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.
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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: \[ h - h_w = d \tan(35°) \tag{1} \]
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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 \( d \).
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Substituting equation (2) into equation (1): \[ h - d \tan(29°) = d \tan(35°) \] Rearranging gives: \[ h = d \tan(35°) + d \tan(29°) \] \[ h = d(\tan(35°) + \tan(29°)) \tag{3} \]
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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 \( h_w \).
- 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): \[ h_w = 100 \cdot 0.5543 = 55.43 \text{ feet, and} \] \[ h = 100 \cdot 1.2545 = 125.45 \text{ feet.} \]
- 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.