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This problem can be solved using trigonometry, specifically the Law of Sines. Here's how we approach it:

### Step 1: Set up the problem
- The distance between the man and the woman is **3.2 miles**, which is the base of the triangle.
- The angles of elevation to the balloon:
  - From the man: \(29^\circ\)
  - From the woman: \(38^\circ\)
- The altitude of the balloon is the height of the triangle, which we will calculate.

### Step 2: Determine the triangle's angles
The triangle has:
- One angle at the man: \(29^\circ\)
- One angle at the woman: \(38^\circ\)
- The remaining angle at the top of the triangle (opposite the base) can be found using:
  \[
  \text{Top angle} = 180^\circ - 29^\circ - 38^\circ = 113^\circ
  \]

### Step 3: Use the Law of Sines to find distances
Using the Law of Sines:
\[
\frac{\text{Base}}{\sin(\text{Top angle})} = \frac{\text{Distance to the balloon from man}}{\sin(\text{Woman's angle})} = \frac{\text{Distance to the balloon from woman}}{\sin(\text{Man's angle})}
\]
Let the distance from the man to the balloon be \(d_m\) and from the woman to the balloon be \(d_w\).

From the Law of Sines:
\[
\frac{3.2}{\sin(113^\circ)} = \frac{d_m}{\sin(38^\circ)} = \frac{d_w}{\sin(29^\circ)}
\]

Calculate \(d_m\):
\[
d_m = \frac{3.2 \cdot \sin(38^\circ)}{\sin(113^\circ)}
\]

### Step 4: Find the altitude (height)
The altitude (height) \(h\) is related to \(d_m\) using:
\[
h = d_m \cdot \sin(29^\circ)
\]

### Step 5: Perform calculations
Let me calculate these values for you.import math

# Given values
base = 3.2  # miles
angle_man = 29  # degrees
angle_woman = 38  # degrees

# Convert angles to radians for trigonometric functions
angle_top = 180 - angle_man - angle_woman  # degrees
angle_top_rad = math.radians(angle_top)
angle_man_rad = math.radians(angle_man)
angle_woman_rad = math.radians(angle_woman)

# Law of Sines: calculate the distance from the man to the balloon (d_m)
d_m = (base * math.sin(angle_woman_rad)) / math.sin(angle_top_rad)

# Calculate the height (altitude) of the balloon
height = d_m * math.sin(angle_man_rad)

# Convert height from miles to feet (1 mile = 5280 feet)
height_feet = height * 5280

height_feet5478.6180745878065478.618074587806successThe altitude of the hot air balloon is approximately **5,479 feet**.

Do you want a breakdown of the calculations or further clarifications?

Here are 5 related questions to explore:
1. How does the Law of Sines apply to triangles with different angles and side measurements?
2. What other scenarios can the Law of Sines solve?
3. How would the solution change if the man and woman were not directly under the balloon?
4. Can we solve this problem using the Law of Cosines instead?
5. How can trigonometry be used to calculate distances in real-world applications?

**Tip:** Always double-check your angle measurements when using trigonometric laws, especially if you're working with radians or degrees.

A man and a woman standing 3.2 miles apart spot a hot air balloon in the sky between them at the same time. If the angle of elevation from the man to the balloon is 29°, and the angle of elevation from the woman to the balloon is 38°, find the altitude of the balloon to the nearest foot.

Solve a trigonometry problem to find the altitude of a hot air balloon spotted 3.2 miles apart by two observers with angles of elevation of 29° and 38°. Learn step-by-step how to use the Law of Sines and height formulas for this real-world scenario. Perfect for Grades 9-12 students.

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