How much area of metal sheet is rquired to make an open cylindrical tank of radius 3.5 m and height of 12 m ?

Options

A. `305m^(2)`
B. `285m^(2)`
C. `292 . 7m^(2)`
D. `302.5m^(2)`

Answer

To find the area of the metal sheet required to make an open cylindrical tank, we need to calculate the total surface area of the cylinder. The formula for the total surface area of an open cylinder (which does not have a top) is given by:

\[ \text{Total Surface Area} = 2\pi rh + \pi r^2 \]

Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder
- \( \pi \) is a constant approximately equal to \( \frac{22}{7} \)

### Step 1: Identify the given values
- Radius \( r = 3.5 \) m
- Height \( h = 12 \) m

### Step 2: Substitute the values into the formula
We will substitute the values of \( r \) and \( h \) into the total surface area formula:

\[ \text{Total Surface Area} = 2\pi(3.5)(12) + \pi(3.5)^2 \]

### Step 3: Calculate the lateral surface area
First, calculate the lateral surface area \( 2\pi rh \):

\[ 2\pi(3.5)(12) = 2 \times \frac{22}{7} \times 3.5 \times 12 \]

Calculating this step-by-step:

1. Calculate \( 2 \times \frac{22}{7} = \frac{44}{7} \)
2. Now calculate \( \frac{44}{7} \times 3.5 \):
- \( 3.5 = \frac{35}{10} = \frac{35}{10} = \frac{35 \times 10}{10 \times 10} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} = \frac{350}{100} \)
- \( \frac{44 \times 35}{7 \times 10} = \frac{1540}{70} = 22 \)
3. Finally, multiply by 12:
- \( 22 \times 12 = 264 \)

So, the lateral surface area is \( 264 \) m².

### Step 4: Calculate the area of the base
Now calculate the area of the base \( \pi r^2 \):

\[ \pi(3.5)^2 = \frac{22}{7} \times (3.5 \times 3.5) = \frac{22}{7} \times 12.25 \]

Calculating this step-by-step:

1. Calculate \( 3.5 \times 3.5 = 12.25 \)
2. Now calculate \( \frac{22 \times 12.25}{7} = \frac{269.5}{7} = 38.5 \)

So, the area of the base is \( 38.5 \) m².

### Step 5: Add the lateral surface area and the base area
Now, add the lateral surface area and the base area to get the total surface area:

\[ \text{Total Surface Area} = 264 + 38.5 = 302.5 \text{ m}^2 \]

### Final Answer
The area of the metal sheet required to make the open cylindrical tank is \( \text{302.5 m}^2 \).

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