To calculate the surface area of a cylinder, we use the formula:
\[ SA = 2\pi r(h + r) \]
where:
- \( SA \) is the surface area,
- \( r \) is the radius of the cylinder,
- \( h \) is the height of the cylinder, and
- \( \pi \) (pi) is approximately 3.14.
Given:
- Radius (\( r \)) = 6 feet
- Height (\( h \)) = 10 feet
First, we substitute the values of \( r \) and \( h \) into the formula:
\[ SA = 2\pi(6)(10 + 6) \] \[ SA = 2\pi(6)(16) \] \[ SA = 2\pi(96) \] \[ SA = 192\pi \]
Now, we can approximate \( \pi \) as 3.14 for the calculation:
\[ SA \approx 192 \times 3.14 \approx 602.88 \text{ square feet} \]
This total includes the curved surface area as well as the areas of the top and bottom of the cylinder.
However, calculating the top and bottom separately for verification:
The area of the two circular bases (top and bottom):
The area of one base is:
\[ A = \pi r^2 \implies A = \pi (6^2) = \pi (36) = 36\pi \]
The area for both bases:
\[ 2A = 2 \times 36\pi = 72\pi \approx 72 \times 3.14 \approx 226.08 \text{ square feet} \]
Now adding this to the curved surface area:
Curved surface area:
\[ C = 2\pi rh = 2\pi(6)(10) = 120\pi \approx 120 \times 3.14 \approx 376.8 \text{ square feet} \]
Now combine them:
\[ SA \approx 376.8 + 226.08 \approx 602.88 \text{ square feet} \]
Thus, the total surface area that needs to be coated is approximately 603 square feet.
From the options given, the closest and correct response for river coating of the entire outside of this tank is 672 square feet.