



 
Great circle distance over radius, $\left(\frac{d}{3963} \right)$ = angular distance in radians. "Highest point" is halfway, $\left(\frac{d}{7916} \right)$ radians. Height is $\left(1  \cos\left(\frac{d}{7916}\right)\right)\times3963$ miles (using radian mode on your calculator!) For short distances (say under 2000 miles) this is rather well approximated by taking $R\approx4000,$ and $\cos(A) \approx 1 \frac{A^2}{2}$ (again, this formula only holds using radians) using which it simplifies to \[h \approx \frac{d^2}{8R} \approx \frac{d^2}{32000}\] So over 10 miles $h \approx \frac{1}{320}$ mile or about sixteen feet. Good Hunting! Jimmy wrote back
RD's expression \[h \approx \frac{d^2}{8R} \approx \frac{d^2}{32000}\] is the easiest to use. For a distance of 100 miles between the bases of the towers the "highest point" between them is \[h \approx \frac{100^2}{32000} = 0.3125 \mbox{ miles, or approximately } 1650 \mbox{ feet.}\] If you want to use the expression \[ \left(1  \cos\left(\frac{d}{7916}\right)\right)\times3963 \] I get $\left(1  \cos\left(\frac{100}{7916}\right)\right)\times3963$ to be 1670 feet using the radian mode on my calculator. If you want to use the angle in degrees then the expression is \[ \left(1  \cos\left(\frac{d}{7916}\times \frac{180}{\pi}\right)\right)\times3963 .\] Harley  


Math Central is supported by the University of Regina and the Imperial Oil Foundation. 