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You are training your dog to catch a frisbee. You are playing in a large field, and you are standing next to your dog when you throw the frisbee. If the path of the frisbee is [tex]$y = -x^2 + 8x + 7$[/tex] and the path of the dog is modeled by [tex]$y = 2x + 7$[/tex], will the dog catch the frisbee? If so, what are the coordinates of the points where they meet?

A. Yes, they intersect at the coordinates [tex]$(0,7)$[/tex] and [tex]$(3,13)$[/tex].
B. Yes, they intersect at the coordinates [tex]$(0,7)$[/tex] and [tex]$(6,19)$[/tex].
C. Yes, they intersect at the coordinates [tex]$(0,7)$[/tex] and [tex]$(7,21)$[/tex].
D. No, the paths do not cross.



Answer :

GinnyAnswer
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].

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