Question 1171450: The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft abouve the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is 43°. She also observes that the angle between the vertical and the line of sight to one of the landmarks is 62° and that to the other landmark is 54°. Find the distance between the two landmarks.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step using trigonometry.
**1. Visualize the Situation**
Imagine the CN Tower as a vertical line. The woman is at the top, and the two landmarks are on the ground. We have a triangle formed by the woman and the two landmarks.
**2. Define Variables**
* Let the height of the observation deck be `h = 1150 ft`.
* Let the angle between the lines of sight to the landmarks be `θ = 43°`.
* Let the angle between the vertical and the line of sight to the first landmark be `α = 62°`.
* Let the angle between the vertical and the line of sight to the second landmark be `β = 54°`.
* Let the distance from the base of the tower to the first landmark be `d1`.
* Let the distance from the base of the tower to the second landmark be `d2`.
* Let the distance between the two landmarks be `D`.
**3. Calculate Distances to Landmarks (d1 and d2)**
We can use the tangent function to relate the height and distances:
* `tan(α) = d1 / h`
* `d1 = h * tan(α) = 1150 * tan(62°) ≈ 1150 * 1.8807 ≈ 2162.8 ft`
* `tan(β) = d2 / h`
* `d2 = h * tan(β) = 1150 * tan(54°) ≈ 1150 * 1.3764 ≈ 1582.9 ft`
**4. Apply the Law of Cosines**
Now, we have a triangle with sides `d1` and `d2`, and the angle between them is `θ = 43°`. We want to find the distance `D` between the landmarks. We can use the Law of Cosines:
* `D² = d1² + d2² - 2 * d1 * d2 * cos(θ)`
* `D² = (2162.8)² + (1582.9)² - 2 * 2162.8 * 1582.9 * cos(43°) `
* `D² ≈ 4677703.84 + 2505572.41 - 2 * 2162.8 * 1582.9 * 0.73135`
* `D² ≈ 7183276.25 - 5005813.1`
* `D² ≈ 2177463.15`
* `D ≈ √2177463.15 ≈ 1475.6 ft`
**Therefore, the distance between the two landmarks is approximately 1475.6 feet.**
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