The angle of elevation from a point on the ground to the top of a pyramid is 36degrees20prime. The angle of elevation from a point 125 feet farther back to the top of the pyramid is 16degrees30prime. Find the height of the pyramid.

We can solve this problem using trigonometry by setting up a system of equations based on the given angles of elevation.

Let's denote the following:

- \( h \): the height of the pyramid (the unknown we want to find),
- \( d \): the distance from the first point to the base of the pyramid,
- \( 125 \, \text{ft} \): the additional distance farther back from the second point compared to the first.

### Step 1: Setup the trigonometric equations

We can form two right triangles based on the points where the angles are measured, both having the same height \( h \), but with different distances from the base of the pyramid.

#### First triangle (closer point):
For the angle of \( 36^\circ 20' \) (converted to decimal: \( 36^\circ 20' = 36 + \frac{20}{60} = 36.333^\circ \)) and distance \( d \) from the pyramid, we can write:
\[
\tan(36.333^\circ) = \frac{h}{d}
\]
\[
h = d \cdot \tan(36.333^\circ)
\]

#### Second triangle (farther point):
For the angle of \( 16^\circ 30' \) (converted to decimal: \( 16^\circ 30' = 16 + \frac{30}{60} = 16.5^\circ \)) and the distance \( d + 125 \) from the pyramid, we can write:
\[
\tan(16.5^\circ) = \frac{h}{d + 125}
\]
\[
h = (d + 125) \cdot \tan(16.5^\circ)
\]

### Step 2: Set the equations equal

Since both equations represent the same height \( h \), we can equate the two expressions:
\[
d \cdot \tan(36.333^\circ) = (d + 125) \cdot \tan(16.5^\circ)
\]

### Step 3: Solve for \( d \)

Now, we solve for \( d \) by expanding and simplifying the equation:
\[
d \cdot \tan(36.333^\circ) = d \cdot \tan(16.5^\circ) + 125 \cdot \tan(16.5^\circ)
\]
\[
d \cdot (\tan(36.333^\circ) - \tan(16.5^\circ)) = 125 \cdot \tan(16.5^\circ)
\]
\[
d = \frac{125 \cdot \tan(16.5^\circ)}{\tan(36.333^\circ) - \tan(16.5^\circ)}
\]

### Step 4: Calculate \( d \)

Let's calculate the values of the tangents and solve for \( d \).

- \( \tan(36.333^\circ) \approx 0.7349 \)
- \( \tan(16.5^\circ) \approx 0.2964 \)

Substitute these values into the equation:
\[
d = \frac{125 \cdot 0.2964}{0.7349 - 0.2964} = \frac{37.05}{0.4385} \approx 84.47 \, \text{ft}
\]

### Step 5: Find the height \( h \)

Now that we have \( d \approx 84.47 \, \text{ft} \), we can substitute this back into the equation for \( h \) from the first triangle:
\[
h = d \cdot \tan(36.333^\circ) = 84.47 \cdot 0.7349 \approx 62.06 \, \text{ft}
\]

### Final Answer:
The height of the pyramid is approximately **62.06 feet**.

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Would you like further details or clarification on any part? Here are some related questions to explore:

1. How do different angles of elevation affect the distance calculations?
2. What are some real-world applications of angle of elevation and trigonometry?
3. How can we use the Law of Sines or Cosines in similar problems?
4. How do we handle cases with obtuse angles in elevation problems?
5. How does this method extend to finding distances or heights in 3D space?

**Tip**: Always double-check angle conversions from degrees and minutes into decimal form, as small errors can impact the final result.

Learn how to determine the height of a pyramid using trigonometry by analyzing angles of elevation from two points on the ground. This problem is suitable for high school students and involves trigonometric ratios and right triangle concepts.

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