While sailing toward a​ statue, a sailor in a boat observed that at a certain​ point, the angle of elevation of the tip of the torch was 27 degrees. After sailing another 100 meters toward the​ statue, the angle of elevation became 38 degrees 50 prime.

HOW TALL IS THE STATUE?

(do not round until the final answer, then round to the nearest integer)

To determine the height of the statue, we can use trigonometry and the concept of right triangles. Here's the approach:

### Step 1: Define variables and diagram the situation.
1. Let \( h \) be the height of the statue.
2. Let \( d \) be the distance from the sailor's initial position to the base of the statue.
3. After sailing 100 meters toward the statue, the new distance becomes \( d - 100 \).

We now have two right triangles:
- The first triangle has an angle of elevation of \( 27^\circ \) and distance \( d \) from the statue.
- The second triangle has an angle of elevation of \( 38^\circ 50' \) (which is \( 38 + \frac{50}{60} = 38.8333^\circ \)) and distance \( d - 100 \).

### Step 2: Apply the tangent function.
For the first triangle:
\[
\tan(27^\circ) = \frac{h}{d}
\]

For the second triangle:
\[
\tan(38.8333^\circ) = \frac{h}{d - 100}
\]

### Step 3: Solve the system of equations.
From the first equation, solve for \( h \):
\[
h = d \cdot \tan(27^\circ)
\]

From the second equation, solve for \( h \) as well:
\[
h = (d - 100) \cdot \tan(38.8333^\circ)
\]

Now, set the two expressions for \( h \) equal to each other:
\[
d \cdot \tan(27^\circ) = (d - 100) \cdot \tan(38.8333^\circ)
\]

### Step 4: Solve for \( d \).
Expanding the right side:
\[
d \cdot \tan(27^\circ) = d \cdot \tan(38.8333^\circ) - 100 \cdot \tan(38.8333^\circ)
\]

Now, collect the \( d \)-terms on one side:
\[
d \cdot (\tan(27^\circ) - \tan(38.8333^\circ)) = -100 \cdot \tan(38.8333^\circ)
\]

Solve for \( d \):
\[
d = \frac{-100 \cdot \tan(38.8333^\circ)}{\tan(27^\circ) - \tan(38.8333^\circ)}
\]

### Step 5: Compute the value of \( d \).
We can now compute the tangent values and solve for \( d \):
\[
\tan(27^\circ) \approx 0.5095, \quad \tan(38.8333^\circ) \approx 0.8011
\]

Substitute these values into the equation:
\[
d = \frac{-100 \cdot 0.8011}{0.5095 - 0.8011}
\]
\[
d = \frac{-80.11}{-0.2916} \approx 274.81 \, \text{meters}
\]

### Step 6: Calculate the height \( h \).
Now that we have \( d \), use the first equation to find \( h \):
\[
h = d \cdot \tan(27^\circ) = 274.81 \cdot 0.5095 \approx 140 \, \text{meters}
\]

### Final Answer:
The height of the statue is approximately **140 meters**.

Would you like further details or clarifications? Here are five related questions:

1. What are other real-life applications of trigonometry in navigation?
2. How does changing the angle of elevation affect the height calculation?
3. What other methods could be used to calculate heights indirectly?
4. How would this problem change if the second angle was measured farther from the statue?
5. What role does approximation play in solving trigonometric problems?

**Tip:** When dealing with trigonometry problems, always try to sketch a diagram to help visualize the situation.

While sailing toward a statue, a sailor in a boat observed that at a certain point, the angle of elevation of the tip of the torch was 27 degrees. After sailing another 100 meters toward the statue, the angle of elevation became 38 degrees 50 prime. HOW TALL IS THE STATUE?

Learn how to determine the height of a statue using trigonometry and angles of elevation. This problem involves applying the tangent function and solving a system of equations. Suitable for students in Grades 10-12.

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