To find the distance from the observer to the base of the tower, we can use the tangent function from trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
In this case:
- The height of the tower (opposite side) is 200 feet.
- The angle of elevation from the observer to the top of the tower is 65°.
Using the tangent function:
\[ \tan(65^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{200}{d} \]
where \( d \) is the distance from the observer to the base of the tower.
We can rearrange this equation to solve for \( d \):
\[ d = \frac{200}{\tan(65^\circ)} \]
Now we can calculate \( \tan(65^\circ) \):
\[ \tan(65^\circ) \approx 2.1445 \]
Now plug this into our equation for \( d \):
\[ d \approx \frac{200}{2.1445} \approx 93.23 \]
So, the distance from the observer to the base of the tower is approximately 93.23 feet.