The height of a right circular cone is 35 cm and the area of its curved surface is four times the area of its base. What is the volume of the cone (in `10^(-3) m^3`and correct up to three decimal places)?

Options

A. 3.316
B. 3.384
C. 2.994
D. 2.625

Answer

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

### Step 1: Understand the given information
We are given:
- Height of the cone (h) = 35 cm
- Curved surface area (CSA) = 4 times the area of the base.

### Step 2: Write the formulas for the areas
The formulas we need are:
- Curved Surface Area (CSA) of a cone: \( CSA = \pi r l \)
- Area of the base: \( A = \pi r^2 \)

### Step 3: Set up the equation based on the given information
From the problem statement, we know:
\[ \pi r l = 4 \cdot \pi r^2 \]

### Step 4: Simplify the equation
Dividing both sides by \( \pi \) (assuming \( r \neq 0 \)):
\[ r l = 4 r^2 \]
Now, divide both sides by \( r \):
\[ l = 4r \]

### Step 5: Use the Pythagorean theorem
In the right triangle formed by the height, radius, and slant height, we can use the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Substituting \( l = 4r \) and \( h = 35 \):
\[ (4r)^2 = r^2 + 35^2 \]
\[ 16r^2 = r^2 + 1225 \]
Now, simplify:
\[ 15r^2 = 1225 \]
\[ r^2 = \frac{1225}{15} \]
\[ r^2 = \frac{245}{3} \]

### Step 6: Calculate the radius
Taking the square root:
\[ r = \sqrt{\frac{245}{3}} \]

### Step 7: Calculate the slant height
Using \( l = 4r \):
\[ l = 4 \sqrt{\frac{245}{3}} \]

### Step 8: Calculate the volume of the cone
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting \( r^2 = \frac{245}{3} \) and \( h = 35 \):
\[ V = \frac{1}{3} \pi \left(\frac{245}{3}\right) \cdot 35 \]
\[ V = \frac{1}{3} \cdot \frac{245 \cdot 35 \cdot \pi}{3} \]
\[ V = \frac{8575 \pi}{9} \]

### Step 9: Calculate the numerical value
Using \( \pi \approx \frac{22}{7} \):
\[ V \approx \frac{8575 \cdot \frac{22}{7}}{9} \]
\[ V \approx \frac{188650}{63} \approx 2994.4 \, \text{cm}^3 \]

### Step 10: Convert to \( 10^{-3} m^3 \)
Since \( 1 \, \text{cm}^3 = 10^{-6} \, \text{m}^3 \):
\[ V \approx 2994.4 \times 10^{-6} \, \text{m}^3 \]
\[ V \approx 2.994 \times 10^{-3} \, \text{m}^3 \]

### Final Answer
The volume of the cone is approximately \( 2.994 \times 10^{-3} \, \text{m}^3 \).

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