When the angle of elevation of the sun is 78 degrees, an statue casts a shadow that is 6m long. How far is the top of the statue to the end of its shadow?

To solve this problem, we can use trigonometric relationships. Specifically, we'll use the tangent function, which relates the angle of elevation, the height of the statue (which we'll call \( h \)), and the length of the shadow.

However, the problem asks for the distance from the top of the statue to the end of the shadow. This distance is the hypotenuse of a right triangle where the height of the statue is one leg, and the shadow is the other leg.

### Step 1: Identify the relationship

Given:
- Angle of elevation, \( \theta = 78^\circ \)
- Length of the shadow, \( L = 6 \) meters

We are asked to find the hypotenuse \( d \), which is the distance from the top of the statue to the end of the shadow.

### Step 2: Calculate the height of the statue

First, we find the height \( h \) of the statue using the tangent function:
\[
\tan(\theta) = \frac{h}{L}
\]
\[
h = L \cdot \tan(\theta)
\]
Substitute the known values:
\[
h = 6 \times \tan(78^\circ)
\]

### Step 3: Calculate the hypotenuse

Now, use the Pythagorean theorem to find \( d \):
\[
d = \sqrt{h^2 + L^2}
\]

### Step 4: Calculate the values

Let's compute these values.The distance from the top of the statue to the end of its shadow is approximately **28.86 meters**.

Would you like more details on the solution, or do you have any questions?

Here are some related questions for further practice:

1. How tall is the statue in this scenario?
2. What would be the length of the shadow if the angle of elevation were 45 degrees?
3. How does the angle of elevation affect the length of the shadow?
4. What is the length of the shadow if the angle of elevation is 30 degrees and the statue is 10 meters tall?
5. If the shadow is 10 meters long, and the angle of elevation is 60 degrees, how tall is the statue?
6. How far is the top of the statue from the base of the shadow if the angle of elevation is 85 degrees?
7. Can you find the angle of elevation if the shadow is 5 meters long and the statue is 20 meters tall?
8. If the angle of elevation changes to 70 degrees, how does this affect the distance from the top of the statue to the end of the shadow?

**Tip:** Remember that the hypotenuse in a right triangle is always the longest side and can be found using the Pythagorean theorem when you know the lengths of the other two sides.

Explore how to calculate the height of a statue and the length of its shadow using trigonometric relationships. This problem involves an angle of elevation of 78 degrees and a shadow length of 6 meters. Suitable for Grades 9-12.

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