To find the height of the tower using the sine ratio, we can use the triangle formed by Ella’s eye level, the top of the tower, and the vertical line from Ella's eye level to the ground directly beneath the top of the tower.
We know:
- The direct distance from her eyes to the top of the tower (hypotenuse) is 46 feet.
- The angle (θ) formed with the eye level and the line of sight to the top of the tower is 21°.
- Ella's eye level is at 5.5 feet above the ground.
Using the sine ratio:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, the "opposite" side is the height difference from Ella's eyes to the top of the tower, and the hypotenuse is the direct distance to the top of the tower.
So we can write:
\[ \sin(21°) = \frac{h}{46} \]
Where \( h \) is the vertical height from Ella's eye level to the top of the tower. First, we can calculate \( h \):
\[ h = 46 \cdot \sin(21°) \]
Now, we calculate \( h \):
Using a calculator for \( \sin(21°) \), we find:
\[ \sin(21°) \approx 0.3584 \]
Now plug this value in:
\[ h = 46 \cdot 0.3584 \approx 16.5 \text{ feet} \]
Now to find the total height of the tower from the ground, we need to add Ella's eye level:
Total height of the tower = \( h + \text{eye level} = 16.5 + 5.5 = 22.0 \text{ feet} \)
So the final answer, using the sine ratio to find the height from her eyes up to the top of the tower, is:
16.5 feet.
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