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In a game of cornhole, Sasha tossed a bean bag and it landed at the edge of the hole. The hole can be represented by the equation [tex]x^2 + y^2 = 5[/tex], and the path of the bean bag can be represented by [tex]y = 0.5x^2 - 4x - 4[/tex]. To which points could she have tossed her bean bag?

A. [tex]\((-1, -2)\)[/tex] or [tex]\((-2, 1)\)[/tex]
B. [tex]\((1, -2)\)[/tex] or [tex]\((2, 1)\)[/tex]
C. [tex]\((-1, 2)\)[/tex] or [tex]\((-2, -1)\)[/tex]
D. [tex]\((1, 2)\)[/tex] or [tex]\((2, -1)\)[/tex]

Sagot :

GinnyAnsweridn
To determine whether Sasha's bean bag landed at any of the given points on the edge of the hole, we need to check which points satisfy the equation of the hole: [tex]\(x^2 + y^2 = 5\)[/tex].

Let's evaluate each set of points:

1. Point [tex]\((-1, -2)\)[/tex]:
[tex]\[ x = -1, \quad y = -2 \][/tex]
[tex]\[ (-1)^2 + (-2)^2 = 1 + 4 = 5 \][/tex]
So, [tex]\((-1, -2)\)[/tex] is on the edge of the hole.

2. Point [tex]\((-2, 1)\)[/tex]:
[tex]\[ x = -2, \quad y = 1 \][/tex]
[tex]\[ (-2)^2 + 1^2 = 4 + 1 = 5 \][/tex]
So, [tex]\((-2, 1)\)[/tex] is on the edge of the hole.

3. Point [tex]\((1, -2)\)[/tex]:
[tex]\[ x = 1, \quad y = -2 \][/tex]
[tex]\[ 1^2 + (-2)^2 = 1 + 4 = 5 \][/tex]
So, [tex]\((1, -2)\)[/tex] is on the edge of the hole.

4. Point [tex]\((2, 1)\)[/tex]:
[tex]\[ x = 2, \quad y = 1 \][/tex]
[tex]\[ 2^2 + 1^2 = 4 + 1 = 5 \][/tex]
So, [tex]\((2, 1)\)[/tex] is on the edge of the hole.

5. Point [tex]\((-1, 2)\)[/tex]:
[tex]\[ x = -1, \quad y = 2 \][/tex]
[tex]\[ (-1)^2 + 2^2 = 1 + 4 = 5 \][/tex]
So, [tex]\((-1, 2)\)[/tex] is on the edge of the hole.

6. Point [tex]\((-2, -1)\)[/tex]:
[tex]\[ x = -2, \quad y = -1 \][/tex]
[tex]\[ (-2)^2 + (-1)^2 = 4 + 1 = 5 \][/tex]
So, [tex]\((-2, -1)\)[/tex] is on the edge of the hole.

7. Point [tex]\((1, 2)\)[/tex]:
[tex]\[ x = 1, \quad y = 2 \][/tex]
[tex]\[ 1^2 + 2^2 = 1 + 4 = 5 \][/tex]
So, [tex]\((1, 2)\)[/tex] is on the edge of the hole.

8. Point [tex]\((2, -1)\)[/tex]:
[tex]\[ x = 2, \quad y = -1 \][/tex]
[tex]\[ 2^2 + (-1)^2 = 4 + 1 = 5 \][/tex]
So, [tex]\((2, -1)\)[/tex] is on the edge of the hole.

Since all the given points:

[tex]\[ (-1, -2), (-2, 1), (1, -2), (2, 1), (-1, 2), (-2, -1), (1, 2), (2, -1) \][/tex]

all satisfy the equation [tex]\(x^2 + y^2 = 5\)[/tex], they all lie on the edge of the hole. Therefore, any of these points could represent where Sasha’s bean bag landed.
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