From the statement of the problem, we know that:
• Brian looks directly across and sees a pier,
,• he walks downstream 400 ft, and looks at the pier again,
,• he is now at an angle θ = 82°.
Using the data of the problem, we make the following diagram:
Where x is the distance across the river.
The diagram constitutes a triangle of:
• angle ,θ = 82°,,
,• opposite side ,OC = 400 ft,,
,• adjacent side ,AC = x,.
From trigonometry, we have the following trigonometric relation:
[tex]\tan \theta=\frac{OC}{AC}\text{.}[/tex]Replacing the data above in the last equation, we have:
[tex]\tan (82^{\circ})=\frac{400ft}{x}.[/tex]Solving for x the last equation, we find that:
[tex]\begin{gathered} x\cdot\tan (82^{\circ})=400ft, \\ x=\frac{400ft}{\tan(82^{\circ})}, \\ x\cong56.21633ft\cong56ft\text{.} \end{gathered}[/tex]Answer
The distance across the river is approximately 56 ft.
