Answer:
**1. Sketch of Leaning Tower of Pisa:**
```
|
|
|
|
__________|__________ Ground
|\
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
|__________\
```
Height: 57 meters
Angle with ground: 86 degrees
**2. Right Triangle Diagram:**
```
|
|\
| \ keys
| \
| \
| \
| \
| \
| \
|________\
| base
```
**3. Known side:** Opposite side (height of tower)
Unknown side: Adjacent side (distance from base to tower)
**4. Trigonometric ratio:** Tan(theta) = Opposite / Adjacent
Calculation: Tan(86°) = 57m / x (where x is the distance from base to tower)
x = 57m / Tan(86°)
x ≈ 9.47 meters
**5. Distance the ant would travel:**
The ant would travel the same distance as the height of the tower, which is 57 meters.
Trigonometric ratio used: None needed, as the distance is directly given by the height of the tower.
**6. Confirming with Pythagorean Theorem:**
Using Pythagorean Theorem,
\( \text{Hypotenuse}^2 = \text{Height}^2 + \text{Base}^2 \)
\( \text{Base}^2 = \text{Hypotenuse}^2 - \text{Height}^2 \)
\( \text{Base} = \sqrt{\text{Hypotenuse}^2 - \text{Height}^2} \)
\( \text{Base} = \sqrt{57^2 - 57^2} \)
\( \text{Base} = \sqrt{0} = 0 \)
This confirms that the ant travels a vertical distance equal to the height of the tower.
**7. Leaning Tower of Niles:**
Height: 94 feet
Angle: 85.5 degrees
Using the same trigonometric ratio as before:
\( x = 94 \text{ feet} / \text{Tan}(85.5°) \)
\( x ≈ 15.51 \text{ feet} \)
So, the keys would land approximately 15.51 feet from the base of the Leaning Tower of Niles.
