Two observers are 200 feet apart,in line with a tree containing a birds nest.The anles of elevation to the birds nest are 30 degrees and 60 degrees.How far is each observer from the base of the tree? Please help me my assigment is due tommorrow!!!!!

To solve this problem, we can use trigonometry. Let's denote the distance between each observer and the base of the tree as x and y.

First, let's consider the observer who has an angle of elevation of 30 degrees. We can start by drawing a triangle with the tree's base, the tree's top (where the bird's nest is), and the observer. This forms a right triangle.

30°
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x

Now, let's consider the observer who has an angle of elevation of 60 degrees. Similarly, we can draw a right triangle with the tree's base, the tree's top, and the observer.

60°
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We know that the sum of the two distances x and y is equal to 200 feet since the observers are 200 feet apart.

x + y = 200 (Equation 1)

Using trigonometry, we know that the length of the tree can be calculated as follows:

Length of the tree = (Distance to the observer) / tan(angle of elevation)

For the observer with an angle of elevation of 30 degrees:

Length of the tree = x / tan(30°) = x / √3

And for the observer with an angle of elevation of 60 degrees:

Length of the tree = y / tan(60°) = y / √3

Since both observers are looking at the same tree, the length of the tree should be the same for both equations.

x / √3 = y / √3

We can simplify this equation to:

x = y (Equation 2)

Now we have two equations (Equation 1 and Equation 2) with two variables (x and y). We can solve this system of equations to find the values of x and y.

First, substitute Equation 2 into Equation 1:

x + x = 200

2x = 200

Solving for x:

x = 200/2

x = 100 feet

Since x = y (from Equation 2), this means y is also 100 feet.

Therefore, each observer is 100 feet away from the base of the tree.

150ft and 50ft