An air show is scheduled for an airport located on a coordinate system measured in miles. The air traffic controllers have closed the airspace, modeled by a quadratic equation, to non-air show traffic. The boundary of the closed airspace starts at the vertex at $(10,6)$ and passes through the point $(12,7)$. A commuter jet has filed a flight plan that takes it along a linear path from $(-18,14)$ to $(16,-13)$. Which system of equations can be used to determine whether the commuter jet's flight path intersects the closed airspace?
$\begin{array}{l}
y=\frac{1}{4}(x-10)^2+6 \\
y=-\frac{27}{34} x-\frac{5}{17}
\end{array}$
$\begin{array}{l}
y=\frac{1}{4}(x-10)^2+6 \\
y=-2 x-22
\end{array}$
$\begin{array}{l}
y=\frac{1}{4}(x-5)^2+10 \\
y=-\frac{1}{2} x+5
\end{array}$
$\begin{array}{l}
y=\frac{1}{4}(x-5)^2+10 \\
y=-2 x-22
\end{array}$
Answer (2)
The correct system of equations representing the closed airspace and the jet's flight path is { y = 4 1 ( x − 10 ) 2 + 6 y = − 34 27 x − 17 5 . This system allows us to determine if the commuter jet intersects the closed airspace. Therefore, the selected multiple-choice option is the first one.
;
Determine the quadratic equation representing the airspace using the vertex form and the given point: y = 4 1 ( x − 10 ) 2 + 6 .
Calculate the slope of the linear path using the two given points: m = − 34 27 .
Find the equation of the line representing the jet's path using the point-slope form: y = − 34 27 x − 17 5 .
The system of equations is: { y = 4 1 ( x − 10 ) 2 + 6 y = − 34 27 x − 17 5 .
Explanation
Problem Analysis We are given that the airspace is modeled by a quadratic equation with a vertex at ( 10 , 6 ) and passing through the point ( 12 , 7 ) . The commuter jet's path is linear, going from ( − 18 , 14 ) to ( 16 , − 13 ) . Our goal is to find the system of equations that represents this scenario, which will allow us to determine if the jet's path intersects the closed airspace.
Finding the Quadratic Equation The vertex form of a quadratic equation is given by y = a ( x − h ) 2 + k , where ( h , k ) is the vertex. In this case, the vertex is ( 10 , 6 ) , so the equation becomes y = a ( x − 10 ) 2 + 6 .
Determining the Value of a Since the quadratic equation passes through the point ( 12 , 7 ) , we can substitute x = 12 and y = 7 into the equation to find the value of a :
7 = a ( 12 − 10 ) 2 + 6 7 = a ( 2 ) 2 + 6 7 = 4 a + 6 1 = 4 a a = 4 1
Thus, the quadratic equation is y = 4 1 ( x − 10 ) 2 + 6 .
Finding the Linear Equation Now, let's find the equation of the line representing the jet's flight path. The line passes through the points ( − 18 , 14 ) and ( 16 , − 13 ) . The slope of the line, m , is given by:
m = x 2 − x 1 y 2 − y 1 = 16 − ( − 18 ) − 13 − 14 = 34 − 27
Simplifying the Linear Equation Using the point-slope form of a linear equation, y − y 1 = m ( x − x 1 ) , and the point ( − 18 , 14 ) , we have:
y − 14 = − 34 27 ( x − ( − 18 )) y = − 34 27 ( x + 18 ) + 14 y = − 34 27 x − 34 27 ⋅ 18 + 14 y = − 34 27 x − 34 486 + 34 476 y = − 34 27 x − 34 10 y = − 34 27 x − 17 5
The System of Equations Therefore, the system of equations that can be used to determine whether the commuter jet's flight path intersects the closed airspace is:
y = 4 1 ( x − 10 ) 2 + 6 y = − 34 27 x − 17 5
Final Answer The system of equations that represents the airspace and the jet's flight path is:
{ y = 4 1 ( x − 10 ) 2 + 6 y = − 34 27 x − 17 5
Examples
Understanding the intersection of paths, like the jet's flight path and the closed airspace, is crucial in many real-world applications. For example, in robotics, it's used to plan collision-free paths for robots moving in a workspace with obstacles. Similarly, in computer graphics and game development, determining intersections between objects is essential for realistic simulations and interactions. By solving systems of equations like this, we can predict and avoid potential collisions or conflicts in various dynamic environments.
