To find the distance between the tourist and her friend on the ground, we can use the concept of right triangles and trigonometric functions.
Given:
- Height of the observation deck (h) = 520 ft
- Angle of depression = 80.2°
The angle of depression from the observation deck to the friend on the ground is the same as the angle of elevation from the friend to the deck.
Let:
- d = horizontal distance from the friend to the base of the Space Needle.
- We can use the tangent function, which relates the angle of elevation (θ) to the opposite side (height) and the adjacent side (horizontal distance).
Using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \]
We can rearrange this equation to find d: \[ d = \frac{h}{\tan(\theta)} \]
Substituting the values: \[ d = \frac{520}{\tan(80.2°)} \]
Calculating \(\tan(80.2°)\): \[ \tan(80.2°) \approx 5.747 \]
Now substituting back into the equation: \[ d = \frac{520}{5.747} \approx 90.5 \text{ ft} \]
To find the straight-line distance (s) from the tourist to her friend, we can use the Pythagorean theorem: \[ s^2 = h^2 + d^2 \]
Calculating: \[ s^2 = (520)^2 + (90.5)^2 \] \[ s^2 = 270400 + 8190.25 \] \[ s^2 = 278590.25 \] \[ s \approx \sqrt{278590.25} \approx 528.2 \text{ ft} \]
Rounding to the nearest tenth, the distance between the tourist and her friend is approximately 528.2 ft.
However, looking at the answer choices provided, it seems like the answer isn't listed exactly. The closest option would be \(527.7\) ft.
Thus, the best choice is: 527.7 ft.
1 answer