How far away is that plane? Using Log tables …

Today I took this phone snapshot of a great plane trail. It was about 3pm, looking about West.  You can click on any image to enlarge it.

A quick check [1] showed it was about 12km up. I’d guess it was about 30 degrees above the horizon. I wondered how far away it was ; meaning what point on the Earth was it directly over when I took this?  So, time for some olde world maths. First – of course – draw it:

I need to find x, the distance in km. Time to go really old school and use logarithms [2]  aka logs.

x =  12/tan(30 degrees). I’m dividing so I need to subtract the logs:   log(x) = log(12) – log(tan 30 degrees)

Off to the PDF of the Formulas and Tables book I found online – from 1968 :-)

log(x) = log(12) - log(tan 30 degrees)
log(x) = 1.0792 - (9.7614 - 10.0000)      
# that's the way the book said to do it for log(tan x) ;  [value in table - 10.0000]

log(x) = 1.0792 - (-0.2386)
log(x) = 1.0792 +   0.2386)
log(x) = 1.3178

Antilog(.3178) = 2075 + 4 = 2079

Now, the first 1 in my 1.3178 means I have (1+1) = 2 digits before the decimal in my final answer.

So it is therefore: 20.79 km

Crosscheck with the very modern Flightaware website [3] which said that at 15:02 (when I took the photo), the flight was at Latitude and Longitude:

Sat 15:02:23    -37.8623    144.6534

So off to Google Earth, draw that point and a point for me in Williamstown. Finally, get Google Earth to measure the ground distance:

With the result being 21.39 km.  Not bad. Not bad at all.

References:

[1] FlightRadar24 app on Android used. BTW this was QF64 from Johannesburg to Sydney.
[2] Have re-discovered how to do Logarithms as part of another exercise involving studying a calculation which was part of an 1860s gold claim dispute at Walhalla. I had learnt them back in the early 70s, then forgot as calculators took over.

[3] http://flightaware.com/live/flight/QFA64/history/20171208/1650Z/FAOR/YSSY/tracklog

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