To find the intersection points of the two parabolic paths given by the equations for pumpkins A and B, we need to set their equations equal to each other. The paths of the pumpkins are given by:
- Pumpkin A: [tex]\( y = -0.01x^2 + 0.32x + 4 \)[/tex]
- Pumpkin B: [tex]\( y = -0.01x^2 + 0.18x + 5 \)[/tex]
Setting the equations equal to each other to find the values of [tex]\( x \)[/tex] where the paths intersect:
[tex]\[ -0.01x^2 + 0.32x + 4 = -0.01x^2 + 0.18x + 5 \][/tex]
Simplifying by canceling out [tex]\(-0.01x^2\)[/tex] from both sides:
[tex]\[ 0.32x + 4 = 0.18x + 5 \][/tex]
Subtract [tex]\(0.18x\)[/tex] and 4 from both sides:
[tex]\[ 0.14x = 1 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1}{0.14} \][/tex]
[tex]\[ x = 7.14285714285714 \][/tex]
Now, substituting [tex]\( x = 7.14285714285714 \)[/tex] back into either of the original equations to find the corresponding [tex]\( y \)[/tex]-coordinate. Using the equation for pumpkin A:
[tex]\[ y = -0.01(7.14285714285714)^2 + 0.32(7.14285714285714) + 4 \][/tex]
[tex]\[ y \approx -0.01(51.0204081632653) + 2.285714285714285 + 4 \][/tex]
[tex]\[ y \approx -0.510204081632653 + 2.285714285714285 + 4 \][/tex]
[tex]\[ y \approx 5.77551020408163 \][/tex]
So, the intersection point is at [tex]\( (7.14285714285714, 5.77551020408163) \)[/tex].
### Interpretation:
The intersection point [tex]\((7.14285714285714, 5.77551020408163)\)[/tex] of the two parabolic paths means:
- At a horizontal distance of approximately 7.14 feet from the launch site, both pumpkins have the same height, which is approximately 5.78 feet.
Thus, the correct interpretation among the given choices is:
The horizontal distance of the pumpkins from the launch site when their heights are the same.
