A man 1.5m tall, standing on top of a mountain 298.5m high, observes the angles of depression of two flying boats D and C to be 28 degrees and 34 degrees respectively. Calculate the distance between the boats, correct to 1 decimal place.
Hint:
My sketch has two right-angled triangles with the same height
Think parallel lines and "transversals"
Can you find the bases of these two triangles?
Locate point A at base of mountain,
Tan34 = (h1+h2)/AC = (298.5+1.5)/AC,
AC*Tan34 = 300,
AC = 445 m. = Distance of boat C from base of mountain.
Tan28 = 300/AD,
AD = 564 m. = Distance of boat D from base of mountain.
AC + CD = AD,
445 + CD = 564,
CD = Distance between the boats.
To find the distance between the boats, we can use the tangent function. Let's assume the distance between the man's position and boat D is x, and the distance between the man's position and boat C is y.
We can set up the following tangent equations based on the angles of depression:
For boat D:
tan(28°) = (298.5 + 1.5) / x
For boat C:
tan(34°) = (298.5 + 1.5) / y
Simplifying these equations, we have:
x = (300 / tan(28°))
y = (300 / tan(34°))
Now we can calculate the distances:
x ≈ (300 / tan(28°)) ≈ 583.8 meters (rounded to 1 decimal place)
y ≈ (300 / tan(34°)) ≈ 497.7 meters (rounded to 1 decimal place)
Finally, we can find the distance between the boats by taking the difference between x and y:
Distance between the boats = |x - y| ≈ |583.8 - 497.7| ≈ 86.1 meters (rounded to 1 decimal place)
Therefore, the distance between boat D and boat C is approximately 86.1 meters.
To solve this problem, we can use trigonometry, specifically the tangent function.
Let's start by drawing a diagram to visualize the situation:
A (Top of mountain)
/ \
/ \
/ \
/ \
/ \
/ \
D C
(Boat D) (Boat C)
We have a right triangle formed by the man on top of the mountain, the boat D, and a line perpendicular to the ground. The angle of depression from the man to boat D is 28 degrees.
We can also see a right triangle formed by the man on top of the mountain, the boat C, and a line perpendicular to the ground. The angle of depression from the man to boat C is 34 degrees.
Now, let's calculate the distance between the boats. We'll use the following steps:
Step 1: Calculate the height of the observer's eye above the ground.
Since the man is 1.5m tall and standing on top of a mountain that is 298.5m high, the total height of the observer's eye above the ground is 298.5m + 1.5m = 300m.
Step 2: Calculate the distance between the observer and each boat.
To do this, we can create separate right triangles for each boat.
In the triangle formed by the observer (A), the perpendicular line (B), and boat D (D), the angle of depression is 28 degrees.
In this triangle, we want to find the adjacent side, which is the distance between the observer and boat D.
Using the tangent function, we can write:
tan(28) = opposite side (boat D) / adjacent side (distance between observer and boat D)
Since the opposite side is the height of the observer's eye above the ground (300m) and the angle of depression is 28 degrees, we can write:
tan(28) = 300 / adjacent side (distance between observer and boat D)
Rearranging the equation to solve for the adjacent side:
adjacent side (distance between observer and boat D) = 300 / tan(28)
Using a scientific calculator, we can calculate:
adjacent side (distance between observer and boat D) ≈ 547.1m (rounded to 1 decimal place)
Similarly, for boat C, we use the same steps, but with the angle of depression of 34 degrees:
adjacent side (distance between observer and boat C) = 300 / tan(34)
adjacent side (distance between observer and boat C) ≈ 436.4m (rounded to 1 decimal place)
Step 3: Calculate the distance between boat D and boat C.
Now that we have the distances between the observer and each boat, we can find the distance between the two boats by subtracting the distance to boat C from the distance to boat D:
Distance between the boats = Distance to boat D - Distance to boat C
= 547.1m - 436.4m
≈ 110.7m (rounded to 1 decimal place)
Therefore, the distance between the boats is approximately 110.7 meters.