Chapter 4.1, Problem 106E

To determine

Find the distance between the two cities San Francisco, California and Seattle, Washington.

Answer to Problem 106E

  $\text{2880}\text{miles}$

Explanation of Solution

Given:

The Latitude of San Francisco, California ${\text{37}}^{0}47\text{'}36\text{'}\text{'}\text{N}$ and Seattle, Washington ${\text{47}}^{0}37\text{'}18\text{'}\text{'}\text{N}$

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 37 minutes and 18 seconds whereas the 2^nd angle has 47 minutes and 36 seconds.

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

  ${\text{47}}^{0}37\text{'}18\text{'}\text{'}\text{N}\text{=}{\text{46}}^0+1^0+36\text{'}+1\text{'}+18\text{'}\text{'}\text{N}\text{=}{\text{46}}^0+60\text{'}+36\text{'}+60\text{'}\text{'}+18\text{'}\text{'}\text{N=}{\text{46}}^{0}96\text{'}78\text{'}\text{'}\text{N}$

We know that       Difference::

  $\begin{array}{l}{1}^0=60\text{minutes}\\ 1\text{minute}=60\text{seconds}\\ 1^0=3600\text{second}\end{array}$ $\begin{array}{l}{46}^{0}96\text{'}78\text{'}\text{'}\\ -\underrightarrow{37{}^{0}47\text{'}39\text{'}\text{'}}\\ 9^{0}49\text{'}42\text{'}\text{'}\end{array}$

  $9^{0}49\text{'}42\text{'}\text{'}=9^0+49\text{'}+42\text{'}\text{'}=8^0+{\frac{49}{60}}^0+{\frac{42}{3600}}^0={9.828333}^0$

Step 2: Now convert the angle ${9.828333}^0$ in radian

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

  ${9.828333}^0=9.828333·0.0174533\text{radian}=0.17153677083\text{radian}\approx 0.172\text{radian}$

Now the distance between two cities: $\text{s}=\text{θ}·\text{r}$ . Given: r = 4000 miles and $\text{θ}=0.172\text{radian}$

Distance: $\text{s}=\text{θ}·\text{r}\text{=}\text{0}\text{.172}·\text{4000}\text{=}\text{2880}\text{miles}$

Thus, the two cities San Francisco, California and Seattle, Washington is $\text{2880}\text{miles}$