The volume of a conical tent is `154 cm^(3)` and the area of its base is `38.5 cm^(2)`. What is the length (in cm.) of canvas required to build the tent, if the canvas is 2 cm is width ?

Options

A. 71.35
B. 68.75
C. 73.25
D. 75.75

Answer

To solve the problem step by step, we need to find the length of the canvas required to build the conical tent given its volume and the area of its base.

### Step 1: Identify the given values
- Volume of the conical tent, \( V = 154 \, \text{cm}^3 \)
- Area of the base of the cone, \( A = 38.5 \, \text{cm}^2 \)
- Width of the canvas, \( w = 2 \, \text{cm} \)

### Step 2: Calculate the radius of the base
The area of the base of the cone (which is circular) is given by the formula:
\[ A = \pi r^2 \]
Substituting the known area:
\[ 38.5 = \frac{22}{7} r^2 \]
To find \( r^2 \):
\[ r^2 = \frac{38.5 \times 7}{22} \]
Calculating this gives:
\[ r^2 = \frac{269.5}{22} = 12.25 \]
Taking the square root to find \( r \):
\[ r = \sqrt{12.25} = 3.5 \, \text{cm} \]

### Step 3: Calculate the height of the cone
The volume of the cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the known values:
\[ 154 = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times h \]
Rearranging to find \( h \):
\[ h = \frac{154 \times 3 \times 7}{22 \times 12.25} \]
Calculating this gives:
\[ h = \frac{3234}{269.5} \approx 12 \, \text{cm} \]

### Step 4: Calculate the slant height of the cone
The slant height \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{(3.5)^2 + (12)^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5 \, \text{cm} \]

### Step 5: Calculate the curved surface area of the cone
The curved surface area \( CSA \) of the cone is given by:
\[ CSA = \pi r l \]
Substituting the values:
\[ CSA = \frac{22}{7} \times 3.5 \times 12.5 \]
Calculating this gives:
\[ CSA = \frac{22 \times 3.5 \times 12.5}{7} = 137.5 \, \text{cm}^2 \]

### Step 6: Calculate the length of the canvas required
The area of the canvas required is equal to the curved surface area. The length \( L \) of the canvas can be calculated using the formula:
\[ L = \frac{CSA}{w} \]
Substituting the values:
\[ L = \frac{137.5}{2} = 68.75 \, \text{cm} \]

### Final Answer
The length of the canvas required to build the tent is \( 68.75 \, \text{cm} \).

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