Step 1 :Given a cylindrical tank with a radius of \(5.5 \mathrm{ft}\) and a height of \(16 \mathrm{ft}\). A gallon of paint will cover \(140 \mathrm{ft}^{2}\) of surface. We need to find out how much paint is needed to put two coats of paint on the entire surface of the tank.
Step 2 :First, we calculate the surface area of the cylindrical tank. The formula for the surface area of a cylinder is \(2\pi r(r + h)\), where \(r\) is the radius and \(h\) is the height. Substituting the given values, we get \(2\pi \times 5.5 \times (5.5 + 16) = 742.986662573986 \mathrm{ft}^{2}\).
Step 3 :Since we are applying two coats of paint, we need to double the surface area. So, the total area to be painted is \(2 \times 742.986662573986 = 1485.973325147972 \mathrm{ft}^{2}\).
Step 4 :Next, we divide the total area to be painted by the area that a gallon of paint can cover to find out how many gallons of paint are needed. So, the gallons of paint needed is \(1485.973325147972 \div 140 = 10.6140958931998\) gallons.
Step 5 :Since we can't buy a fraction of a gallon of paint, we round up to the nearest whole number. So, we need \(\boxed{11}\) full gallons of paint.