Answer: 191.5 feet
Step-by-step explanation:
You can observe two right triangles in the figure attached: ABC and DBC.
Where the width of the river is "x".
Remember the trigonometric identity:
[tex]tan\alpha=\frac{opposite}{adjacent}[/tex]
Then, for the triangle ABC:
[tex]tan(22\°)=\frac{CB}{AB}\\\\tan(22\°)=\frac{CB}{50+x}[/tex]
Solve for CB:
[tex](50+x)*tan(22\°)=CB[/tex] [Equation 1]
For the triangle DBC:
[tex]tan(27\°)=\frac{CB}{DB}\\\\tan(27\°)=\frac{CB}{x}[/tex]
Solve for CB:
[tex]xtan(27\°)=CB[/tex] [Equation 2]
Making [Equation 1]=[Equation 2] and solving for "x", you get:
[tex](50+x)*tan(22\°)=xtan(27\°)\\\\50tan(22\°)+xtan(22\°)=xtan(27\°)\\\\50tan(22\°)=xtan(27\°)-xtan(22\°)\\\\50tan(22\°)=(tan(27\°)-tan(22\°))x\\\\x=\frac{50tan(22\°)}{tan(27\°)-tan(22\°)}\\\\x=191.5ft[/tex]
The width of the river can be found using trigonometric ratios. Therefore, the width of the river is 191.5 ft
The situation forms a right angle triangle
A right angle triangle has one of its angles as 90 degrees. The sides and angles can be found using trigonometric ratios.
Therefore, the opposite side is the height of the tree. The width of the river can be found using trigonometric ratios:
tan ∅ = opposite / adjacent
tan 27° = h / x
tan 22° = h / x + 50
where
x = width of the river
h = height of tree
Therefore,
x tan 27° = h
x + 50 tan 22° = h
Hence,
x tan 27° = (x + 50) tan 22°
x tan 27° = x tan 22° + 50 tan 22°
x tan 27° - x tan 22° = 50 tan 22°
x(0.10549922365) = 20.2013112918
x = 20.2013112918 / 0.10549922365
x = 191.483032698
x = 191.5 ft
learn more on right triangle here: https://brainly.com/question/6322314
