Question 611188
You are being asked to compare 2 right triangles, both
of which have a side in common, which is the height.
At the surveyor's 1st position, call his distance from 
the base of the mountain {{{ d }}} miles
Call the height of the mountain {{{ h }}} miles
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(1) {{{ tan( 23 ) = h/d }}}
and
(2) {{{ tan( 43 ) = h/( d - .25 ) }}}
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On my calculator,
{{{ tan( 23 ) = .42447 }}}
{{{ tan( 43 ) = .93252 }}}
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(1) {{{ .42447 = h/d }}}
(1) {{{ h = .42447d }}}
and
(2) {{{ .93252 = h/( d - .25 ) }}}
(2) {{{ h = .93252d - .233130 }}}
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Substitute (1) into (2)
(2) {{{ .42447d = .93252d - .233130 }}}
(2) {{{ .93252d - .42447d = .233130 }}}
(2) {{{ .50805d = .233130 }}}
(2) {{{ d = .45887 }}}
and, since
(1) {{{ h = .42447d }}}
(1) {{{ h = .42447*.45887 }}}
(1) {{{ h = .19478 }}}
The mountain is {{{ .19478 }}} miles high, or
{{{ .19478*5280 = 1028.42 }}} ft high
check answer:
(1) {{{ h = .42447d }}}
(1) {{{ .19478 = .42447*.45887 }}}
(1) {{{ .19478 = .19478 }}}
OK
(2) {{{ .93252 = h/( d - .25 ) }}}
(2) {{{ .93252 = .19478/( .45887 - .25 ) }}}
(2) {{{ .93252 = .19478/.20887 }}}
(2) {{{ .93252 = .93254 }}}
Error is due to rounding off