[imardmd1]

My differential calculus says that ΔC = 2πΔr = πΔD; but you want the diameter increased by two feet (that is, the radius increased by 1 foot), so that would require 6 feet 3 3/8 inches, plus a smidgin. Hmm?

[MosesKnows]

so that would require 6 feet 3 3/8 inches

Close enough but neither calculus nor differential calculus is required.

If the diameter of the earth is 8,000 miles then it requires 132.633,600 feet of rope to encircle the earth.

C = 2 π R
π = 3.14
R in feet = 4,000 miles x 5280 feet = 21,120,000 feet

C = 2 x 3.14 x 21,120,000 feet
C = 132,633,600 feet of rope

Raise the rope 1 foot.

C = 2 x 3.14 x 21,120,001 feet
C = 132,633,606.28 feet of rope

132,633,606.28 – 132,633,600 = 6.28 feet, which is how much longer the rope must be to raise it 1 foot off the surface along the circumference.