bernhardt and julia are observing an eagals nest in a tree. Julia is 75 m from the tree, and sees it an angle of elevation of 42 degrees.
A) How high up the tree is the nest?
B) bernhardt is standing 30 m behind Julia. At what angle of elevation does he see the nest?
To solve this problem, we can use trigonometry. Let's start with Part A:
A) How high up the tree is the nest?
Step 1: Draw a diagram to visualize the situation.
Let's represent the height of the tree and the distance between Julia and the tree with variables.
Let h represent the height of the tree.
Let d represent the distance between Julia and the tree.
E
/|
/ |
h / | d
/___|
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Step 2: Identify the relevant trigonometric ratio to solve the problem.
We are given that Julia sees the tree at an angle of elevation of 42 degrees. Since we are looking for the height of the tree, we can use the tangent ratio.
Step 3: Set up the trigonometric equation.
The tangent of an angle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the tree and the adjacent side is the distance between Julia and the tree.
Using the formula: tan(angle) = opposite/adjacent
tan(42 degrees) = h/d
Step 4: Calculate the height of the tree.
Rearranging the equation, we can solve for h:
h = d * tan(42 degrees)
Since Julia is 75 m away from the tree, we can substitute d = 75 m into the equation:
h = 75 m * tan(42 degrees)
Calculate the value of tan(42 degrees) using a calculator:
h ≈ 75 m * 0.9004
h ≈ 67.53 m
Therefore, the nest is approximately 67.53 meters high up the tree.
Now let's move on to Part B.
B) Bernhardt is standing 30 m behind Julia. At what angle of elevation does he see the nest?
Step 5: Determine the relevant trigonometric ratio for Part B.
Bernhardt is standing 30 meters behind Julia, so the distance between Bernhardt and the tree is d + 30 m.
We can use the tangent ratio again to find the angle of elevation for Bernhardt.
Step 6: Set up the trigonometric equation.
tan(angle) = opposite/adjacent
We want to find the angle of elevation for Bernhardt. The opposite side is still the height of the tree, and the adjacent side is now d + 30 m.
So, tan(angle) = h / (d + 30)
Step 7: Calculate the angle of elevation for Bernhardt.
Rearrange the equation to solve for the angle:
angle = arctan(h / (d + 30))
Substitute the values of h and d into the equation:
angle = arctan(67.53 m / (75 m + 30 m))
Calculating this with a calculator, we get:
angle ≈ arctan(67.53 / 105)
angle ≈ 33.2 degrees
Therefore, Bernhardt sees the nest at an angle of elevation of approximately 33.2 degrees.
To find the height of the nest in the tree, we can use trigonometry. Let's use the right triangle formed by the tree, the height of the nest, and Julia's line of sight.
A) To determine the height of the nest, we need to find the length of the side opposite the angle of elevation. We know the distance between Julia and the tree is 75 m, and the angle of elevation is 42 degrees.
Using the trigonometric function tangent (tan), we can set up the following equation:
tan(42 degrees) = height of the nest / 75 m
Now, we can solve for the height of the nest:
height of the nest = 75 m * tan(42 degrees)
Plugging this into a calculator, we find that the height of the nest is approximately 65.51 m.
B) Now, let's find the angle of elevation at which Bernhardt sees the nest. Bernhardt is standing 30 m behind Julia. Since Bernhardt is behind Julia, the distance between Bernhardt and the tree is 75 m + 30 m = 105 m.
Let's say the angle of elevation Bernhardt sees is θ. We can use the same trigonometric function, tangent (tan), to set up the following equation:
tan(θ) = height of the nest / 105 m
We want to solve for θ, so we need to rearrange the equation:
θ = arctan(height of the nest / 105 m)
Plugging in the previously calculated height of the nest (65.51 m), we find that the angle of elevation Bernhardt sees is approximately 32.73 degrees.