A company packages its milk powder in cylindrical containers whose base has a diameter of 16.8 cm and height 20.5 cm. Company places a label around the curved surface of the container. If the label is placed 1.5 cm from the top and the bottom, what is the surface area of the label? `[ "Take" pi = (22)/(7) ]`

Answer

To find the surface area of the label placed around the cylindrical container, we will follow these steps:

### Step 1: Determine the radius of the cylinder
The diameter of the cylinder is given as 16.8 cm. The radius (r) can be calculated using the formula:
\[ r = \frac{\text{diameter}}{2} \]
Substituting the value:
\[ r = \frac{16.8 \, \text{cm}}{2} = 8.4 \, \text{cm} \]

### Step 2: Calculate the height of the label
The total height of the cylinder is given as 20.5 cm. Since the label is placed 1.5 cm from the top and 1.5 cm from the bottom, we need to subtract these two distances from the total height:
\[ \text{Height of the label (h)} = \text{Total height} - 1.5 \, \text{cm} - 1.5 \, \text{cm} \]
Calculating this gives:
\[ h = 20.5 \, \text{cm} - 3 \, \text{cm} = 17.5 \, \text{cm} \]

### Step 3: Calculate the curved surface area of the label
The curved surface area (CSA) of a cylinder can be calculated using the formula:
\[ \text{CSA} = 2 \pi r h \]
Substituting the values we have:
- \( r = 8.4 \, \text{cm} \)
- \( h = 17.5 \, \text{cm} \)
- \( \pi = \frac{22}{7} \)

Now substituting these values into the CSA formula:
\[ \text{CSA} = 2 \times \frac{22}{7} \times 8.4 \times 17.5 \]

### Step 4: Simplify the expression
Calculating step-by-step:
1. Calculate \( 2 \times \frac{22}{7} = \frac{44}{7} \)
2. Now calculate \( 8.4 \times 17.5 \):
\[ 8.4 \times 17.5 = 147 \]
3. Now substitute back into the CSA formula:
\[ \text{CSA} = \frac{44}{7} \times 147 \]

### Step 5: Calculate the final CSA
Now simplify:
\[ \text{CSA} = \frac{44 \times 147}{7} \]
Calculating \( 44 \times 147 = 6468 \):
\[ \text{CSA} = \frac{6468}{7} = 924 \, \text{cm}^2 \]

### Final Answer
The surface area of the label is \( 924 \, \text{cm}^2 \).

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