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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 0.6 x squared plus 4 x plus 11−0.6x2+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 

enter your response here 

feet

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

To determine how far the ball travels horizontally before its first bounce, we need to find the point where the ball intersects with the bridge. This occurs when the height of the ball (given by the first equation) equals the height of the bridge (given by the second equation). Therefore, we need to solve the following system of equations:

### Given Equations:
\[
y = -0.6x^2 + 4x + 11
\]
\[
y = -0.2x + 8
\]

### Step 1: Set the equations equal to each other.
Since both equations equal \( y \), we set them equal to each other:
\[
-0.6x^2 + 4x + 11 = -0.2x + 8
\]

### Step 2: Rearrange the equation to set it equal to zero.
Move all terms to one side of the equation:
\[
-0.6x^2 + 4x + 11 + 0.2x - 8 = 0
\]
Simplify:
\[
-0.6x^2 + 4.2x + 3 = 0
\]

### Step 3: Multiply the entire equation by \(-10\) to eliminate decimals.
\[
6x^2 - 42x - 30 = 0
\]

### Step 4: Simplify the equation by dividing by 6.
\[
x^2 - 7x - 5 = 0
\]

### Step 5: Solve the quadratic equation using the quadratic formula.
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( x^2 - 7x - 5 = 0 \), \( a = 1 \), \( b = -7 \), and \( c = -5 \).

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-5)}}{2(1)}
\]
\[
x = \frac{7 \pm \sqrt{49 + 20}}{2}
\]
\[
x = \frac{7 \pm \sqrt{69}}{2}
\]
\[
x = \frac{7 \pm 8.31}{2}
\]

This gives two potential solutions for \( x \):
\[
x_1 = \frac{7 + 8.31}{2} = \frac{15.31}{2} = 7.66 \quad (\text{approximately})
\]
\[
x_2 = \frac{7 - 8.31}{2} = \frac{-1.31}{2} = -0.66 \quad (\text{not physically meaningful, since } x \text{ cannot be negative in this context})
\]

### Step 6: Interpret the result.
The horizontal distance before the first bounce is approximately \( \boxed{7.66} \) feet.

Would you like further details or have any questions?

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Here are five related questions:
1. How do you use the quadratic formula to solve quadratic equations?
2. What is the significance of the vertex in a parabola?
3. How do you determine if a quadratic equation has real or complex roots?
4. What are the real-life applications of quadratic equations?
5. How do you find the point of intersection between two curves?

**Tip:** When solving a quadratic equation, always check the sign of the discriminant to know the nature of the roots (real or complex).

Learn how to calculate the horizontal distance traveled by a ball before its first bounce using quadratic equations. Detailed step-by-step solution provided with a focus on solving for intersections of curves. Suitable for high school students.

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