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To find the height of the Eiffel Tower using the given trigonometric information, let's analyze the problem step by step.

### Given Information:
- The distances and angles are provided:
  - The angle of elevation at 30° from a point 100 meters from the base.
  - The angle of elevation at 45° from another point closer to the base.

### Steps to Solve:

1. **Label the Diagram**:  
   - Let \( h \) be the height of the Eiffel Tower.
   - Let \( d \) be the horizontal distance from the point directly below the Eiffel Tower where the 45° angle is measured.
   - The total distance from the 30° measurement point to the base of the Eiffel Tower is \( d + 100 \) meters.

2. **Using Trigonometry at 45°**:
   - From the point where the angle of elevation is 45°:
     \[
     \sin 45^\circ = \frac{h}{d}
     \]
     Given \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), we have:
     \[
     \frac{\sqrt{2}}{2} = \frac{h}{d}
     \]
     Therefore, \( h = \frac{d\sqrt{2}}{2} \).

3. **Using Trigonometry at 30°**:
   - From the point where the angle of elevation is 30°:
     \[
     \sin 30^\circ = \frac{h}{d + 100}
     \]
     Given \( \sin 30^\circ = \frac{1}{2} \), we have:
     \[
     \frac{1}{2} = \frac{h}{d + 100}
     \]
     Therefore, \( h = \frac{d + 100}{2} \).

4. **Equating Heights**:
   - From the two equations, equate the two expressions for \( h \):
     \[
     \frac{d\sqrt{2}}{2} = \frac{d + 100}{2}
     \]
     Simplify this equation:
     \[
     d\sqrt{2} = d + 100
     \]
     Solve for \( d \):
     \[
     d(\sqrt{2} - 1) = 100
     \]
     \[
     d = \frac{100}{\sqrt{2} - 1}
     \]
     Rationalize the denominator:
     \[
     d = \frac{100(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} = \frac{100(\sqrt{2} + 1)}{2 - 1} = 100(\sqrt{2} + 1)
     \]

5. **Calculate Height**:
   - Substitute \( d = 100(\sqrt{2} + 1) \) back into the expression for \( h \) derived from the 45° angle:
     \[
     h = \frac{d\sqrt{2}}{2} = \frac{100(\sqrt{2} + 1)\sqrt{2}}{2}
     \]
     Simplify the expression:
     \[
     h = 50\sqrt{2}(\sqrt{2} + 1) = 50(2 + \sqrt{2}) = 100 + 50\sqrt{2}
     \]

### Final Answer:
The height of the Eiffel Tower is \( 100 + 50\sqrt{2} \) meters in surd form.

Would you like further details or have any questions?

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**Related Questions:**
1. How do you derive the trigonometric identities used in this problem?
2. What are the steps to rationalize the denominator in a surd expression?
3. How do different angles of elevation affect the calculation of height in trigonometric problems?
4. Can you solve a similar problem involving a different set of angles and distances?
5. How does the Pythagorean Theorem relate to trigonometric calculations in right triangles?

**Tip:** When dealing with trigonometric problems, always try to express unknown quantities in terms of one variable using known trigonometric identities and then solve the equations systematically.

What is the height of the Eiffel Tower given the angles of elevation and distance, and express the answer in surd form?

Learn how to calculate the height of the Eiffel Tower using angles of elevation and trigonometric identities. The problem involves applying basic trigonometric ratios and expressing the final answer in surd form. Suitable for students in Grades 10-12.

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