Chapter 4.1, Problem 101E

To determine

Find the difference in the Latitude of Syracuse, New York and Annapolis, Maryland.

Answer to Problem 101E

  $2^{0}30\text{'}36\text{'}\text{'}\text{N}$

Explanation of Solution

Given:

The radius of Earth is 6378 miles and Syracuse is 450 km due to north of Annapolis.

Concept Used:

Radian: The radian measure, $\text{θ}$ of a central angle is defined as the ration of the length of the arc the angle subtends, $\text{s}$ divided by the radius of the circle, $\text{r}$

$\text{θ}=\frac{\text{length}\text{of}\text{subtended}\text{arc}}{\text{length}\text{of}\text{radius}}=\frac{\text{s}}{\text{r}}$

Calculation:

Given: The radius of Earth is 6378 miles and Syracuse is 450 km due to north of Annapolis.

Convert $450\text{km}$ into miles:

  $\begin{array}{l}1\text{km}=0.621371\text{miles}\\ 450\text{km}=450·0.621371\text{miles}=279.617\text{miles}\approx 279.62\text{miles}\end{array}$

Now we get $\text{r}\text{=}\text{6378}\text{miles}\text{and}\text{s}\text{=}279.62\text{miles}$

Step 1: Find the Latitude between the cities in radian.

  $\text{θ}=\frac{\text{s}}{\text{r}}=\frac{279.62}{6378}\text{radian}\text{=}\text{0}\text{.0438413296}\text{radian}$

Step 2: Convert $\text{0}\text{.0438413296}\text{radian}$ into deg.

  $\begin{array}{l}1\text{radian}=57.2958\mathrm{deg}\\ 0.0438413296\text{radian}=(0.0438413296)·(57.2958)\mathrm{deg}=2.511923154\mathrm{deg}\approx 2.51\mathrm{deg}\end{array}$

Step 3: Convert $2.51\mathrm{deg}$ in ${\text{D}}^{\text{0}}\text{M'}\text{S''}$ form $[1^0=60\text{minutes};1\text{'}=60\text{'}\text{'}]$

  ${2.51}^0=2^0+{0.51}^0=2^0+0.51·60\text{'}=2^0+30.6\text{'}=2^0+30\text{'}+0.6\text{'}=2^0+30\text{'}+0.6·60\text{'}\text{'}=2^{0}30\text{'}36\text{'}\text{'}$

Thus, the difference in the Latitude of Syracuse, New York and Annapolis, Maryland is $2^{0}30\text{'}36\text{'}\text{'}\text{N}$ .