To solve this problem, let's break it down step by step.

Step 1: Identify the Situation

• A telephone pole has a height of 10 meters.
• It casts a shadow of 15 meters on the slope of a hill.
• The angle of elevation of the sun (angle $\angle BAC$) is 52°.
• We are tasked with finding the angle of elevation $\angle DAC$ of the ground, which is the slope of the hill.

Step 2: Understand the Geometry

We can model this situation using trigonometry.

• Angle BAC is the angle formed between the line of sight from the top of the pole to the sun and the ground, which is 52°.
• Angle DAC is the angle between the slope of the hill (the ground) and the horizontal, which is the angle we're trying to find.

Step 3: Using the Tan Function for the Pole and the Shadow

First, if the ground were level, we could find the length of the shadow using the tangent of the angle $\angle BAC$ (since tangent relates the height of the pole and the length of the shadow).

The formula for the tangent of an angle in a right triangle is:

$\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}$

In this case:

$\tan(52^\circ) = \frac{10}{\text{shadow length on level ground}}$

Now, solve for the shadow length on level ground:

$\text{shadow length on level ground} = \frac{10}{\tan(52^\circ)}$

Using a calculator:

$\text{shadow length on level ground} \approx \frac{10}{1.2799} \approx 7.82 \text{ meters}$

Step 4: Relationship Between the Angles

Now, let's compare the situation on the slope. The shadow on the slope is 15 meters, which is longer than the shadow on the level ground. This suggests that the ground is tilted upward.

Using the relationship between the horizontal ground (shadow length on level ground), the slope's shadow (15 meters), and the angle $\angle DAC$, we can apply trigonometry to find the angle of the slope. The ratio between the shadow length on the slope and the shadow length on level ground is related to the angle $\angle DAC$.

Let $\theta = \angle DAC$ be the angle of the slope. The length of the shadow on the slope is related to the shadow on level ground by the cosine of the slope angle:

$\frac{\text{shadow on slope}}{\text{shadow on level ground}} = \frac{1}{\cos(\theta)}$

Substitute the known values:

$\frac{15}{7.82} = \frac{1}{\cos(\theta)}$

Solving for $\cos(\theta)$:

$\cos(\theta) = \frac{7.82}{15} \approx 0.521$

Now, solve for $\theta$:

$\theta = \cos^{-1}(0.521)$

Using a calculator:

$\theta \approx 58^\circ$

Final Answer:

The angle of elevation of the ground $\angle DAC$ is approximately 58°.

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Would you like more details on any of these steps? Here are some related questions:

1. How does the tangent of an angle relate to the shadow of a vertical object?
2. What are some other real-world applications of using trigonometry to solve for angles in slopes?
3. How can we adjust for angles of elevation in problems involving inclined planes?
4. How does the cosine function help when working with inclined surfaces?
5. What would change if the height of the pole were different? How would that affect the shadow length?

Tip: When working with shadows, always remember to break the problem into smaller right triangles, and use trigonometric ratios (sine, cosine, or tangent) depending on the known and unknown values!