To find the height of a flagpole, a student stands a random distance away from it and measures the angle ofelevation from her feet to the top of the flagpole to be 70o.  The student then moves 2m further away from the flagpole in the same line of sight as before, and notices that the new angle of elevationas measured before isnow60o.  Draw a diagram depicting the situation, then calculate the height of the flagpole

To solve this problem, we'll use trigonometry, specifically the tangent function, which relates the angle of elevation to the height and distance of the object. Let's break down the steps and draw the diagram.

### Step 1: Draw the Diagram

1. Draw a vertical line representing the flagpole with height \( h \).
2. Mark two points on the ground along the same line of sight:
   - The first point \( A \), where the student is initially standing, is a distance \( x \) away from the base of the flagpole.
   - The second point \( B \), where the student moves 2 meters further away, is a distance \( x + 2 \) from the base of the flagpole.
3. Draw the horizontal ground line connecting the base of the flagpole to both points \( A \) and \( B \).
4. Draw two lines from points \( A \) and \( B \) to the top of the flagpole, forming angles of elevation \( 70^\circ \) and \( 60^\circ \) respectively.

This will create two right triangles, \( \triangle ABC \) and \( \triangle ABD \), where:
- \( h \) is the height of the flagpole,
- \( x \) is the distance from the flagpole to point \( A \),
- \( x + 2 \) is the distance from the flagpole to point \( B \),
- \( \theta_1 = 70^\circ \) (angle of elevation from point \( A \)),
- \( \theta_2 = 60^\circ \) (angle of elevation from point \( B \)).

### Step 2: Set Up Equations Using Tangent

For the first position (at point \( A \)):
\[
\tan(70^\circ) = \frac{h}{x}
\]
So, 
\[
h = x \cdot \tan(70^\circ)
\]

For the second position (at point \( B \)):
\[
\tan(60^\circ) = \frac{h}{x + 2}
\]
So,
\[
h = (x + 2) \cdot \tan(60^\circ)
\]

### Step 3: Set Up the Equation to Solve for \( x \) and \( h \)

Since both expressions represent the height \( h \), we can set them equal to each other:
\[
x \cdot \tan(70^\circ) = (x + 2) \cdot \tan(60^\circ)
\]

Expanding and solving for \( x \):
\[
x \cdot \tan(70^\circ) = x \cdot \tan(60^\circ) + 2 \cdot \tan(60^\circ)
\]
\[
x \cdot (\tan(70^\circ) - \tan(60^\circ)) = 2 \cdot \tan(60^\circ)
\]
\[
x = \frac{2 \cdot \tan(60^\circ)}{\tan(70^\circ) - \tan(60^\circ)}
\]

### Step 4: Calculate \( x \) and then \( h \)

Substituting the values of the tangents:
\[
\tan(70^\circ) \approx 2.747
\]
\[
\tan(60^\circ) \approx 1.732
\]

\[
x = \frac{2 \cdot 1.732}{2.747 - 1.732}
\]
\[
x \approx \frac{3.464}{1.015} \approx 3.413 \, \text{meters}
\]

Now, calculate \( h \):
\[
h = x \cdot \tan(70^\circ) \approx 3.413 \cdot 2.747 \approx 9.374 \, \text{meters}
\]

### Final Answer:

The height of the flagpole is approximately **9.37 meters**.

---

Would you like further details on any part of the solution? Here are some related questions:

1. What other trigonometric identities could be used to solve similar problems?
2. How would the problem change if the student moved closer to the flagpole?
3. Can this method be applied to non-right triangles? How?
4. How does the accuracy of the angle measurements affect the result?
5. What if the ground was not level, how would that affect the calculations?

**Tip:** Always ensure that your calculator is set to the correct mode (degrees/radians) when solving trigonometric problems.

To find the height of a flagpole, a student stands a random distance away from it and measures the angle of elevation from her feet to the top of the flagpole to be 70°. The student then moves 2m further away from the flagpole in the same line of sight as before, and notices that the new angle of elevation as measured before is now 60°. Draw a diagram depicting the situation, then calculate the height of the flagpole.

Learn how to calculate the height of a flagpole using trigonometry. This problem involves angles of elevation and the tangent function, suitable for students in Grades 10-12. Follow the detailed steps to solve the problem with a clear diagram.

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