A football player punts a football from a height of 2 feet. The ball reaches a maximum height of 50 feet after traveling 56 feet horizontally, and is caught 111 feet from where the ball was kicked. The path of the ball can be modeled by a parabola, where $y$ is the height (in feet) and $x$ is the horizontal distance traveled (in feet). At what height is the ball caught? Round your answer to the nearest tenth.

To solve this problem, we will model the path of the football using the general equation of a parabola:

\[
y = ax^2 + bx + c
\]

### Step 1: Setting up the known points
We know three points on the parabola:
- The ball is kicked from a height of 2 feet when $ x = 0 $. So, $ (0, 2) $ is a point on the parabola. This gives the equation $ c = 2 $.
- The ball reaches a maximum height of 50 feet when $ x = 56 $. This gives the point $ (56, 50) $.
- The ball is caught at $ x = 111 $. We need to find the height $ y $ at this point.

The vertex of the parabola is at $ x = 56 $ since that’s where the ball reaches its maximum height. Therefore, we can write the equation in vertex form:

\[
y = a(x - 56)^2 + 50
\]

### Step 2: Using the initial condition
We know that the ball was kicked from a height of 2 feet at $ x = 0 $. Plugging $ x = 0 $ and $ y = 2 $ into the vertex form equation:

\[
2 = a(0 - 56)^2 + 50
\]

Simplifying:

\[
2 = a(56)^2 + 50
\]

\[
2 = 3136a + 50
\]

\[
3136a = 2 - 50
\]

\[
3136a = -48
\]

\[
a = \frac{-48}{3136} = -\frac{3}{196}
\]

So the equation of the parabola is:

\[
y = -\frac{3}{196}(x - 56)^2 + 50
\]

### Step 3: Finding the height when $ x = 111 $
We substitute $ x = 111 $ into the equation to find the height at which the ball is caught:

\[
y = -\frac{3}{196}(111 - 56)^2 + 50
\]

Simplifying:

\[
y = -\frac{3}{196}(55)^2 + 50
\]

\[
y = -\frac{3}{196}(3025) + 50
\]

\[
y = -\frac{9075}{196} + 50
\]

\[
y \approx -46.3 + 50
\]

\[
y \approx 3.7
\]

### Final Answer:
The ball is caught at a height of approximately 3.7 feet.

---

Would you like more details or have any questions?

Here are some related questions to deepen understanding:
1. How do we find the vertex of a parabola given its equation?
2. What does the parameter $ a $ signify in the equation of a parabola?
3. How do horizontal shifts affect the equation of a parabola?
4. What is the significance of completing the square in deriving the vertex form?
5. How would the height change if the maximum height were different?

**Tip:** Always verify if the vertex form or standard form of a parabola is more convenient based on the given data.

A football player punts a football from a height of 2 feet. The ball reaches a maximum height of 50 feet after traveling 56 feet horizontally, and is caught 111 feet from where the ball was kicked. The path of the ball can be modeled by a parabola, where y is the height (in feet) and x is the horizontal distance traveled (in feet). At what height is the ball caught? Round your answer to the nearest tenth.

Learn how to determine the height at which a football is caught by modeling its path using a parabola. This problem involves algebraic concepts such as quadratic equations and properties of parabolas, suitable for students in Grades 9-11. Follow a step-by-step solution to find the height when the ball is caught 111 feet from where it was kicked.

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