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I have taken this question from keith williams book. I know that we can find the maximum range of DME by line of sight formula which is R=1.23(Ht + Hr). But I don't know how to determine it's minimum range.

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    $\begingroup$ Minimum "range" meaning the lowest number that will be shown on the DME display? Note that even when the aircraft is directly over the ground station it will still take a nonzero time for a signal to get from the aircraft to the station and back, which will read as a distance. $\endgroup$
    – randomhead
    Apr 1 at 22:17
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    $\begingroup$ Note also that this is a poorly-written question because flight level 150 is NOT a set distance above sea level. Fifteen thousand feet is always 15000' above sea level, but FL150 goes up and down relative to sea level depending on atmospheric conditions. $\endgroup$
    – randomhead
    Apr 1 at 22:19

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Aside from the issues others have pointed out in the comments, the method of determining the minimum DME range display is straightforward.

Since DME measures slant range, the minimum will be observed when directly above the DME antenna. So the minimum is (aircraft altitude - antenna altitude) measured in nautical miles rounded to the nearest 0.1 NM. The key point others pointed out is that both altitudes have to use the same reference.

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