A woman wishing to know the height of a mountain measures the angle of elevation of the mountaintop as 12.6° After walking 0.96 km closer to the mountain on level ground, she finds the angle to be 14.7° Find the height of the mountain

To find the height of the mountain, we can solve this problem using trigonometry, specifically using the tangent function. Let's define the variables involved:

- Let \( h \) be the height of the mountain (what we need to find).
- Let \( d \) be the initial distance from the mountain before the woman walks closer.
- After walking 0.96 km, the new distance from the mountain is \( d - 0.96 \) km.
- The angle of elevation from the original position is \( 12.6^\circ \).
- The angle of elevation from the new position is \( 14.7^\circ \).

### Step 1: Using the tangent function for the first position

From the first position, we know:
\[
\tan(12.6^\circ) = \frac{h}{d}
\]
This equation gives us:
\[
h = d \cdot \tan(12.6^\circ)
\]

### Step 2: Using the tangent function for the second position

From the second position, after walking 0.96 km closer, we have:
\[
\tan(14.7^\circ) = \frac{h}{d - 0.96}
\]
This equation gives us:
\[
h = (d - 0.96) \cdot \tan(14.7^\circ)
\]

### Step 3: Solving the system of equations

Now we have two expressions for \( h \). We can set them equal to each other:
\[
d \cdot \tan(12.6^\circ) = (d - 0.96) \cdot \tan(14.7^\circ)
\]

We can now solve this equation for \( d \), and then use \( d \) to find \( h \).

Let me compute this.It appears that the computed values for the distance and height are negative, which is incorrect for this physical scenario. This suggests there may be a mistake in handling the equation or assumptions. Let me double-check the setup and resolve the issue.There seems to still be an issue with the values, leading to unrealistic results (negative distance). I will revisit the equation structure and correct the method for solving this. Let me resolve this step-by-step by isolating \(d\) properly and avoiding potential sign errors.The correct result shows that the woman was initially about \(6.49 \, \text{km}\) away from the mountain, and the height of the mountain is approximately \(1.45 \, \text{km}\).

Would you like further details or clarifications?

Here are 5 related questions:
1. How is the tangent function related to angles and distances in trigonometry?
2. Why are degrees converted to radians in trigonometric calculations?
3. How would the height of the mountain change if the angles were larger?
4. What would happen if the woman walked a shorter or longer distance?
5. How could this method be applied to other real-world elevation problems?

**Tip:** Always make sure to double-check the sign and physical meaning of values, especially when dealing with distances and heights in word problems.

A woman wishing to know the height of a mountain measures the angle of elevation of the mountaintop as 12.6° After walking 0.96 km closer to the mountain, she finds the angle to be 14.7° Find the height of the mountain

Learn how to determine the height of a mountain using trigonometry. This problem involves calculating the height based on angles of elevation and distance changes. Detailed step-by-step solution using the tangent function. Suitable for students in Grades 10-12.

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