The radius and height of a right circular cone are in the ratio 1: (2.4). If its curved surface area is 2502.5 `cm^(2)`, then what is its volume? (Take `pi= ""_(7)^(22)`)

Options

A. A) 8085 `cm^(3)`
B. B) 8820 `cm^(3)`
C. C) 11550 `cm^(3)`
D. D) 13475 `cm^(3)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Ratio of Radius and Height
The radius (r) and height (h) of the cone are given in the ratio of 1:2.4. We can express this ratio in terms of a variable \( x \):
- Let \( r = 1x \) and \( h = 2.4x \).

### Step 2: Convert the Ratio to Simpler Form
To simplify the ratio, we can multiply both parts by 10:
- \( r = 10 \) and \( h = 24 \).
Thus, we can express them as:
- \( r = 5x \) and \( h = 12x \).

### Step 3: Find the Slant Height
Using the Pythagorean theorem, we can find the slant height (l) of the cone:
- \( l = \sqrt{r^2 + h^2} = \sqrt{(5x)^2 + (12x)^2} = \sqrt{25x^2 + 144x^2} = \sqrt{169x^2} = 13x \).

### Step 4: Use the Curved Surface Area Formula
The formula for the curved surface area (CSA) of a cone is given by:
- \( \text{CSA} = \pi r l \).
Substituting the known values:
- \( 2502.5 = \frac{22}{7} \times (5x) \times (13x) \).

### Step 5: Solve for \( x \)
Now we will solve for \( x \):
1. Substitute the values into the equation:
\[ 2502.5 = \frac{22}{7} \times 65x^2 \]
2. Multiply both sides by 7 to eliminate the fraction:
\[ 2502.5 \times 7 = 22 \times 65x^2 \]
\[ 17517.5 = 1430x^2 \]
3. Divide both sides by 1430:
\[ x^2 = \frac{17517.5}{1430} = 12.25 \]
4. Taking the square root gives:
\[ x = \sqrt{12.25} = 3.5 \]

### Step 6: Calculate Radius and Height
Now we can find the radius and height:
- \( r = 5x = 5 \times 3.5 = 17.5 \, \text{cm} \)
- \( h = 12x = 12 \times 3.5 = 42 \, \text{cm} \)

### Step 7: 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:
1. Calculate \( r^2 \):
\[ r^2 = (17.5)^2 = 306.25 \]
2. Substitute into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 306.25 \times 42 \]
3. Calculate:
\[ V = \frac{1}{3} \times \frac{22 \times 306.25 \times 42}{7} \]
\[ = \frac{1}{3} \times \frac{22 \times 12862.5}{7} \]
\[ = \frac{1}{3} \times 40425 \]
\[ = 13475 \, \text{cm}^3 \]

### Final Answer
The volume of the cone is \( 13475 \, \text{cm}^3 \).

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