Question

12.) from a 200 - foot observation tower on the beach, a man sights a whale in difficulty. the angle of depression of the whale is 7°. how far is the whale from the shoreline?

Explanation:

Step1: Identify right - triangle relationship

We have a right - triangle where the height of the tower is the opposite side and the distance from the shoreline to the whale is the adjacent side with respect to the angle of depression. The angle of depression is equal to the angle of elevation from the whale to the top of the tower. Let the distance from the shoreline to the whale be $x$. The height of the tower $h = 200$ feet and the angle of elevation $\theta=7^{\circ}$.
We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Apply the tangent formula

Using the formula $\tan\theta=\frac{h}{x}$, where $\theta = 7^{\circ}$ and $h = 200$. So, $\tan(7^{\circ})=\frac{200}{x}$.
We can solve for $x$: $x=\frac{200}{\tan(7^{\circ})}$.
Since $\tan(7^{\circ})\approx0.1228$, then $x=\frac{200}{0.1228}\approx1628.7$.

Answer:

The whale is approximately $1628.7$ feet from the shoreline.