Final answer:
The surface area of a cylinder is found by adding the areas of two base circles and the rectangle formed by the side of the cylinder. For a cylinder with a height of 7 feet and a base radius of 4 feet, the surface area is 88π square feet, which is approximately 276.46 square feet.
Step-by-step explanation:
To find the surface area of a cylinder with a height of 7 feet and base radius of 4 feet, we'll need to calculate the areas of two parts: the sides and the ends. The surface area of the sides is calculated just like the area of a rectangle, which is its height times its width. For a cylinder, the 'width' of this rectangle is the circumference of the base circle, which is 2πr. The ends of the cylinder are simply two circles, and their areas are calculated using the formula πr².
To summarize the calculation:
- Calculate the circumference of the base circle: C = 2πr = 2π(4) = 8π feet
- Calculate the area of the side of the cylinder (also known as the lateral surface area): A_side = perimeter × height = C × 7 = 8π × 7 = 56π square feet
- Calculate the area of one base circle: A_base = π × r² = π × 4² = 16π square feet
- Since there are two bases, you double the base area: Total base area = 2 × A_base = 2 × 16π = 32π square feet
- Add the lateral surface area and the total base area to find the total surface area of the cylinder: Total surface area = A_side + Total base area = 56π + 32π = 88π square feet.
Therefore, the total surface area of the cylinder is 88π square feet. If we use the approximation π ≈ 3.1416, we can calculate this as approximately 276.46 square feet.