Chapter 9.3, Problem 23WE

To determine

To Find: the earth distance rounded to nearest $100\text{km}$ in the figure shown.

Answer to Problem 23WE

  $3800\text{km}$

Explanation of Solution

Given:

Figure is drawn as.

  

  $XE=\text{Radius}=6400\text{km}$, $ST=300\text{km}$

Two chords whose lengths are in the ratio $1:2$ and distances from the center are in the ratio $2:1$ .

Calculation:

Applying trigonometry in triangle XES.

  $\begin{array}{l}\mathrm{Cos}\left(\frac{n}{2}\right)=\frac{XE}{SE}\\ \mathrm{Cos}\left(\frac{n}{2}\right)=\frac{6400}{\text{ST+ET}}\\ \mathrm{Cos}\left(\frac{n}{2}\right)=\frac{6400}{\text{300+6400}}\\ n=2\times {\cos }^{-1}\left(\frac{6400}{\text{300+6400}}\right)\\ n={34.42}^{\circ }\end{array}$

Earth distance is the arc length XTY.

  $\begin{array}{l}\stackrel{⌢}{XTY}=2\pi r\times \frac{n}{360}\\ \stackrel{⌢}{XTY}=2\pi \times 6400\times \frac{34.42}{360}\\ \stackrel{⌢}{XTY}=3842.80\text{ km}\\ \boxed{\stackrel{⌢}{XTY}=3800\text{km}}\end{array}$

Therefore, the answer is $3800\text{km}$