Chapter 8.8, Problem 31E

(a)

To determine

To Prove: The expression for the correlation is $C=R\sec \left(\frac{L}{R}\right)-R$ .

Explanation of Solution

Given:

The radius is R and the length of the highway is L.

Calculation:

Consider L as the length of the arc that is subtended by the angle $\theta$ .

Then,

  $\theta =\frac{L}{R}$

Consider the expression,

  $\begin{array}{l}R\sec \theta =R+C\\ C=R\sec \theta -R\end{array}$

Then,

  $C=R\sec \frac{L}{R}-R$

Hence, proved.

(b)

To determine

To Prove: The expression for the correlation is $C=\frac{L^2}{2R}+\frac{5L^4}{24R^2}$ .

Explanation of Solution

Given:

The radius is R and the length of the highway is L.

Calculation:

Substitute $\sec \theta =T_4\left(\theta \right)$ that is $1+\frac{1}{2}{\theta }^2+\frac{5}{24}{\theta }^4$ in the equation $C=R\sec \frac{L}{R}-R$ then,

  $\begin{array}{c}C=R\left[1+\frac{1}{2}{\theta }^2+\frac{5}{24}{\theta }^4\right]-R\\ =R\left[1+\frac{1}{2}{\left(\frac{L}{R}\right)}^2+\frac{5}{24}{\left(\frac{L}{R}\right)}^4\right]-R\\ =\frac{L^2}{2R}+\frac{5L^4}{24R^2}\end{array}$

(c)

To determine

To Find: The comparison for the formula given formulas in part (a) and (b).

Answer to Problem 31E

The difference between the two results is of $8.08\times {10}^{-9}\text{km}$ .

Explanation of Solution

Given:

The length of the highway that is 100 km long.

Consider the radius of the Earth 6370 km.

Calculation:

Consider the formula for part (a) is,

  $C=R\sec \frac{L}{R}-R$

Then,

  $\begin{array}{c}C=\left(6370\right)\sec \frac{\left(100\right)}{\left(6370\right)}-6370\\ =0.7400996544\text{km}\end{array}$

Consider the formula,

  $\begin{array}{c}C=\frac{\left(100\right)}{2\left(6370\right)}\sec \frac{\left(100\right)}{\left(6370\right)}-6370\\ =0.78500995736\text{km}\end{array}$

Thus, the difference between the two results is of $8.08\times {10}^{-9}\text{km}$ .