Questions: 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 x^2+y^2=5, and the path of the bean bag can be represented by y=0.5x^2+1.5x-4. To which points could she have tossed her bean bag? (-1,-2) or (-2,1) (1,-2) or (2,1) (-1,2) or (-2,-1) (1,2) or (2,-1)

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 x^2+y^2=5, and the path of the bean bag can be represented by y=0.5x^2+1.5x-4. To which points could she have tossed her bean bag? (-1,-2) or (-2,1) (1,-2) or (2,1) (-1,2) or (-2,-1) (1,2) or (2,-1)
Transcript text: 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 $x^{2}+y^{2}=5$, and the path of the bean bag can be represented by $y=0.5 x^{2}+1.5 x-4$. To which points could she have tossed her bean bag? $(-1,-2)$ or $(-2,1)$ $(1,-2)$ or $(2,1)$ $(-1,2)$ or $(-2,-1)$ $(1,2)$ or $(2,-1)$
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Solution

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Solution Steps

Step 1: Define the Equations

We start with the equations representing the hole and the path of the bean bag. The hole is represented by the equation of a circle: \[ x^2 + y^2 = 5 \] The path of the bean bag is represented by the equation of a parabola: \[ y = 0.5x^2 + 1.5x - 4 \]

Step 2: Substitute and Solve

To find the points where the bean bag could have landed, we substitute the expression for \(y\) from the parabola into the circle's equation: \[ x^2 + (0.5x^2 + 1.5x - 4)^2 = 5 \] Solving this equation yields the \(x\)-coordinates of the intersection points.

Step 3: Find Corresponding \(y\)-Values

After solving for \(x\), we find the corresponding \(y\)-values using the parabola equation. The solutions yield the points: \[ (1, -2) \quad \text{and} \quad (2, 1) \]

Step 4: Check Given Points

We compare the calculated intersection points with the provided options:

  • \((-1, -2)\) or \((-2, 1)\)
  • \((1, -2)\) or \((2, 1)\)
  • \((-1, 2)\) or \((-2, -1)\)
  • \((1, 2)\) or \((2, -1)\)

The points \((1, -2)\) and \((2, 1)\) match the calculated intersection points.

Final Answer

The possible points where Sasha could have tossed her bean bag are \((1, -2)\) and \((2, 1)\). Thus, the answer is: \[ \boxed{(1, -2) \text{ and } (2, 1)} \]

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