The figure below represents a circular courtyard with a radius of 100 feet that is surrounded by a high stone wall. A floodlight located at E shines into the courtyard. If a person walks x feet away from the center along DC, express the distance d that the persons shadow travels as a function of x.

In this problem, we are asked to express the distance \( d \) that a person's shadow travels as a function of \( x \), where \( x \) is the distance the person walks along line \( DC \), away from the center of the circular courtyard.

Let’s break down the situation:

1. **Given Parameters**:
   - The radius of the circular courtyard is 100 feet.
   - \( DC \) is a straight line along the diameter of the circle.
   - A person walks \( x \) feet away from the center of the circle along this line \( DC \).
   - The floodlight is located at a point \( E \), which I assume to be outside the courtyard, potentially far enough that the light rays can be treated as parallel.

2. **Relationship Between the Person and Their Shadow**:
   - The person's shadow will depend on their position on \( DC \), and how the light rays from the floodlight intersect the circular courtyard.
   - Since the floodlight is shining from point \( E \), we can assume that the further the person moves along \( DC \), the further the shadow extends along the circumference of the courtyard.

### Mathematical Representation

To express the distance \( d \), the shadow’s position as a function of \( x \), we need to consider the geometry of the situation.

- The total length of the diameter is 200 feet (since the radius is 100 feet).
- When the person is standing at point \( x \) along \( DC \), their shadow will extend along the arc formed by the intersection of the light ray with the circular boundary.
- We can use trigonometry to relate \( x \) to the angle subtended by the shadow.

#### Step-by-Step Function Derivation

Let the center of the circle be \( O \), and the person is standing at point \( P \) along \( DC \), where \( OP = x \). 

The equation of the circle is:
\[
x^2 + y^2 = 100^2 = 10000
\]

Let’s assume the light rays form an angle \( \theta \) at the center \( O \), which determines how far along the circumference the shadow reaches.

From the geometry of the situation, the angle \( \theta \) is related to the person’s position \( x \) along \( DC \) by:
\[
\cos(\theta) = \frac{x}{100}
\]
This gives:
\[
\theta = \cos^{-1}\left(\frac{x}{100}\right)
\]

The distance \( d \) along the circular boundary (which is the arc length corresponding to the angle \( \theta \)) can be found using the formula for arc length:
\[
d = r \theta = 100 \cdot \cos^{-1}\left(\frac{x}{100}\right)
\]

Thus, the distance \( d \) that the person's shadow travels is:
\[
d(x) = 100 \cdot \cos^{-1}\left(\frac{x}{100}\right)
\]

### Final Answer:
\[
d(x) = 100 \cdot \cos^{-1}\left(\frac{x}{100}\right)
\]

This is the function that gives the distance the person’s shadow travels as a function of their position \( x \) along \( DC \).

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Would you like further clarification or details on any step?

Here are 5 related questions to deepen your understanding:
1. How would the function change if the radius of the courtyard were different?
2. How does the shadow’s length change as the person moves closer to the center?
3. What happens to the shadow when the person reaches the edge of the courtyard?
4. How would the distance function change if the light source was not infinitely far away?
5. Can you express the rate of change of the shadow’s distance as the person moves?

**Tip**: Always try to visualize the problem with a diagram; it can make relationships clearer, especially in geometry.

The figure below represents a circular courtyard with a radius of 100 feet that is surrounded by a high stone wall. A floodlight located at E shines into the courtyard. If a person walks x feet away from the center along DC, express the distance d that the person's shadow travels as a function of x.

Explore how to express the distance a person's shadow travels in a circular courtyard as a function of their position. Learn to apply geometric and trigonometric concepts like the inverse cosine function and arc length formula. Suitable for students in Grades 10-12.

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