Solved: On the surface of a globe or, more precisely, on the surface of the Earth, the

Chapter 9, Problem 66

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On the surface of a globe or, more precisely, on the surface of the Earth, the boundaries of the states of Colorado and Wyoming are both "spherical rectangles." (In this problem we assume that the Earth is a perfect sphere.) Colorado is bounded by the lines of longitude \(102^{\circ} \mathrm{W}\) and \(109^{\circ} \mathrm{W}\) and the lines of latitude \(37^{\circ} \mathrm{N}\) and \(41^{\circ} \mathrm{N}\). Wyoming is bounded by longitudes \(104^{\circ} \mathrm{W}\) and \(111^{\circ} \mathrm{W}\) and latitudes \(41^{\circ} \mathrm{N}\) and \(45^{\circ} \mathrm{N}\). See FIGURE 9.R.5.

(a) Without explicitly computing their areas, determine which state is larger and explain why.

(b) By what percentage is Wyoming larger (or smaller) than Colorado? [Hint: Suppose the radius of the Earth is R. Project a spherical rectangle in the Northern Hemisphere that is determined by latitudes \(\theta_{1}\) and \(\theta_{2}\) and longitudes \(\phi_{1}\) and \(\phi_{2}\) onto the xy-plane.]

(c) One reference book gives the areas of the two states as 104,247 and \(97,914 \mathrm{mi}^{2}\). How does this answer compare with the answer in part (b)?