Your school’s marching band is performing at halftime during a football game. In the last formation, the band members form a circle 100 feet wide in the center of the field. You start at a point on the circle 100 feet from the goal line, march $300^{\circ}$ around the circle, and then walk toward the goal line to exit the field. How far from the goal line are you at the point where you leave the circle? Click here for the image related to exercise.

To solve this problem, we need to break it down step by step:

### Step 1: Understanding the Setup
- The band forms a **circle** that is 100 feet in diameter, which means the **radius** $r$ of the circle is:
  \[
  r = \frac{100}{2} = 50 \text{ feet}.
  \]
- You start at a point on the circle that is **100 feet from the goal line**.
- You march **300°** around the circle, which leaves **60°** for the remaining arc since $360^\circ - 300^\circ = 60^\circ$.
- After marching 300°, you exit the circle by walking directly toward the goal line.

### Step 2: Coordinate System
We can visualize this by placing the circle on a coordinate system:
- Place the center of the circle at the origin $(0, 0)$.
- Since you start 100 feet from the goal line and the diameter of the circle is 100 feet, the **distance from the center of the circle to the goal line** is 50 feet. So, the goal line is at $x = -50$ on the coordinate plane.
- Initially, you are at the rightmost point of the circle, $(50, 0)$.

### Step 3: Calculating the Position After 300°
Marching **300°** around the circle means you end up at an angle of **300°** from the positive x-axis. In polar coordinates, the position on the circle after moving 300° is given by the following formulas for $x$ and $y$ coordinates:

\[
x = r \cos(\theta)
\]
\[
y = r \sin(\theta)
\]

Substituting $r = 50$ feet and $\theta = 300^\circ$, we convert the angle to radians first:

\[
\theta = 300^\circ = \frac{5\pi}{3} \text{ radians}.
\]

Now, compute $x$ and $y$:

\[
x = 50 \cos\left(\frac{5\pi}{3}\right) = 50 \times 0.5 = 25 \text{ feet},
\]
\[
y = 50 \sin\left(\frac{5\pi}{3}\right) = 50 \times (-\frac{\sqrt{3}}{2}) = -25\sqrt{3} \approx -43.3 \text{ feet}.
\]

So, after marching 300°, you are at the point $(25, -43.3)$ on the coordinate plane.

### Step 4: Distance from the Goal Line
The goal line is at $x = -50$, so to find how far you are from the goal line, we need to calculate the horizontal distance between your final position $x = 25$ and the goal line $x = -50$:

\[
\text{Distance from goal line} = |25 - (-50)| = |25 + 50| = 75 \text{ feet}.
\]

### Final Answer:
At the point where you leave the circle, you are **75 feet from the goal line**.

---

Would you like more details or explanations on any step? Here are some related questions to explore further:

1. How do you calculate distances in polar coordinates?
2. What is the relationship between degrees and radians?
3. How can you calculate the arc length you walked?
4. What is the significance of the angle 300° in trigonometry?
5. How does marching along different angles affect your final position on a circle?

**Tip:** Always visualize problems involving circles on a coordinate plane to better understand the relationships between angles and distances!

Your school’s marching band is performing at halftime during a football game. In the last formation, the band members form a circle 100 feet wide in the center of the field. You start at a point on the circle 100 feet from the goal line, march 300° around the circle, and then walk toward the goal line to exit the field. How far from the goal line are you at the point where you leave the circle?

Solve a real-world trigonometry problem where you march 300° around a 100-foot wide circle and calculate how far you are from the goal line after leaving the circle. The solution covers circular geometry, trigonometric functions, and distance calculations. Suitable for Grades 10-12.

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