To find the surface area of a closed cylindrical drum, we need to calculate the surface area of both the circular ends and the curved surface area of the cylinder.
### Step 1: Determine the Radius
The radius [tex]\( r \)[/tex] of the cylinder can be found using the diameter.
[tex]\[
r = \frac{\text{diameter}}{2} = \frac{2.1}{2} = 1.05 \text{ m}
\][/tex]
### Step 2: Recall the Formula for the Surface Area of a Cylinder
The formula to calculate the total surface area [tex]\( A \)[/tex] of a closed cylindrical drum is:
[tex]\[
A = 2\pi r (r + h)
\][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the cylinder,
- [tex]\( h \)[/tex] is the height of the cylinder,
- [tex]\( \pi \)[/tex] is a constant [tex]\( \left(\pi = \frac{22}{7}\right) \)[/tex].
### Step 3: Substitute the Values
Now, substitute the known values into the surface area formula:
[tex]\[
r = 1.05 \text{ m}, \quad h = 4 \text{ m}, \quad \pi = \frac{22}{7}
\][/tex]
### Step 4: Perform the Calculation
[tex]\[
A = 2 \cdot \left(\frac{22}{7}\right) \cdot 1.05 \cdot (1.05 + 4)
\][/tex]
[tex]\[
A = 2 \cdot \left(\frac{22}{7}\right) \cdot 1.05 \cdot 5.05
\][/tex]
[tex]\[
A \approx 33.33 \text{ square metres}
\][/tex]
Thus, the surface area of the metallic sheet used in the closed cylindrical drum is [tex]\( 33.33 \)[/tex] square meters.
