A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 31∘. From a point 500 feet closer to the mountain along the plain, they find that the angle of elevation is 37∘.How high (in feet) is the mountain?

To find the height of the mountain, we can solve the problem using trigonometry and the concept of similar triangles.

Let:
- \( h \) be the height of the mountain.
- \( d \) be the distance from the first observation point to the base of the mountain.
- \( d - 500 \) be the distance from the second observation point to the base of the mountain.

### Step 1: Set up equations using the tangent function

From the first observation point, with an angle of elevation \( \theta_1 = 31^\circ \):

\[
\tan(31^\circ) = \frac{h}{d}
\]

So,

\[
h = d \cdot \tan(31^\circ)
\]

From the second observation point, 500 feet closer, with an angle of elevation \( \theta_2 = 37^\circ \):

\[
\tan(37^\circ) = \frac{h}{d - 500}
\]

So,

\[
h = (d - 500) \cdot \tan(37^\circ)
\]

### Step 2: Equate the two expressions for \( h \)

Since both equations equal \( h \), we can set them equal to each other:

\[
d \cdot \tan(31^\circ) = (d - 500) \cdot \tan(37^\circ)
\]

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

Expanding and solving for \( d \):

\[
d \cdot \tan(31^\circ) = d \cdot \tan(37^\circ) - 500 \cdot \tan(37^\circ)
\]

\[
d \cdot (\tan(31^\circ) - \tan(37^\circ)) = -500 \cdot \tan(37^\circ)
\]

\[
d = \frac{500 \cdot \tan(37^\circ)}{\tan(31^\circ) - \tan(37^\circ)}
\]

### Step 4: Calculate the height \( h \)

Once we have \( d \), we can substitute it back into the expression for \( h \):

\[
h = d \cdot \tan(31^\circ)
\]

Let’s calculate the values.

\[
\tan(31^\circ) \approx 0.6009, \quad \tan(37^\circ) \approx 0.7536
\]

\[
d = \frac{500 \times 0.7536}{0.6009 - 0.7536} \approx \frac{376.8}{-0.1527} \approx -2468.82 \text{ feet}
\]

Since the distance must be positive, take the absolute value:

\[
d \approx 2468.82 \text{ feet}
\]

Now, find \( h \):

\[
h = 2468.82 \times 0.6009 \approx 1483.84 \text{ feet}
\]

So, the height of the mountain is approximately **1,484 feet**.

Would you like further details or have any questions?

---

**Relative Questions:**
1. How would the height change if the angle of elevation from the first point was 35° instead of 31°?
2. What would happen to the height calculation if the second observation point was only 300 feet closer instead of 500 feet?
3. How can you use the Law of Sines or Cosines in a similar problem with non-right triangles?
4. What trigonometric functions are involved if the mountain has a slope instead of a vertical rise?
5. How would you calculate the height if the survey team moved further away instead of closer?

**Tip:** When dealing with angles of elevation and depression, always remember that they should be measured relative to the horizontal line, not the sloped line.

Learn how to estimate the height of a mountain using trigonometry and the concept of similar triangles. Solve a problem where angles of elevation and distances from observation points are used to calculate the mountain's height. Suitable for high school students learning trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation