Question

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? StartLayout Enlarged Left-Brace 1st Row y minus 2 (x minus 4) squared = 2 2nd Row 5 x 11 y = 62 EndLayout StartLayout Enlarged Left-Brace 1st Row y minus (x minus 4) squared = 2 2nd Row 5 x 11 y = 62 EndLayout StartLayout Enlarged Left-Brace 1st Row y minus (x minus 4) squared = 2 2nd Row 11 x 5 y = 2 EndLayout StartLayout Enlarged Left-Brace 1st Row y minus 2 (x minus 4) squared = 2 2nd Row 11 x 5 y = 2 EndLayout.

Answer 1

The system of equations that can be used to determine whether the road intersects the boundary of the tower’s signal is

 $y = 2x^2 - 16x + 34\\11y = -5x + 62$

What is 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 vertex point(peak point) is on (h,k)

For the given situation, the quadratic equation has vertex (h,k) = (4,2) and it passes through (5,4).

If we consider the equation to be of the form $y=a(x-h)^2 + k$,

then, as h = 4, k = 2, it becomes $y = a(x-4)^2 + 2$

Now since it passes through (5,4), thus at x = 5, and y = 4, the equation must be true, or:

$y = a(x-4)^2 + 2|_{(x,y) = (5,4)}\\\\4 = a(5-4)^2 + 2\\4 = a + 2\\a = 2$

Thus, the quadratic equation in consideration is: $y = 2(x-4)^2 + 2 = 2x^2 + 32 -16x + 2\\y = 2x^2 - 16x + 34$

The straight line in consideration connects points (–3, 7) and (8, 2).

Equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$  is

$(y - y_1) = \dfrac{y_2 - y_1}{x_2 - x_1} (x -x_1)$

Since we've got $(x_1, y_1) = (-3,7) \: \rm and \: \: (x_2, y_2) = (8,2)$

We get the equation of the straight line in consideration as:

$(y - y_1) = \dfrac{y_2 - y_1}{x_2 - x_1} (x -x_1)\\\\(y-7) = \dfrac{2-7}{8 - (-3) } (x-(-3))\\\\y - 7 = -\dfrac{5}{11}(x+3)\\\\11y = -5x + 62$

Therefore, the system of equations that can be used to determine whether the road intersects the boundary of the tower’s signal is

$y = 2x^2 - 16x + 34\\11y = -5x + 62$

These equations' graphs' intersection is the solution to this system

Learn more about system of equations here:

https://brainly.com/question/2825832

Answer 2

Answer:

y-2(x-4)^2=2

5x+11y=62

Step-by-step explanation:

a on edge 22