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nika2105 [10]
3 years ago
6

Two boats depart from a port located at (–8, 1) in a coordinate system measured in kilometers and travel in a positive x-directi

on. The first boat follows a path that can be modeled by a quadratic function with a vertex at (1, 10), whereas the second boat follows a path that can be modeled by a quadratic function with a vertex at (0, –7). Which system of equations can be used to determine whether the paths of the boats cross?

Mathematics
2 answers:
miss Akunina [59]3 years ago
7 0

Answer:

\left\{\begin{array}{l}y=-\dfrac{1}{9}x^2 +\dfrac{2}{9}x+\dfrac{89}{9}\\ \\y=\dfrac{1}{8}x^2 -7\end{array}\right.

Step-by-step explanation:

1st boat:

Parabola equation:

y=ax^2 +bx+c

The x-coordinate of the vertex:

x_v=-\dfrac{b}{2a}\Rightarrow -\dfrac{b}{2a}=1\\ \\b=-2a

Equation:

y=ax^2 -2ax+c

The y-coordinate of the vertex:

y_v=a\cdot 1^2-2a\cdot 1+c\Rightarrow a-2a+c=10\\ \\c-a=10

Parabola passes through the point (-8,1), so

1=a\cdot (-8)^2-2a\cdot (-8)+c\\ \\80a+c=1

Solve:

c=10+a\\ \\80a+10+a=1\\ \\81a=-9\\ \\a=-\dfrac{1}{9}\\ \\b=-2a=\dfrac{2}{9}\\ \\c=10-\dfrac{1}{9}=\dfrac{89}{9}

Parabola equation:

y=-\dfrac{1}{9}x^2 +\dfrac{2}{9}x+\dfrac{89}{9}

2nd boat:

Parabola equation:

y=ax^2 +bx+c

The x-coordinate of the vertex:

x_v=-\dfrac{b}{2a}\Rightarrow -\dfrac{b}{2a}=0\\ \\b=0

Equation:

y=ax^2+c

The y-coordinate of the vertex:

y_v=a\cdot 0^2+c\Rightarrow c=-7

Parabola passes through the point (-8,1), so

1=a\cdot (-8)^2-7\\ \\64a-7=1

Solve:

a=-\dfrac{1}{8}\\ \\b=0\\ \\c=-7

Parabola equation:

y=\dfrac{1}{8}x^2 -7

System of two equations:

\left\{\begin{array}{l}y=-\dfrac{1}{9}x^2 +\dfrac{2}{9}x+\dfrac{89}{9}\\ \\y=\dfrac{1}{8}x^2 -7\end{array}\right.

UNO [17]3 years ago
4 0

Answer:

The answer is A

Step-by-step explanation:

Just took the test

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