The volume of the space inside a right circular conical tent is `138(2)/(7)m^(3)` and its vertical height is 4 m. Find the canvas required to make the tent and also find the cost of the canvas at the rate of Rs.120 per `m^(2)`.

Options

A. `126.3 m^(2)`, Rs. 15164.16
B. `126.3 m^(2)`, Rs. 15156
C. `136.2 m^(2)`, Rs. 16344
D. `142.3 m^(2)`, Rs. 17076

Answer

To solve the problem step by step, we will first find the radius of the cone using the volume formula, then calculate the curved surface area (CSA) of the cone to determine the amount of canvas required, and finally calculate the cost of the canvas.

### Step 1: Find the radius of the cone

The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( V = 138\frac{2}{7} \, m^3 = \frac{968}{7} \, m^3 \) (converting mixed fraction to improper fraction)
- \( h = 4 \, m \)

Substituting the known values into the volume formula:
\[ \frac{968}{7} = \frac{1}{3} \pi r^2 (4) \]

### Step 2: Simplify the equation

Rearranging the equation:
\[ \frac{968}{7} = \frac{4}{3} \pi r^2 \]

Multiplying both sides by \( \frac{3}{4} \):
\[ r^2 = \frac{968 \times 3}{7 \times 4 \pi} \]

### Step 3: Calculate \( r^2 \)

Substituting \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{968 \times 3}{7 \times 4 \times \frac{22}{7}} = \frac{968 \times 3}{88} \]

Calculating:
\[ r^2 = \frac{2904}{88} = 33 \]

### Step 4: Find the radius \( r \)

Taking the square root:
\[ r = \sqrt{33} \approx 5.74 \, m \]

### Step 5: Find the slant height \( l \)

Using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} = \sqrt{33 + 4^2} = \sqrt{33 + 16} = \sqrt{49} = 7 \, m \]

### Step 6: Calculate the curved surface area (CSA) of the cone

The formula for the CSA of a cone is:
\[ \text{CSA} = \pi r l \]

Substituting the values:
\[ \text{CSA} = \pi \times \sqrt{33} \times 7 \]

Using \( \pi \approx \frac{22}{7} \):
\[ \text{CSA} = \frac{22}{7} \times 5.74 \times 7 = 22 \times 5.74 = 126.28 \, m^2 \]

### Step 7: Calculate the cost of the canvas

The cost of the canvas is given by:
\[ \text{Cost} = \text{CSA} \times \text{Rate} \]
Where the rate is Rs. 120 per \( m^2 \):
\[ \text{Cost} = 126.28 \times 120 \]

Calculating:
\[ \text{Cost} = 15153.6 \, \text{Rs.} \]

### Final Answers:
- Canvas required: \( 126.28 \, m^2 \)
- Cost of the canvas: \( 15153.6 \, \text{Rs.} \)