To find the surface area of a cylindrical tank, we need to calculate the lateral surface area and the area of the top and bottom (bases) of the cylinder.
The formula for the surface area \( A \) of a cylinder is given by:
\[ A = 2\pi rh + 2\pi r^2 \]
where:
- \( r \) is the radius of the cylinder's base,
- \( h \) is the height of the cylinder.
Given:
- The diameter of the cylinder is 15 ft, so the radius \( r = \frac{15}{2} = 7.5 \) ft.
- The height \( h = 4.5 \) ft.
Now, we can substitute the values into the formula:
-
Calculate the lateral surface area: \[ \text{Lateral surface area} = 2\pi rh = 2 \times 3.14 \times 7.5 \times 4.5 \]
-
Calculate the area of the top and bottom: \[ \text{Area of top and bottom} = 2\pi r^2 = 2 \times 3.14 \times (7.5)^2 \]
Now, let's compute each component.
Lateral surface area: \[ 2 \times 3.14 \times 7.5 \times 4.5 = 2 \times 3.14 \times 33.75 = 211.875 , \text{ft}^2 \]
Area of the top and bottom: \[ 2 \times 3.14 \times (7.5)^2 = 2 \times 3.14 \times 56.25 = 352.5 , \text{ft}^2 \]
Total Surface Area: \[ A = 211.875 + 352.5 = 564.375 , \text{ft}^2 \]
Rounded to the nearest tenth, the surface area of the tank is: \[ \text{Surface area} \approx 564.4 , \text{ft}^2 \]
Now, let's determine how many gallons of paint are needed for two coats of paint:
- One gallon covers 140 ft².
- For two coats, we'll double the surface area: \[ \text{Total area to paint} = 564.4 \times 2 = 1128.8 , \text{ft}^2 \]
Now, calculate how many gallons are needed: \[ \text{Gallons required} = \frac{1128.8}{140} \approx 8.136 \]
Since you can only buy full gallons, you'll need to round up: \[ \text{Gallons needed} = 9 , \text{gallons} \]
In summary:
- Surface area of the tank: 564.4 ft²
- Full gallons of paint needed for two coats: 9 gallons