https://file.aiquickdraw.com/mathbot/file/172487497973993padxl3.jpg

The problem involves estimating the height of a hill using angles of elevation. Let's extract the key details from the image:

1. **Scenario**: 
   - You start by observing the top of a hill from a certain point. At that point, the angle of elevation to the hilltop is 20°.
   - You then move 300 meters closer to the hill, and the angle of elevation increases to 35°.

2. **Objective**: 
   - Find:
     a. \( h \): the height of the hill.
     b. \( x \): the distance between your second position and the hill.

### Solution Approach:

We can use trigonometry to solve for the height \( h \) of the hill and the distance \( x \) from your second position to the hill. Let's denote:

- \( x \) = distance from the second position to the hill.
- \( h \) = height of the hill.
- \( x + 300 \) = distance from the first position to the hill.

**Step 1: Use the tangent function for both positions.**

For the first position:
\[
\tan(20^\circ) = \frac{h}{x + 300}
\]

For the second position:
\[
\tan(35^\circ) = \frac{h}{x}
\]

**Step 2: Solve for \( h \) in terms of \( x \) from the second equation:**

\[
h = x \cdot \tan(35^\circ)
\]

**Step 3: Substitute the expression for \( h \) into the first equation:**

\[
\tan(20^\circ) = \frac{x \cdot \tan(35^\circ)}{x + 300}
\]

**Step 4: Solve for \( x \):**

Rearrange the equation:

\[
x \cdot \tan(20^\circ) + 300 \cdot \tan(20^\circ) = x \cdot \tan(35^\circ)
\]

\[
x (\tan(35^\circ) - \tan(20^\circ)) = 300 \cdot \tan(20^\circ)
\]

\[
x = \frac{300 \cdot \tan(20^\circ)}{\tan(35^\circ) - \tan(20^\circ)}
\]

**Step 5: Calculate the values of \( x \) and \( h \):**

Let me calculate the values for you.The solution yields the following:

1. The distance between your second position and the hill (\(x\)) is approximately **324.74 meters**.
2. The height of the hill (\(h\)) is approximately **227.39 meters**.

Would you like a detailed explanation of the steps, or do you have any specific questions?

Here are 5 related questions for further exploration:
1. How does changing the distance moved affect the height calculation?
2. What would the height of the hill be if the angles were different?
3. How do you find the height using only one observation point and distance?
4. Can you determine the height if only one angle and distance are known?
5. How does trigonometry apply to real-world surveying and navigation?

**Tip**: Understanding how trigonometric ratios relate angles to distances is key for solving elevation problems!

Learn how to estimate the height of a hill using trigonometry. Solve a problem involving angles of elevation, calculating both the height of the hill and the distance to the hilltop from different observation points. Detailed step-by-step solution provided.

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