Sure! We have two equations that represent the paths of the frisbee and the dog:
1. The equation of the frisbee's path is [tex]\( y = -x^2 + 8x + 7 \)[/tex].
2. The equation of the dog's path is [tex]\( y = 2x + 7 \)[/tex].
To determine if the dog will catch the frisbee, we need to find the points where these two paths intersect. This is done by setting the two equations equal to each other and solving for [tex]\( x \)[/tex]:
[tex]\[
-x^2 + 8x + 7 = 2x + 7
\][/tex]
First, we simplify the equation by moving all terms to one side of the equation:
[tex]\[
-x^2 + 8x + 7 - 2x - 7 = 0
\][/tex]
[tex]\[
-x^2 + 6x = 0
\][/tex]
Next, we factor the equation:
[tex]\[
-x(x - 6) = 0
\][/tex]
This gives us the solutions:
[tex]\[
x = 0 \quad \text{or} \quad x = 6
\][/tex]
Now we need to find the corresponding [tex]\( y \)[/tex]-coordinates for these [tex]\( x \)[/tex]-values using either of the original equations. We'll use the equation [tex]\( y = 2x + 7 \)[/tex]:
For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 2(0) + 7 = 7
\][/tex]
So, one intersection point is [tex]\( (0, 7) \)[/tex].
For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = 2(6) + 7 = 19
\][/tex]
So, the other intersection point is [tex]\( (6, 19) \)[/tex].
Therefore, the coordinates where the dog will catch the frisbee are [tex]\( (0, 7) \)[/tex] and [tex]\( (6, 19) \)[/tex].
The correct answer is:
Yes, they intersect at the coordinates [tex]\((0, 7)\)[/tex] and [tex]\((6, 19)\)[/tex].
