The volume of a conical tent is 154 cm3 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 in width?

Options

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

Answer

To find the length of canvas required to build the conical tent, we need to follow these steps:

### Step 1: Understand the given information
We have the following information:
- Volume of the conical tent (V) = 154 cm³
- Area of the base (A) = 38.5 cm²

### Step 2: Use the formula for the volume of a cone
The formula for the volume of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where \( r \) is the radius of the base and \( h \) is the height of the cone.

### Step 3: Use the formula for the area of the base
The area of the base of the cone is given by:
\[ A = \pi r^2 \]
From the given area, we can find \( r^2 \):
\[ \pi r^2 = 38.5 \implies r^2 = \frac{38.5}{\pi} \]
Using \( \pi \approx 3.14 \):
\[ r^2 \approx \frac{38.5}{3.14} \approx 12.25 \implies r \approx \sqrt{12.25} = 3.5 \text{ cm} \]

### Step 4: Substitute \( r \) into the volume formula to find \( h \)
Now we can substitute \( r \) back into the volume formula:
\[ 154 = \frac{1}{3} \pi (3.5)^2 h \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
So, substituting this value:
\[ 154 = \frac{1}{3} \pi (12.25) h \]
Now, multiply both sides by 3:
\[ 462 = \pi (12.25) h \]
Now, divide by \( \pi (12.25) \):
\[ h = \frac{462}{\pi \times 12.25} \]
Using \( \pi \approx 3.14 \):
\[ h \approx \frac{462}{3.14 \times 12.25} \approx \frac{462}{38.525} \approx 12 \text{ cm} \]

### Step 5: Calculate the slant height \( L \)
The slant height \( L \) of the cone can be calculated using the Pythagorean theorem:
\[ L = \sqrt{r^2 + h^2} \]
Substituting the values of \( r \) and \( h \):
\[ L = \sqrt{(3.5)^2 + (12)^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5 \text{ cm} \]

### Step 6: Calculate the curved surface area (CSA)
The curved surface area (CSA) of the cone is given by:
\[ CSA = \pi r L \]
Substituting the values of \( r \) and \( L \):
\[ CSA = \pi \times 3.5 \times 12.5 \]
Using \( \pi \approx 3.14 \):
\[ CSA \approx 3.14 \times 3.5 \times 12.5 \approx 123.875 \text{ cm}^2 \]

### Step 7: Calculate the length of canvas required
Since the canvas is 2 cm wide, the length of canvas required can be calculated by dividing the CSA by the width of the canvas:
\[ \text{Length of canvas} = \frac{CSA}{\text{Width}} = \frac{123.875}{2} \approx 61.9375 \text{ cm} \]

### Final Answer
The length of canvas required to build the tent is approximately **61.94 cm**.