The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft above 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 θ = 36°. She also observes that the angle between the vertical and the line of sight to one of the landmarks is a = 61° and to the other landmark is b = 49°. Find the distance between the two landmarks. (Round your answer to the nearest foot.)

To find the distance between the two landmarks, we can use trigonometry and the concept of similar triangles.

Let's label the distance between the woman on the observation deck and one of the landmarks as x, and the distance between the woman and the other landmark as y.

Since the CN Tower is a tall structure and the landmarks are on the ground, we can assume that the lines of sight from the woman to the landmarks are parallel to each other.

Now, let's consider the angle of elevation from the woman to the first landmark. This angle, a = 61°, is the angle between the vertical and the line of sight. Similarly, the angle of elevation from the woman to the second landmark, b = 49°.

From the information given, we can form two right triangles, one for each landmark. In these triangles, the side opposite to the angle of elevation (x and y) represents the height difference between the woman's observation deck and the ground at each landmark.

We can now write the following trigonometric ratios:

For the first landmark:
tan(a) = x / 1150

For the second landmark:
tan(b) = y / 1150

To find x and y, we can rearrange these equations:
x = 1150 * tan(a)
y = 1150 * tan(b)

Now, let's find the horizontal distance between the two landmarks. We know that the angle formed by the lines of sight to the landmarks is θ = 36°. The horizontal distance between the two landmarks can be calculated using the following trigonometric ratio:

tan(θ) = (x - y) / h

Since the horizontal distance between the landmarks is the same, we can set x - y = h.

Therefore, we have:
tan(36°) = h / h

Since tan(36°) = 0.7265 (approximately), we can write:
0.7265 = h / h

Simplifying this equation, we get:
1 = h / h

So, the horizontal distance h is equal to 1.

To find the distance between the two landmarks, we need to calculate the magnitude of x - y.

Using the values obtained earlier, we have:
x - y = 1150 * tan(a) - 1150 * tan(b)

Substituting the given values for angles a and b, we get:
x - y = 1150 * tan(61°) - 1150 * tan(49°)

Now, we can substitute the values of these tangents into the equation above:
x - y = 1150 * 1.9548 - 1150 * 1.1918

Simplifying this equation, we find:
x - y ≈ 1230.82 - 1373.27

Therefore,
x - y ≈ -142.45 ft

Since we are interested in the distance between the two landmarks, we take the absolute value:
|x - y| ≈ |-142.45| ≈ 142.45 ft

Hence, the distance between the two landmarks is approximately 142.45 feet.

approximate 1802 ft