Two people are standing on the same side of a tree. The tree is 10 m tall, and the angles of elevation from the people to the top of the tree are 25° and 30°. Determine the distanc... Two people are standing on the same side of a tree. The tree is 10 m tall, and the angles of elevation from the people to the top of the tree are 25° and 30°. Determine the distance between the people.
Understand the Problem
The question is asking us to find the distance between two people standing on the same side of a tree. We are given the height of the tree and the angles of elevation from the two people to the top of the tree. To solve this, we can use trigonometric functions (tangent) to determine the horizontal distances of each person from the base of the tree and then calculate the difference to find the distance between them.
Answer
$$ D = \left| \frac{h}{\tan(\theta_A)} - \frac{h}{\tan(\theta_B)} \right| $$
Answer for screen readers
The distance between the two people is given by:
$$ D = \left| \frac{h}{\tan(\theta_A)} - \frac{h}{\tan(\theta_B)} \right| $$
Steps to Solve
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Identify the variables Let the height of the tree be $h$. The angles of elevation from person A and person B to the top of the tree are $\theta_A$ and $\theta_B$, respectively. Let the horizontal distances from the base of the tree to person A and person B be $d_A$ and $d_B$.
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Set up the tangent equations Using the tangent function, we can relate the height of the tree to the distances:
For person A: $$ \tan(\theta_A) = \frac{h}{d_A} $$
For person B: $$ \tan(\theta_B) = \frac{h}{d_B} $$
This allows us to solve for $d_A$ and $d_B$.
- Solve for the distances Rearranging the equations from step 2 gives us:
For person A: $$ d_A = \frac{h}{\tan(\theta_A)} $$
For person B: $$ d_B = \frac{h}{\tan(\theta_B)} $$
- Calculate the distance between the two people The distance between the two people is the absolute difference of their distances from the tree:
$$ \text{Distance} = |d_A - d_B| = \left| \frac{h}{\tan(\theta_A)} - \frac{h}{\tan(\theta_B)} \right| $$
- Substitute the values Now, substitute the actual values for $h$, $\theta_A$, and $\theta_B$ to find the distance.
The distance between the two people is given by:
$$ D = \left| \frac{h}{\tan(\theta_A)} - \frac{h}{\tan(\theta_B)} \right| $$
More Information
Using trigonometry to find distances based on angles of elevation is a common application in fields such as surveying and navigation. This method demonstrates how triangles can be used to simplify real-world problems.
Tips
- Forgetting to convert angles from degrees to radians if necessary.
- Misapplying the tangent formula by mixing up the opposite and adjacent sides.
- Not taking the absolute value when calculating the distance, leading to negative distances.
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