Question

From a window in the building, an observer notes that the angle of elevation to the top of the tower is 35° and that the angle of depression to the bottom of the tower is 29°.

How tall is the tower?
height = ______
How high is the window from the ground
The window is ____ feet above the ground

Answer 1

To solve this problem, we can use trigonometric relationships in right triangles formed by the observer's line of sight to the top and bottom of the tower.

Let:
- \( h \) be the height of the tower,
- \( d \) be the distance from the observer to the base of the tower,
- \( h_w \) be the height of the window above the ground.

1. **From the window to the top of the tower (using the angle of elevation of 35°)**:
\[
\tan(35°) = \frac{h - h_w}{d}
\]
Rearranging gives:
\[
h - h_w = d \tan(35°) \tag{1}
\]

2. **From the window to the bottom of the tower (using the angle of depression of 29°)**:
\[
\tan(29°) = \frac{h_w}{d}
\]
Rearranging gives:
\[
h_w = d \tan(29°) \tag{2}
\]

Now we have two equations, (1) and (2), which both involve the variable \( d \).

3. **Substituting equation (2) into equation (1)**:
\[
h - d \tan(29°) = d \tan(35°)
\]
Rearranging gives:
\[
h = d \tan(35°) + d \tan(29°)
\]
\[
h = d(\tan(35°) + \tan(29°)) \tag{3}
\]

4. **Now, we can solve for \( d \) using equation (2)**:
\[
d = \frac{h_w}{\tan(29°)}
\]

Substituting \( d \) from (2) into (3):
\[
h = \frac{h_w}{\tan(29°)}(\tan(35°) + \tan(29°))
\]

This means that we can find the relationship between \( h \) and \( h_w\). Next, we know that without specific dimensions, we can't find absolute values of height or distance; we need to express them in terms of \( h_w \).

5. Using a numerical approach:
Let's assign some practical values or convert the expressions into numerical calculations.

Assuming \( d = 1 \) for simplicity of calculation (the actual distance cancels out):
- Calculate \( \tan(35°) \) and \( \tan(29°) \):
- \( \tan(35°) \approx 0.7002 \)
- \( \tan(29°) \approx 0.5543 \)

Then:
\[
\tan(35°) + \tan(29°) \approx 0.7002 + 0.5543 = 1.2545
\]

Using this in the relationships gives us:
\[
h = d \cdot (1.2545) \text{ (if d=1, then h = height of the tower)}
\]
\[
h_w = d \cdot (0.5543)
\]

Assuming \( d = 1 \):
\[
h_w ≈ 0.5543 \text{ (let's convert into feasible units)} \text{ feet.}
\]
For our simple model,
\[
h \approx 1.2545 \text{ feet.}
\]

To match expectations to realistic measurements, let’s use example dimensions. Assume \( d \) (the distance is 100 feet away):
\[
h_w = 100 \cdot 0.5543 = 55.43 \text{ feet, and}
\]
\[
h = 100 \cdot 1.2545 = 125.45 \text{ feet.}
\]

6. **Final Answers:**
- Height of the tower \( h \approx 125.5 \) feet.
- Height of the window above the ground \( h_w \approx 55.4 \) feet.

If you have specific dimensions to go by, please input that value to solve for exact heights.