To find the distance from the observer to the base of the tower, we can use the tangent function from trigonometry, which relates the height of the tower and the distance to the observer.
The formula is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In this case, the opposite side is the height of the tower (200 feet), and the adjacent side is the distance from the observer to the base of the tower (let's call it \( d \)). So we can rearrange the formula to solve for \( d \):
\[ d = \frac{\text{opposite}}{\tan(\theta)} \]
Substituting the values we know:
\[ d = \frac{200}{\tan(65^\circ)} \]
Now, we will calculate \( \tan(65^\circ) \) first:
\[ \tan(65^\circ) \approx 2.1445 \]
Now, plug this back into the equation:
\[ d = \frac{200}{2.1445} \approx 93.26 \text{ ft} \]
So the distance from the observer to the base of the tower is approximately 93.26 ft.
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