To calculate the surface area of a cylinder, we need to determine both the lateral surface area and the area of the top and bottom faces.
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Lateral Surface Area: The lateral surface area \(A_{lateral}\) of a cylinder is given by the formula: \[ A_{lateral} = 2\pi rh \] where \(r\) is the radius and \(h\) is the height.
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Area of the Top and Bottom Faces: The area of one circular face is given by the formula: \[ A_{circle} = \pi r^2 \] Since there are two circular faces (top and bottom), the total area of the faces is: \[ A_{faces} = 2\pi r^2 \]
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Total Surface Area: The total surface area \(A_{total}\) of the cylinder is the sum of the lateral surface area and the area of the two faces: \[ A_{total} = A_{lateral} + A_{faces} \]
Now, substituting the given values \(r = 4\) feet and \(h = 3\) feet into the formulas:
Lateral Surface Area: \[ A_{lateral} = 2\pi rh = 2\pi(4)(3) = 24\pi \]
Area of the Top and Bottom Faces: \[ A_{faces} = 2\pi r^2 = 2\pi(4^2) = 2\pi(16) = 32\pi \]
Total Surface Area: \[ A_{total} = A_{lateral} + A_{faces} = 24\pi + 32\pi = 56\pi \]
To find the total surface area in numeric form, we can substitute \(\pi\) (approximately 3.14) if needed: \[ A_{total} \approx 56 \times 3.14 \approx 175.84 \]
Rounded to two decimal places, the total surface area of the cylinder is: \[ \boxed{175.84 \text{ square feet}} \]
However, if we need to provide the answer in terms of \(\pi\), the surface area remains: \[ A_{total} = 56\pi \text{ square feet} \]