I2

In leveling a sufficiently long roadway of length L, an allowance must be made for the curvature of the Earth. In this problem, we develop how this is done. (a) In the figure below, R is the radius of the Earth and y is the needed leveling correction. Find a formula for sec(x ) in terms of R and y. IR (b) Find the fourth-degree Taylor polynomial for sec(x). (c) For small values of x, we may use the Taylor polynomial Pz(x) for sec(x). Use P2(x) and your result from part (a) to show that the leveling correction is y = L2/(2R). (d) Find the number of inches of leveling correction needed for a 1 mile leng of roadway. Use R = 4000 miles