To find the total surface area of the cylindrical metallic barrel that is closed at one end, we need to consider the area of the circular base and the lateral surface area of the cylinder. Here is a step-by-step solution to determine the total surface area, rounded to one decimal place.
### Step 1: Determine the area of the circular base
The formula for the area of a circle is:
[tex]\[ \text{Base Area} = \pi r^2 \][/tex]
where [tex]\( r \)[/tex] is the radius of the base.
Given:
[tex]\[ \pi = \frac{22}{7} \][/tex]
[tex]\[ r = 0.245 \, \text{m} \][/tex]
So, we calculate:
[tex]\[ \text{Base Area} = \frac{22}{7} \times (0.245)^2 \][/tex]
[tex]\[ \text{Base Area} \approx 0.1886 \, \text{m}^2 \][/tex]
### Step 2: Determine the lateral surface area of the cylinder
The formula for the lateral surface area of a cylinder is:
[tex]\[ \text{Lateral Surface Area} = 2 \pi r h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cylinder.
Given:
[tex]\[ \pi = \frac{22}{7} \][/tex]
[tex]\[ r = 0.245 \, \text{m} \][/tex]
[tex]\[ h = 2.5 \, \text{m} \][/tex]
So, we calculate:
[tex]\[ \text{Lateral Surface Area} = 2 \times \frac{22}{7} \times 0.245 \times 2.5 \][/tex]
[tex]\[ \text{Lateral Surface Area} \approx 3.85 \, \text{m}^2 \][/tex]
### Step 3: Determine the total surface area
Since the barrel is closed at one end, the total surface area is the sum of the area of the circular base and the lateral surface area.
[tex]\[ \text{Total Surface Area} = \text{Base Area} + \text{Lateral Surface Area} \][/tex]
[tex]\[ \text{Total Surface Area} \approx 0.1886 + 3.85 \][/tex]
[tex]\[ \text{Total Surface Area} \approx 4.03865 \, \text{m}^2 \][/tex]
### Step 4: Round the total surface area to one decimal place
[tex]\[ \text{Total Surface Area} \approx 4.0 \, \text{m}^2 \][/tex]
Hence, the total surface area of the cylindrical barrel, correct to one decimal place, is [tex]\( 4.0 \, \text{m}^2 \)[/tex].
