The angle of elevation from a point on the ground to the top of a pyramid is 39degrees40prime. The angle of elevation from a point 129 feet farther back to the top of the pyramid is 32degrees40prime. Find the height of the pyramid.

Let’s solve this problem using trigonometry and the tangent function. We will define the necessary variables and equations based on the given information:

1. Let \( h \) be the height of the pyramid.
2. Let \( x \) be the distance from the first point on the ground to the base of the pyramid.
3. The angle of elevation from the first point is \( 39^\circ 40' \).
4. The angle of elevation from the second point (129 feet farther back) is \( 32^\circ 40' \).

### Step 1: Convert Angles to Decimal Degrees
First, convert the angles from degrees and minutes to decimal degrees:

- \( 39^\circ 40' = 39 + \frac{40}{60} = 39.6667^\circ \).
- \( 32^\circ 40' = 32 + \frac{40}{60} = 32.6667^\circ \).

### Step 2: Set Up Trigonometric Equations
Using the tangent function, which relates the angle of elevation to the height and distance, we have two equations:

1. From the first point:
   \[
   \tan(39.6667^\circ) = \frac{h}{x}
   \]

2. From the second point (129 feet farther back):
   \[
   \tan(32.6667^\circ) = \frac{h}{x + 129}
   \]

### Step 3: Solve the System of Equations
Now, we need to solve this system of equations to find \( h \). Let’s calculate the values step-by-step:

\[
x = \frac{h}{\tan(39.6667^\circ)}
\]

\[
x + 129 = \frac{h}{\tan(32.6667^\circ)}
\]

Substituting the value of \( x \) from the first equation into the second:

\[
\frac{h}{\tan(39.6667^\circ)} + 129 = \frac{h}{\tan(32.6667^\circ)}
\]

Let’s solve this equation for \( h \).The height of the pyramid is approximately \( 364.69 \) feet.

Would you like a detailed step-by-step explanation, or do you have any further questions?

Here are some related questions for further exploration:

1. How would the solution change if the second point were closer instead of farther back?
2. What trigonometric function would you use if the problem involved the angle of depression instead?
3. How can we verify the correctness of our answer using a different method?
4. What role does the tangent function play in solving real-world elevation problems?
5. How would this problem change if the height of the observer was also given?

**Tip:** When dealing with angles in degrees and minutes, always convert to decimal form before using trigonometric functions!

The angle of elevation from a point on the ground to the top of a pyramid is 39 degrees 40 minutes. The angle of elevation from a point 129 feet farther back to the top of the pyramid is 32 degrees 40 minutes. Find the height of the pyramid.

Learn how to calculate the height of a pyramid by using the angles of elevation from two points. This trigonometry problem involves applying the tangent function and converting degrees and minutes to decimal form. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation