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?

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To determine whether the commuter jet's flight path intersects the closed airspace, we need to find the equation of the boundary of the closed airspace and then check if any part of the flight path lies on that equation.

First, let's find the equation of the boundary. Since the vertex of the quadratic is given as (10, 6), we know that the equation is in the form of:

y = a(x - h)^2 + k

where (h, k) is the vertex. Substituting the given vertex (10, 6), we get:

y = a(x - 10)^2 + 6

Next, we can use the fact that the boundary passes through the point (12, 7). Substituting these values, we get:

7 = a(12 - 10)^2 + 6
7 = 4a + 6
4a = 7 - 6
4a = 1
a = 1/4

So, the equation of the boundary of the closed airspace is:

y = (1/4)(x - 10)^2 + 6

Now, let's check if the flight path intersects this equation. The flight path is a linear equation given by:

y = mx + b

where m is the slope and b is the y-intercept. We can find the slope (m) using the two given points (-18, 14) and (16, -13):

m = (y2 - y1) / (x2 - x1)
m = (-13 - 14) / (16 - (-18))
m = -27 / 34

So, the equation of the flight path becomes:

y = (-27/34)x + b

To determine whether the flight path intersects the closed airspace, we need to solve the system of equations:

y = (1/4)(x - 10)^2 + 6
y = (-27/34)x + b

This can be written as the following system of equations:

(1/4)(x - 10)^2 + 6 = (-27/34)x + b

Simplifying this equation will allow us to determine whether the flight path intersects the closed airspace.

Did you find the equation for the flight path of the plane ?

Did you find the equation of the parabola?
Did you solve the two equations and got a real answer?
If you have a solution, the the plane intersects the closed airspace.
If you have no solution ...... etc

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?

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?