l^2 = r^2 + h^2 \]
Rearranging this gives us:
\[ h^2 = l^2 - r^2 \]

### Step 5: Substitute the values of l and r to find h.
Substituting \( l = 70 \) cm and \( r = 56 \) cm:
\[ h^2 = 70^2 - 56^2 \]
Calculating \( 70^2 \) and \( 56^2 \):
\[ h^2 = 4900 - 3136 \]
\[ h^2 = 1764 \]
Taking the square root to find \( h \):
\[ h = \sqrt{1764} \approx 42 \text{ cm} \]

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

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