From the top of a cliff 150m high, the angles of depression of two boats are 60 and 30. Find the distance between the boats if they are on the opposite sides of the cliff.
x+y
tan 30 = x/150
x = 150 tan 30
tan 60 = y/150
y 150 tan 60
so
d = 150 (tan 30 + tan 60)
d = 150 (1/sqrt 3 + sqrt 3 /2)
= 150 (2 + 3)/2sqrt 3
= 375/sqrt 3
= 216.5
To find the distance between the two boats, we can use trigonometry.
Let's start by visualizing the situation. We have a cliff, and the angles of depression of the two boats are given as 60 degrees and 30 degrees. The angles of depression are the angles formed between the line of sight from the top of the cliff to the boats and the horizontal line.
To solve this question, we'll use the tangent function. The tangent of an angle can be defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the height of the cliff (150m) and the adjacent side is the distance between the boats.
For the boat with a 60-degree angle of depression, we can construct a right triangle with the adjacent side being the distance between the boats and the opposite side being the height of the cliff. Similarly, for the boat with a 30-degree angle of depression, we can construct another right triangle with the same height of the cliff but a different adjacent side (the distance between the boats).
Let's label the distance between the boats as 'x'.
In the first right triangle (with a 60-degree angle of depression), the tangent of 60° is equal to the opposite side (150m) divided by the adjacent side (x). So we have:
tan(60°) = 150 / x
Similarly, in the second right triangle (with a 30-degree angle of depression), the tangent of 30° is equal to the opposite side (150m) divided by the adjacent side (x + distance between the boats). We assume that the boats are on opposite sides of the cliff, so the distance between them is x. Thus, we have:
tan(30°) = 150 / (x + x)
To solve for x, we need to find the value of x that satisfies both equations. Let's go ahead and solve the equations simultaneously.
tan(60°) = 150 / x
tan(30°) = 150 / (2x)
Using trigonometric identities, we know that tan(60°) = sqrt(3), and tan(30°) = 1/sqrt(3). Substituting these values, we have:
sqrt(3) = 150 / x
1/sqrt(3) = 150 / (2x)
Now, let's solve these equations:
From the first equation, we can rearrange to solve for x:
x = 150 / sqrt(3)
From the second equation, we can rearrange to solve for x:
x = 150 / (2 * 1/sqrt(3))
x = (150 * sqrt(3)) / 2
Simplifying the equation, we get:
x = 75 * sqrt(3)
So, the distance between the boats, when they are on opposite sides of the cliff, is 75 * sqrt(3) meters.