LaToya throws a ball from the top of a bridge. Her throw is modeled by the equation yequalsnegative 0.6 x squared plus 3 x plus 8​, and the bridge is modeled by the equation yequalsnegative 0.1 x plus 5. About how far does the ball travel horizontally before its first​ bounce?
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enter your response here feet
​(Round to two decimal places as​ needed.)

To solve this problem, we need to find the point where the ball first intersects the bridge after it is thrown. This point of intersection will represent where the ball first bounces.

### Step-by-Step Solution:

1. **Given Equations:**
   - The equation of the ball's path:  
     \[
     y = -0.6x^2 + 3x + 8
     \]
   - The equation of the bridge:  
     \[
     y = -0.1x + 5
     \]

2. **Set the Equations Equal to Each Other:**
   To find the intersection, set the two equations equal to each other:
   \[
   -0.6x^2 + 3x + 8 = -0.1x + 5
   \]

3. **Rearrange the Equation:**
   Bring all terms to one side:
   \[
   -0.6x^2 + 3x + 8 + 0.1x - 5 = 0
   \]
   Simplify:
   \[
   -0.6x^2 + 3.1x + 3 = 0
   \]

4. **Solve the Quadratic Equation:**
   To solve for \(x\), use the quadratic formula:  
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]  
   where \(a = -0.6\), \(b = 3.1\), and \(c = 3\).

Let's calculate the roots for \(x\).The solutions for \(x\) are approximately:

\[
x_1 \approx -1.09, \quad x_2 \approx 4.59
\]

Since \(x\) represents the horizontal distance and must be non-negative, we take the positive value:

\[
x \approx 4.59
\]

### Conclusion:

The ball travels approximately **4.59 feet** horizontally before its first bounce. 

Would you like more details on this solution or have any questions?

### Related Questions:
1. What is the maximum height reached by the ball?
2. How long does it take for the ball to reach its maximum height?
3. What is the vertical distance between the bridge and the ball at its highest point?
4. How far does the ball travel horizontally before hitting the ground if there were no bridge?
5. What would happen if the initial speed of the throw was doubled?

**Tip:** Always check if a quadratic equation has real roots before applying the quadratic formula.

Calculate the horizontal distance traveled by a ball before its first bounce using quadratic equations. Detailed step-by-step solution included. Learn about the intersection of functions in real-world scenarios.

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