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ANSWER:
191.5 feet
STEP-BY-STEP EXPLANATION:
According to the statement, we can plan the following graph that represents the situation:
We can see two right triangles in the figure graph ABC and DBC. Where the width of the river would be x.
We can calculate the value of x with the help of the tangent trigonometric ratio, just like that.
[tex]\begin{gathered} \tan \theta=\frac{\text{ opposite}}{\text{ adjacent}} \\ \text{For ABC} \\ \tan 22=\frac{CB}{AB} \\ \tan 22=\frac{CB}{50+x}\rightarrow CB=\tan 22\cdot(50+x)\text{ (1)} \\ \text{For DBC} \\ \tan 27=\frac{CB}{DB} \\ \tan 27=\frac{CB}{x}\rightarrow CB=\tan 27\cdot x\text{ (2)} \end{gathered}[/tex]We match both equations so that you can calculate the value of x:
[tex]\begin{gathered} \tan 22\cdot(50+x)=\tan 27\cdot x \\ 50\tan 22+x\tan 22=x\tan 27 \\ x\tan 27-x\tan 22=50\tan 22 \\ x\cdot(\tan 27-\tan 22)=50\tan 22 \\ x=\frac{50\tan 22}{(\tan 27-\tan 22)} \\ x=191.5\text{ ft} \end{gathered}[/tex]Therefore, the wide of the river is 191.5 feet.
