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To calculate the total surface area of the cylinder, we cut off the two ends to obtain two circles of radius $3.$



The two ends combined have an area of $$2\pi r^2=2\pi (3^2)=18\pi .$$ Next, we must calculate the lateral surface area. To do this, we make a vertical cut through this area, and unroll the lateral surface. When we do this, we obtain a rectangle with height $10.$ The width of the rectangle (ie. the length of the top edge) is equal to the circumference of one of the ends, since the top edge of this rectangle lay exactly along the circumference of the top end.



The circumference of one of the ends is $$2\pi r=2\pi (3)=6\pi ,$$ so the width of the rectangle is $6\pi .$ Therefore, the area of this rectangle is $10\times 6\pi =60\pi .$ So the total surface area of the cylinder is $18\pi +60\pi =\boxed{78\pi }.$