Solution:
iven the figure below:
Where
[tex]\begin{gathered} S\Rightarrow satellite \\ B\Rightarrow Bora-Bora \\ G\Rightarrow Greenland \\ R\Rightarrow Rapa\text{ Nui} \\ H\Rightarrow St.\text{ Helen} \end{gathered}[/tex]
Using the secant-secant theorem expressed as
[tex]SB\left(SB+BG\right)=SR\left(SR+RH\right)\text{ ----- equation 1}[/tex]
Given that th e satellite signal reaches Bora-Bora at 38,000 km and Greenland which is an additional 20,000km, this implies that
[tex]\begin{gathered} SB=38000 \\ BG=20000 \end{gathered}[/tex]
If the distance from the satellite to Rapa Nui is 45,000 km, this implies that
[tex]SR=45000[/tex]
The total distance from the satelite to St. Helen is expressed as
[tex]SH=SR+RH\text{ ----- equation 2}[/tex]
From equation 1,
[tex]\begin{gathered} \begin{equation*} SB\left(SB+BG\right)=SR\left(SR+RH\right)\text{ } \end{equation*} \\ substitute\text{ equation 2 into equation 1} \\ thus,\text{ we have} \\ SB\left(SB+BG\right)=SR\left(SH\right)\text{ ---- equation 3} \end{gathered}[/tex]
Substituting the values, we have
[tex]\begin{gathered} 38000\left(38000+20000\right)=45000\left(SH\right) \\ \Rightarrow38000\times58000=45000\left(SH\right) \end{gathered}[/tex]
divide both sides by 45000,
[tex]\begin{gathered} SH=\frac{38000\times58000}{45000} \\ \Rightarrow SH=48977.77777 \end{gathered}[/tex]
Hence, the signal will travel a distance of 48977.77777 km from the satellite to St. Helen.
