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To solve this problem, we can use trigonometry. Here’s a breakdown of the information given and the steps to solve it:

1. **Given Information:**
   - Distance between the observer and the base of the tower (horizontal distance): \( 90 \) feet.
   - Angle of elevation from the observer to the top of the tower: \( 25^\circ \).
   - We need to find the distance between the observer's eye and the top of the tower, which is the hypotenuse of the right triangle formed.

2. **Using Trigonometry:**
   Let:
   - \( d \) be the distance we need to find (the hypotenuse of the triangle).
   - \( \theta = 25^\circ \) is the angle of elevation.
   - \( \text{adjacent side} = 90 \) feet.

   Since we are dealing with the hypotenuse and the adjacent side, we use the **cosine** function:
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
   \]
   Plugging in the values:
   \[
   \cos(25^\circ) = \frac{90}{d}
   \]

3. **Solving for \( d \):**
   Rearrange the equation to isolate \( d \):
   \[
   d = \frac{90}{\cos(25^\circ)}
   \]

4. **Calculate the Value:**
   Using a calculator to evaluate \( \cos(25^\circ) \):
   \[
   d = \frac{90}{\cos(25^\circ)} \approx \frac{90}{0.9063} \approx 99.29
   \]

So, the distance between the observer's eye and the top of the tower, rounded to the nearest hundredth, is approximately:
\[
\boxed{99.29 \text{ feet}}
\]

Would you like more details or have any questions? Here are some related questions to consider:

1. What would be the height of the tower?
2. How would the solution change if the angle of elevation were different?
3. What trigonometric function would we use if we only had the opposite side?
4. How can we find the angle of elevation if we only have distances?
5. Can this problem be solved using the Pythagorean theorem if we know the height?

**Tip:** When solving right triangle problems, choose the trigonometric function (sine, cosine, or tangent) that involves the sides you know and the side you need to find.

When an observer looks at the top of a tower from 90 feet away, the angle of elevation is 25°. Determine the distance between the observer’s eye and the top of the tower, rounded to the nearest 100th.

Learn how to calculate the distance between an observer's eye and the top of a tower using trigonometry. This problem involves a 25° angle of elevation from 90 feet away and applies cosine functions to find the hypotenuse of a right triangle. Suitable for students in Grades 9-10.

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