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Solve using the most quick simplest method

To solve the problem of finding the height of the pyramid using the quickest and simplest method, let's break it down step-by-step.

### Given Information:
- **Angle of elevation from the first point (A):** \( 35^\circ 30' \)
- **Angle of elevation from the second point (B):** \( 21^\circ 30' \)
- **Distance between the two points (A to B):** \( 135 \text{ ft} \)

### Step 1: Convert the angles to decimal form.
- \( 35^\circ 30' = 35.5^\circ \)
- \( 21^\circ 30' = 21.5^\circ \)

### Step 2: Set up the tangent equations for the heights.
Let \( h \) be the height of the pyramid and \( d \) be the horizontal distance from point A to the base of the pyramid.

- From point A: 
  \[
  \tan(35.5^\circ) = \frac{h}{d}
  \]
  \[
  h = d \cdot \tan(35.5^\circ)
  \]

- From point B (135 feet further back):
  \[
  \tan(21.5^\circ) = \frac{h}{d + 135}
  \]
  \[
  h = (d + 135) \cdot \tan(21.5^\circ)
  \]

### Step 3: Set the two equations for height equal to each other.
\[
d \cdot \tan(35.5^\circ) = (d + 135) \cdot \tan(21.5^\circ)
\]

### Step 4: Solve for \( d \).
\[
d \cdot \tan(35.5^\circ) = d \cdot \tan(21.5^\circ) + 135 \cdot \tan(21.5^\circ)
\]
\[
d \cdot (\tan(35.5^\circ) - \tan(21.5^\circ)) = 135 \cdot \tan(21.5^\circ)
\]
\[
d = \frac{135 \cdot \tan(21.5^\circ)}{\tan(35.5^\circ) - \tan(21.5^\circ)}
\]

### Step 5: Calculate \( d \).
\[
d \approx \frac{135 \cdot 0.3939}{0.7123 - 0.3939} \approx \frac{53.1765}{0.3184} \approx 167.04 \text{ ft}
\]

### Step 6: Calculate the height \( h \).
\[
h = d \cdot \tan(35.5^\circ)
\]
\[
h \approx 167.04 \cdot 0.7123 \approx 119.01 \text{ ft}
\]

### Final Answer:
The height of the pyramid is approximately **119.01 feet**.

Would you like a more detailed explanation or any further clarification on any of these steps?

### Related Questions:
1. How do you convert angles from degrees and minutes to decimal form?
2. What is the significance of using the tangent function in solving problems like this?
3. How would the height calculation change if the distance between points A and B were different?
4. Can you apply this method to find the height of other structures?
5. What are some real-world applications of these types of trigonometric problems?

### Tip:
Always double-check the angle conversions and calculations, as small errors can lead to significant differences in the final result.

The angle of elevation from a point on the ground to the top of a pyramid is 35° 30'. The angle of elevation from a point 135 ft farther back to the top of the pyramid is 21° 30'. Find the height of the pyramid.

Learn how to find the height of a pyramid by using angles of elevation from two different points. This problem covers trigonometric concepts such as tangent functions and is suitable for high school students.

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