Doug is looking at a cliff. From where he is standing he determines that the angle of elevation to the top is 54°. 50 m away from Doug, Gary estimates the angle between the base of the cliff, himself, and Doug to be 26°. Doug estimates the angle between the
base of the cliff, himself, and Gary to be 70°.
What is the height, h, of the cliff to the nearest tenth of a meter? Draw a diagram!

Let's break down the problem step by step and then solve it using trigonometry. 

### Step 1: Understand the problem and draw a diagram
We have two people, Doug and Gary, looking at a cliff. The problem involves three angles:
- The angle of elevation from Doug to the top of the cliff: \( 54^\circ \)
- The angle between the base of the cliff, Doug, and Gary (as seen from Gary): \( 26^\circ \)
- The angle between the base of the cliff, Gary, and Doug (as seen from Doug): \( 70^\circ \)

We need to find the height of the cliff, \( h \), to the nearest tenth of a meter.

### Step 2: Label the points on the diagram
Let's denote the following:
- \( A \) as the top of the cliff
- \( B \) as the base of the cliff
- \( D \) as Doug's position
- \( G \) as Gary's position

### Step 3: Apply the law of sines to the triangle \( \triangle DBG \)

1. **Law of Sines**:
   \[
   \frac{\sin(\angle DGB)}{DG} = \frac{\sin(\angle GDB)}{GB} = \frac{\sin(\angle BGD)}{BD}
   \]
   From the problem:
   \[
   \angle GDB = 70^\circ, \quad \angle DGB = 26^\circ, \quad \angle BGD = 180^\circ - (70^\circ + 26^\circ) = 84^\circ
   \]
   
   Distance between Doug and Gary: \( DG = 50 \) m.

2. **Finding \( BD \) using the law of sines**:
   \[
   \frac{BD}{\sin(26^\circ)} = \frac{50}{\sin(84^\circ)}
   \]
   Simplifying for \( BD \):
   \[
   BD = \frac{50 \times \sin(26^\circ)}{\sin(84^\circ)}
   \]

3. **Use trigonometry to find the height \( h \)**:
   The height \( h \) can be found using the right triangle \( \triangle ABD \), where \( h = AB \). Using the sine of the angle \( \angle BAD = 54^\circ \):
   \[
   \sin(54^\circ) = \frac{h}{BD}
   \]
   Simplifying for \( h \):
   \[
   h = BD \times \sin(54^\circ)
   \]

### Step 4: Perform calculations

Now let's calculate these step by step:

1. Calculate \( BD \):
\[
BD = \frac{50 \times \sin(26^\circ)}{\sin(84^\circ)} 
\]

2. Calculate \( h \):
\[
h = BD \times \sin(54^\circ)
\]

Let's compute these values now.The height of the cliff is approximately **17.8 meters** to the nearest tenth.

Would you like a detailed explanation or have any further questions?

Here are some related questions to consider:
1. How would the height of the cliff change if Doug moved closer to the cliff?
2. What would happen to the height calculation if Gary was positioned further away?
3. How can you apply the Law of Cosines to this problem instead of the Law of Sines?
4. What is the significance of the angle of elevation in determining height?
5. How would a different angle of elevation impact the height of the cliff?
6. Can you determine the distance from Gary to the base of the cliff?
7. How would the height change if the angles were different but Doug remained 50m away?
8. How do trigonometric identities simplify the process of solving such problems?

**Tip:** Always check if a right triangle is formed in problems involving elevation angles to simplify calculations.

Learn how to calculate the height of a cliff using trigonometry. This problem involves angles of elevation, the Law of Sines, and right triangle trigonometry. Discover step-by-step solutions and insights for similar math problems.

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