The volume of a conical tent is `1232 m ^(3)` and the area of the base floor is `154 m^(2).` Calculate the :
length of the canvas required to cover this conical tent if its width is 2 m.

Answer

To solve the problem step by step, we will follow the necessary calculations to find the length of the canvas required to cover the conical tent.

### Step 1: Find the Radius of the Base
The area of the base of the conical tent is given as \(154 \, m^2\). The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
Setting this equal to the area of the base:
\[ \pi r^2 = 154 \]
Using \(\pi \approx \frac{22}{7}\):
\[ \frac{22}{7} r^2 = 154 \]
To find \(r^2\), multiply both sides by \(\frac{7}{22}\):
\[ r^2 = 154 \times \frac{7}{22} \]
Calculating this gives:
\[ r^2 = 49 \quad \Rightarrow \quad r = \sqrt{49} = 7 \, m \]

### Step 2: Find the Height of the Cone
The volume of the conical tent is given as \(1232 \, m^3\). The volume of a cone is given by the formula:
\[ \text{Volume} = \frac{1}{3} \pi r^2 h \]
Substituting the known values:
\[ 1232 = \frac{1}{3} \times \frac{22}{7} \times 7^2 \times h \]
This simplifies to:
\[ 1232 = \frac{1}{3} \times \frac{22}{7} \times 49 \times h \]
Calculating the constants:
\[ 1232 = \frac{22 \times 49}{21} h \]
\[ 1232 = \frac{1078}{21} h \]
Now, solving for \(h\):
\[ h = \frac{1232 \times 21}{1078} \]
Calculating this gives:
\[ h = 24 \, m \]

### Step 3: Find the Slant Height of the Cone
The slant height \(l\) of the cone can be found using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the known values:
\[ l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \, m \]

### Step 4: Calculate the Surface Area of the Tent
The lateral surface area \(A\) of the conical tent is given by:
\[ A = \pi r l \]
Substituting the values:
\[ A = \frac{22}{7} \times 7 \times 25 \]
Calculating this gives:
\[ A = 22 \times 25 = 550 \, m^2 \]

### Step 5: Find the Length of the Canvas
The canvas required to cover the tent is represented by the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given that the width of the canvas is \(2 \, m\):
\[ 550 = \text{Length} \times 2 \]
Solving for Length:
\[ \text{Length} = \frac{550}{2} = 275 \, m \]

### Final Answer
The length of the canvas required to cover the conical tent is \(275 \, m\).

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