The path of a punted football is given by the function
f(x) = −
16
2025
x2 +
9
5
x + 1.5
where
f(x)
is the height (in feet) and x is the horizontal distance (in feet) from the point at which the ball is punted.
(a) How high is the ball when it is punted?

Correct: Your answer is correct.
ft

(b) What is the maximum height of the punt? (Round your answer to two decimal places.)

Incorrect: Your answer is incorrect.
ft

(c) How long is the punt? (Round your answer to two decimal places.)

Let's break down each part of the problem step by step using the function provided:

The function describing the path of the football is:

\[
f(x) = -\frac{16}{2025}x^2 + \frac{9}{5}x + 1.5
\]

Where:
- \( f(x) \) is the height of the ball in feet,
- \( x \) is the horizontal distance in feet.

### (a) How high is the ball when it is punted?

This is asking for the height when the ball is at \( x = 0 \) (the point where the ball is punted).

To find this, substitute \( x = 0 \) into the equation:

\[
f(0) = -\frac{16}{2025}(0)^2 + \frac{9}{5}(0) + 1.5 = 1.5
\]

So, the ball is **1.5 feet high** when punted.

### (b) What is the maximum height of the punt?

The function \( f(x) \) is a quadratic equation of the form \( ax^2 + bx + c \), which represents a parabola. Since the coefficient of \( x^2 \) is negative, the parabola opens downward, and the vertex of the parabola gives the maximum point.

To find the maximum height, we use the formula for the x-coordinate of the vertex of a parabola:

\[
x_{\text{vertex}} = \frac{-b}{2a}
\]

In our equation:
- \( a = -\frac{16}{2025} \),
- \( b = \frac{9}{5} \).

Substitute these values into the vertex formula:

\[
x_{\text{vertex}} = \frac{-\frac{9}{5}}{2 \times -\frac{16}{2025}} = \frac{-\frac{9}{5}}{-\frac{32}{2025}} = \frac{9}{5} \times \frac{2025}{32} = \frac{18225}{160} = 113.91
\]

So, the horizontal distance where the maximum height occurs is \( x = 113.91 \) feet.

Now, substitute \( x = 113.91 \) back into the original equation to find the maximum height:

\[
f(113.91) = -\frac{16}{2025}(113.91)^2 + \frac{9}{5}(113.91) + 1.5
\]

Let me calculate that.

The maximum height of the punt is approximately **104.02 feet**.

### (c) How long is the punt?

To find the total length of the punt, we need to determine when the ball hits the ground, i.e., when \( f(x) = 0 \).

This is done by solving the quadratic equation:

\[
-\frac{16}{2025}x^2 + \frac{9}{5}x + 1.5 = 0
\]

We can solve this using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Let me calculate the solution for you.The total length of the punt is approximately **228.64 feet**.

### Summary:
- (a) The ball is **1.5 feet high** when it is punted.
- (b) The maximum height of the punt is **104.02 feet**.
- (c) The total length of the punt is **228.64 feet**.

Do you want more details on any of these calculations? 

Here are 5 related questions for further exploration:
1. How would the equation change if the ball were kicked from a different height?
2. How can we determine the time the ball takes to reach its maximum height?
3. What is the significance of the discriminant in the quadratic formula?
4. How does changing the coefficient of \(x^2\) affect the shape of the parabola?
5. How would air resistance affect the path of the ball?

**Tip:** The quadratic formula is useful for solving any quadratic equation, even if the context is different from projectile motion.

The path of a punted football is given by the function f(x) = -(16/2025)x^2 + (9/5)x + 1.5 where f(x) is the height (in feet) and x is the horizontal distance (in feet) from the point at which the ball is punted. (a) How high is the ball when it is punted? (b) What is the maximum height of the punt? (c) How long is the punt?

Learn how to analyze the path of a punted football using a quadratic equation. This step-by-step solution covers key algebraic concepts such as finding the maximum height and total distance using the vertex and quadratic formulas. Suitable for Grades 10-12.

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