Chapter 12.7, Problem 35E

Changes in torus surface area The surface area of a torus with an inner radius r and an outer radius R > r is S = 4π²(R²– r²).

a.    If r increases and R decreases, does S increase or decrease, or is it impossible to say?

b.    If r increases and R increases, does S increase or decrease, or is it impossible to say?

c.    Estimate the change in the surface area of the torus when r changes from r = 3.00 to r = 3.05 and R changes from R = 5.50 to R = 5.65.

d.    Estimate the change in the surface area of the torus when r changes from r = 3.00 to r = 2.95 and R changes from R = 7.00 to R = 7.04.

e.    Find the relationship between the changes in r and R that leaves the surface area (approximately) unchanged.