A radio tower is located on a coordinate system measured in miles. The range of a signal in a particular direction is modeled by a quadratic function where the boundary of the signal starts at the vertex at (4, 2). It passes through the point (5, 4). A linear road connects points (–3, 7) and (8, 2). Which system of equations can be used to determine whether the road intersects the boundary of the tower’s signal?

Given that the range of the signal is a quadratic function: (x-h)^2 = 4a (y - k) (5-4)^2 = 4a (4 - 2) 4a =1/2 therefore, (x -4)^2 = (1/2)(y – 2) The road is a line. Slope of the road is (7 -2)/(-3 -8) =-5/11 Therefore the equation of the line: 5x + 11y = 5(8) + 11(2) 5x + 11y = 62

Answer Link

shivishivangi1679 shivishivangi1679 · Answer 2 · 2022-07-04T11:53:04+02:00

The system of equations can be used to determine whether the road intersects the boundary of the tower’s signal; y = 2( x - 4 )² + 2 and 5 x + 11 y = 62.

What is the vertex form of a quadratic equation?

If a quadratic equation is written in the form

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

then it is called to be in vertex form.

It is called so because when you plot this equation's graph, you will see the vertex point is on (h,k).

To determine a quadratic function in vertex form:

y = a ( x - 4 )² + 2

4 = a ( 5 - 4 )² + 2

4 = a + 2

a = 2

y = 2 ( x - 4 )² + 2

Then we will determine a linear function in standard form, that passes through the points ( -3, 7 ) and ( 8, 2 ).

7 = -3 m + b

2 = 8 m + b / ·(-1)

7 = - 3 m + b

- 2 = - 8 m - b

5 = - 11 m

m = -5/11,

2 = -5/11 · 8 + b,

b= 2 + 40/11,

b = 62/11

Substitue,

y = - 5/11 x + 62/11 / ·11

11 y = - 5 x + 62

5 x + 11 y = 62

Finally the system of equations is:

y = 2( x - 4 )² + 2

5 x + 11 y = 62

Learn more about vertex form of a quadratic equation here:

https://brainly.com/question/9912128

#SPJ5