To determine the height of the tree based on the given angles of elevation and depression, we can break it into two segments:

1. The height of the tree above Mariela's eye level ($h_1$) using the angle of elevation ($42^\circ$).
2. The height of the tree below Mariela's eye level ($h_2$) using the angle of depression ($31^\circ$).

The formula for the height based on an angle and distance is: $h = d \cdot \tan(\theta)$ Where:

• $h$ is the height,
• $d$ is the horizontal distance,
• $\theta$ is the angle in degrees.

Step 1: Height above eye level ($h_1$):

Using the angle of elevation $42^\circ$: $h_1 = 20 \cdot \tan(42^\circ)$

Step 2: Height below eye level ($h_2$):

Using the angle of depression $31^\circ$: $h_2 = 20 \cdot \tan(31^\circ)$

Step 3: Total height of the tree ($H$):

The total height is the sum of $h_1$ and $h_2$: $H = h_1 + h_2$

Let's calculate this. math

Given values

distance = 20 # feet angle_elevation = 42 # degrees angle_depression = 31 # degrees

Calculating heights

h1 = distance * math.tan(math.radians(angle_elevation)) # height above eye level h2 = distance * math.tan(math.radians(angle_depression)) # height below eye level

Total height of the tree

total_height = h1 + h2 total_heightThe total height of the tree is approximately 30.03 feet.

Would you like a detailed breakdown of the steps or further clarifications? Here are five related questions to deepen your understanding:

1. How is the tangent function used to calculate heights in right triangles?
2. What happens if the horizontal distance changes? How does it affect the height calculation?
3. Can you apply the same method to find the height of a building from ground level?
4. How would the calculation change if Mariela were at the base of the tree instead?
5. Why is it important to split the height calculation into two components?

Tip: Always ensure angles are converted to radians when performing trigonometric calculations in programming!