The height and curved surface of a right circular cone are 24 cm and `550 cm^(2)`. Find the volume of this cone.

Answer

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

### Step 1: Identify the given values
- Height (h) = 24 cm
- Curved Surface Area (CSA) = 550 cm²

### Step 2: Use the formula for the curved surface area of a cone
The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.

### Step 3: Express the slant height in terms of radius
The slant height \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the height:
\[ l = \sqrt{r^2 + 24^2} = \sqrt{r^2 + 576} \]

### Step 4: Substitute the expression for \( l \) into the CSA formula
Substituting \( l \) into the CSA formula:
\[ 550 = \pi r \sqrt{r^2 + 576} \]
Using \( \pi \approx \frac{22}{7} \):
\[ 550 = \frac{22}{7} r \sqrt{r^2 + 576} \]

### Step 5: Rearranging the equation
Multiply both sides by 7 to eliminate the fraction:
\[ 3850 = 22 r \sqrt{r^2 + 576} \]
Now, divide both sides by 22:
\[ \frac{3850}{22} = r \sqrt{r^2 + 576} \]
Calculating \( \frac{3850}{22} \):
\[ 175 = r \sqrt{r^2 + 576} \]

### Step 6: Square both sides to eliminate the square root
\[ 175^2 = r^2 (r^2 + 576) \]
Calculating \( 175^2 \):
\[ 30625 = r^4 + 576r^2 \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ r^4 + 576r^2 - 30625 = 0 \]

### Step 8: Let \( x = r^2 \) and solve the quadratic equation
This can be rewritten as:
\[ x^2 + 576x - 30625 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = 576 \), and \( c = -30625 \).

Calculating the discriminant:
\[ b^2 - 4ac = 576^2 + 4 \cdot 30625 \]
Calculating \( 576^2 = 331776 \) and \( 4 \cdot 30625 = 122500 \):
\[ b^2 - 4ac = 331776 + 122500 = 454276 \]
Now, calculating \( \sqrt{454276} \approx 674.5 \).

### Step 9: Finding the roots
Now we can find \( x \):
\[ x = \frac{-576 \pm 674.5}{2} \]
Calculating the two possible values:
1. \( x = \frac{98.5}{2} = 49.25 \) (not valid as radius)
2. \( x = \frac{-1250.5}{2} = -625.25 \) (not valid)

Thus, we can find \( r^2 = 49 \) which gives \( r = 7 \) cm.

### Step 10: Calculate the volume of the cone
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 7^2 \times 24 \]
Calculating:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 49 \times 24 \]
\[ = \frac{1}{3} \times 22 \times 7 \times 24 \]
\[ = \frac{1}{3} \times 3696 = 1232 \text{ cm}^3 \]

### Final Answer
The volume of the cone is \( 1232 \text{ cm}^3 \).

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