While camping by the Merced River in Yosemite Valley, a group of friends were admiring a particular tree on the opposite bank. Maria claimed that the height of the tree could be determined from the group’s side of the river by the following method:
• Measure a 50 meter segment,, on this shore.
• Considering the tree to be located at point C, measure and .
• From A, measure the angle of evaluation to the top of the tree.
• Use the measurement below with some trigonometric ratio to calculate the height.
The friends measured to be and to be . The angle of elevation from point A was .
Draw a sketch of the situation and determine whether Maria was correct. If she was, compute the height; if not, explain why not.
Based on the given information, Maria claimed that the height of the tree can be determined using some trigonometric ratios. In order to verify her claim, we need to draw a sketch of the situation and calculate the height of the tree.
Let's label the points on the sketch:
B C
|-----------------T
| /
| /θ
| /
A
- A represents the location where the angle of elevation to the top of the tree is measured.
- B represents the starting point of the 50-meter segment on the shore.
- C represents the location of the tree.
- T represents the top of the tree.
- θ represents the angle of evaluation from point A.
Now, let's calculate the height of the tree using trigonometric ratios. We are given the following measurements:
- AB = 50 meters (the segment on the shore)
- AC = 400 meters (the distance from point A to the tree)
- θ = 60 degrees (the angle of elevation from point A)
To calculate the height of the tree, we need to set up the following trigonometric ratio:
tan(θ) = height of the tree / distance from point A to the tree
Using the given measurements:
tan(60 degrees) = height of the tree / 400 meters
tan(60 degrees) ≈ 1.732 (approximately)
height of the tree = 1.732 * 400 meters
height of the tree ≈ 692.8 meters
Therefore, Maria was correct, and the height of the tree is approximately 692.8 meters.
To determine if Maria's claim is correct, we need to analyze the situation and use trigonometric ratios.
First, let's draw a sketch of the situation. We have a river running horizontally, with two shores: the one where the group is standing and the opposite shore where the tree is located. Let's label the group's side of the shore as "Shore A" and the opposite shore as "Shore B". Point C represents the location of the tree.
B
------------------------------------------
| /\
| C / \
| / \
| / A \
------------------------------------------
Merced River
The group measures a 50-meter segment on Shore A and finds that segment AB to be 60 meters, and segment BC to be 20 meters. The angle of elevation from point A to the top of the tree is 30 degrees.
Now, let's determine whether Maria's claim is correct.
We can use the trigonometric ratio tangent to calculate the height of the tree. The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the height of the tree (h), and the adjacent side is segment BC (20 meters).
So, the equation becomes:
tan(30 degrees) = h / 20
To solve for h, we can rearrange the equation:
h = tan(30 degrees) * 20
Using a scientific calculator, we can find that the tangent of 30 degrees is approximately 0.5774.
So, h ≈ 0.5774 * 20
h ≈ 11.548 meters
Therefore, Maria's claim is correct. The height of the tree is approximately 11.548 meters.