Question

Standing on one bank of a river, an explorer measures the angle to the top of a tree on the opposite bark to be 27 degrees. He backs up 50 feet and measures theangle to the top of the tree to now be 22 degrees. How wide is the river? Round to the nearest tenth of a foot202 feet18.7 feet1873 feet1915 feet

Answer 1

`x=191.48\approx191.5\text{ ft}`

Explanation:

To approach this situation we can draw the diagram that represents it:

Where the wide of the river would be x. We can solve this situation using trigonometric ratios, in this case, the tangent of 22 degrees to find the height of the tree:

`\begin{gathered} \tan (\theta)=(opposite)/(adjacent) \\ \tan (22)=(h)/(50+x) \end{gathered}`

Solve for the height.

`h=\tan (22)\cdot(50+x)\text{ (Equation 1)}`

Now, find the second equation with the other triangle and solve the system.

`\begin{gathered} \tan (27)=(h)/(x) \\ h=x\cdot\tan (27)\text{ (Equation 2)} \end{gathered}`

To solve the system, equalize the 2 equations and solve for x:

`\begin{gathered} \tan (22)\cdot(50+x)=x\cdot\tan (27) \\ 50\cdot\tan (22)+x\tan (22)=x\tan (27) \\ 50\cdot\tan (22)=x\tan (27)-x\tan (22) \\ 50\cdot\tan (22)=x(\tan (27)-\tan (22)) \\ x=(50\cdot\tan (22))/(\tan (27)-\tan (22)) \\ x=191.48\approx191.5\text{ ft} \end{gathered}`