Chapter 4.1, Problem 105E

To determine

Find the distance between the two cities Dallas, Texas and Omaha, Nebraska.

Answer to Problem 105E

  $\text{592}\text{miles}$

Explanation of Solution

Given:

The Latitude of Dallas, Texas ${\text{32}}^{0}47\text{'}39\text{'}\text{'}\text{N}$ and Omaha, Nebraska ${\text{41}}^{0}15\text{'}50\text{'}\text{'}\text{N}$

The radius of Earth is 4000 miles.

Concept Used:

Formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. The result is the difference.

We can write: minuend - subtrahend = difference.

For subtraction we can use the formulas: $1^0=60\text{minutes};1\text{minute}=60\text{seconds};1^0=3600\text{second}$

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, $\text{r}$

$\begin{array}{l}\text{θ}=\frac{\text{length}\text{of}\text{subtended}\text{arc}}{\text{length}\text{of}\text{radius}}=\frac{\text{s}}{\text{r}}\\ \\ \text{s}=\text{θ}·\text{r}\end{array}$

Calculation:

Step 1: Find the difference of Latitude between the two cities.

In this Problem Minuend (the first angle) has 15 minutes whereas the 2^nd angle has 47 minutes.

We can rewrite the first angle in ${\text{D}}^{\text{0}}\text{M'}\text{S''}$ . $[1^0=60\text{'}]$

  ${\text{41}}^{0}15\text{'}50\text{'}\text{'}\text{N}\text{=}{\text{40}}^0+1^0+15\text{'}+50\text{'}\text{'}\text{N}\text{=}{\text{40}}^0+60\text{'}+15\text{'}+50\text{'}\text{'}\text{N=}{\text{40}}^{0}75\text{'}50\text{'}\text{'}\text{N}$

We know that       Difference::

  $\begin{array}{l}{1}^0=60\text{minutes}\\ 1\text{minute}=60\text{seconds}\\ 1^0=3600\text{second}\end{array}$ $\begin{array}{l}{40}^{0}75\text{'}50\text{'}\text{'}\\ -\underrightarrow{32{}^{0}47\text{'}39\text{'}\text{'}}\\ 8^{0}28\text{'}11\text{'}\text{'}\end{array}$

  $8^{0}28\text{'}11\text{'}\text{'}=8^0+28\text{'}+11\text{'}\text{'}=8^0+{\frac{28}{60}}^0+{\frac{11}{3600}}^0={8.469722}^0$

Step 2: Now convert the angle ${8.469722}^0$ in radian

We know that: $1^0=0.0174533\text{radian}$

  ${8.469722}^0=8.469722·0.0174533\text{radian}=0.14782453563\text{radian}\approx 0.148\text{radian}$

Now the distance between two cities: $\text{s}=\text{θ}·\text{r}$ . Given: r = 4000 miles and $\text{θ}=0.148\text{radian}$

Distance: $\text{s}=\text{θ}·\text{r}\text{=}\text{0}\text{.148}·\text{4000}\text{=}\text{592}\text{miles}$

Thus, the distance between Dallas, Texas and Omaha, Nebraska is $\text{592}\text{miles}$