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Part 1

LaToya throws a ball from the top of a bridge. Her throw is modeled by the equation 

yequals=negative 0.5 x squared plus 4 x plus 11−0.5x2+4x+11​, 

and the bridge is modeled by the equation 

yequals=negative 0.2 x plus 8−0.2x+8. 

About how far does the ball travel horizontally before its first​ bounce?

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Part 1

About 

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feet

​(Round to two decimal places as​ needed.)

About(Round to two decimal places as needed.)

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To determine how far the ball travels horizontally before its first bounce, we need to find the point where the ball intersects the bridge, i.e., where the two equations for the height \( y \) are equal. 

### Step 1: Set the two equations equal to each other.
The equations given are:

\[
y = -0.5x^2 + 4x + 11
\]

\[
y = -0.2x + 8
\]

Set the equations equal to each other:

\[
-0.5x^2 + 4x + 11 = -0.2x + 8
\]

### Step 2: Simplify the equation.
Move all terms to one side to set the equation to zero:

\[
-0.5x^2 + 4x + 11 + 0.2x - 8 = 0
\]

Combine like terms:

\[
-0.5x^2 + 4.2x + 3 = 0
\]

### Step 3: Solve the quadratic equation.
To solve this quadratic equation, we can use the quadratic formula:

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

Here, \( a = -0.5 \), \( b = 4.2 \), and \( c = 3 \). Substitute these values into the quadratic formula:

\[
x = \frac{-4.2 \pm \sqrt{(4.2)^2 - 4(-0.5)(3)}}{2(-0.5)}
\]

Simplify the equation step by step:

\[
x = \frac{-4.2 \pm \sqrt{17.64 + 6}}{-1}
\]

\[
x = \frac{-4.2 \pm \sqrt{23.64}}{-1}
\]

\[
x = \frac{-4.2 \pm 4.86}{-1}
\]

### Step 4: Calculate the roots.
Now calculate the two possible values for \( x \):

\[
x_1 = \frac{-4.2 + 4.86}{-1} = -0.66
\]

\[
x_2 = \frac{-4.2 - 4.86}{-1} = 9.06
\]

Since we're interested in the horizontal distance where the ball first touches the bridge, the relevant positive solution is \( x = 9.06 \) feet.

### Final Answer:
The ball travels approximately **9.06 feet** horizontally before its first bounce.

Would you like any additional explanations or details on the steps involved?

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Here are five related questions you might consider:

1. How would the distance change if the bridge equation had a steeper slope?
2. What if the ball was thrown from a higher position? How would the equations change?
3. How can you find the maximum height of the ball during its flight?
4. What is the significance of the discriminant in the quadratic equation in this context?
5. How would the solution change if the bridge equation was quadratic instead of linear?

**Tip:** When solving quadratic equations, always check both roots to determine which one makes sense in the context of the problem.

Learn how to calculate the horizontal distance traveled by a ball before its first bounce using quadratic and linear equations. This problem involves solving equations and finding intersections, suitable for Grades 7-9.

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