Final answer:
The surface area of the grain bin consists of the surface area of the cone and the surface area of the cylinder. The surface area of a cone can be calculated using the formula A = πr(r + s), where r is the radius of the base and s is the slant height. The surface area of a cylinder is given by the formula A = 2πrh, where r is the radius of the base and h is the height. By calculating the surface area of both the cone and cylinder, and then adding them together, the total surface area of the grain bin is 216π sq. ft.
Step-by-step explanation:
To calculate the surface area of the grain bin, we need to find the surface areas of both the cone and the cylinder and then add them together. The surface area of a cone can be calculated using the formula A = πr(r + s), where r is the radius of the cone's base and s is the slant height. In this case, the radius is equal to the radius of the cylinder, which is half of its diameter, so r = 6 ft. The slant height can be found using the Pythagorean theorem: s = √(r^2 + h^2) = √(6^2 + 8^2) = 10 ft. Plugging in these values, we have A = π(6 ft)(6 ft + 10 ft) = 96π sq. ft.
The surface area of a cylinder is given by the formula A = 2πrh, where r is the radius of the base and h is the height. Again, using the radius of the cylinder as 6 ft, and the height as 10 ft, we have A = 2π(6 ft)(10 ft) = 120π sq. ft.
Finally, we can calculate the total surface area by adding the surface areas of the cone and the cylinder: 96π sq. ft + 120π sq. ft = 216π sq. ft. Therefore, the correct option is Option 3: 216π sq. ft.