Final answer:
The lateral area of the cylinder is 676π square feet and the surface area is 1014π square feet.
Step-by-step explanation:
To find the lateral area of the cylinder, we need to calculate the perimeter of the base (which is the circumference of the circle) and multiply it by the height of the cylinder. In this case, the base has an area of 169π square feet, so the radius squared is 169π/π, which simplifies to 169 square feet.
Therefore, the radius is the square root of 169, which is 13 feet. The height of the cylinder is twice the radius, so it is 26 feet. Now we can calculate the perimeter of the base, which is the circumference of the circle: 2πr = 2π(13) = 26π feet. Finally, to find the lateral area, we multiply the perimeter of the base by the height: 26π × 26 = 676π square feet.
To find the surface area of the cylinder, we need to add the areas of the two bases to the lateral area. The area of each base is πr², which is π(13)² = 169π square feet. So the total base area is 2(169π) = 338π square feet. Therefore, the surface area of the cylinder is the sum of the base area and the lateral area: 338π + 676π = 1014π square feet.