Question 180794
Let the width of the river = {{{w}}} ft
Let the height of the tree = {{{h}}} ft
given:
(1) {{{tan(23) = h/w}}}
(2) {{{tan(18) = h/(45 + w)}}}
These are 2 equations with 2 unknowns, so 
they should be solvable
{{{tan(23) = .4245}}}
{{{tan(18) = .3249}}}
(1) {{{.4245 = h/w}}}
(2) {{{.3249 = h/(45 + w)}}}
----------------------------
Multiply both sides of (1) by {{{w}}}
Multiply both sides of (2) by {{{45 + w}}}
(1) {{{.4245w = h}}}
(2) {{{.3249*(45 + w) = h}}}
(2) {{{14.6214 + .3249w = h}}}
Subtract (2) from (1)
(1) {{{.4245w = h}}}
(2) {{{-14.6214 - .3249w = -h}}}
{{{.0996w - 14.6214 = 0}}}
{{{.0996w = 14.6214}}}
{{{w = 14.6214/.0996}}}
{{{w = 146.8012}}}
{{{.8012*12 = 9.61}}}
The river is 146 ft 9 in wide
check answer:
(2) {{{tan(18) = h/(45 + w)}}}
{{{.3249 = h/191.8012}}}
{{{h = .3249*191.8012}}}
{{{h = 62.3162}}} ft the height of the tree
(1) {{{tan(23) = h/w}}}
{{{.4245 = 62.3162/146.8012}}}
{{{.4245 = .424494}}} close enough