Chapter 9.3, Problem 24WE

To determine

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

Answer to Problem 24WE

  $5300\text{km}$

Explanation of Solution

Given:

Figure is drawn as.

  

  $XE=\text{Radius}=6400\text{km}$, $ST=600\text{km}$

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.

  $\begin{array}{l}\mathrm{Cos}\left(\frac{n}{2}\right)=\frac{XE}{SE}\\ \mathrm{Cos}\left(\frac{n}{2}\right)=\frac{6400}{\text{ST+ET}}\\ \mathrm{Cos}\left(\frac{n}{2}\right)=\frac{6400}{\text{600+6400}}\\ n=2\times {\cos }^{-1}\left(\frac{6400}{\text{600+6400}}\right)\\ n={47.79}^{\circ }\end{array}$

Earth distance is the arc length XTY.

  $\begin{array}{l}\stackrel{⌢}{XTY}=2\pi r\times \frac{n}{360}\\ \stackrel{⌢}{XTY}=2\pi \times 6400\times \frac{47.79}{360}\\ \stackrel{⌢}{XTY}=5335.488\text{ km}\\ \boxed{\stackrel{⌢}{XTY}=5300\text{km}}\end{array}$

Therefore, the answer is $5300\text{km}$