Question

PLEASE HELP!!!!

A sailboat with a sick passenger aboard is following a parabolic path given by the equation y=-2x+8. The boat’s captain sent out a distress signal. A speedboat is trying to catch the sailboat to provide medical aid and is on a path given by the linear equation y=x^2-6x+8. Do the paths of the two boats cross?

solve in Graphically and Algebraically!!!!

please answer correctly!!

Answer 1

• Yes, the paths of the two boat cross and the coordinates of the crosses are found below.

Step-by-step explanation:

1. Graphical solution

First, understand that the descriptions of the two equations are exchanged, i.e. the linear equation is y = - 2x + 8 and the quadratic equation is y = x² - 6x + 8, but that does not change the solution.

The graphical method consist in drawing the two equations in the same coordinate plane and the intersection of the curves are the solutions to the system of equations.

a) Graph of the linear equation, y = - 2x + 8

A line may be easily drawn using two points. You can choose the intercepts. You do it in this way:

• y-intercept: x = 0 ⇒ y = -2(0) + 8 = 8 ⇒ point (0,8)

• x-intercept: y = 0 ⇒ 0 = - 2x + 8 ⇒ 2x = 8 ⇒ x = 4 ⇒ point (4, 0).

• Draw those two points and the line that join them

• See the line in the attached graph.

b) Graph of the quadratic equation, y = x² - 6x + 8

• The shape of a quadratic equation is a parabola

• The positive sign of the leading coefficient (1) tells that the parabola opens upward and its vertex is a minimum.

• Find the y-intercept: x = 0 ⇒ y = 0² - 6(0) + 8 = 8 ⇒ point (0,8)

• Find the x intercepts: y = 0

x² - 6x + 8 = 0

(x - a) (x - b) = 0, where - a - b = - 6 and (- a) × (- b) = 8

(x - 4) (x - 2) = 0 ⇒ x = 4 and x = 2.

⇒ points (2, 0) and (4, 0)

• The axis of symetry is half-way between the two x incercepts:

(2 + 4) / 2 = 3 ⇒ x = 3 is the axis of symetry.

• Find the vertex: the vertex is on the axis of symmetry. Find the correspoinding y-coordinate for x = 3

y = 3² - 6(3) + 8 = 9 - 18 + 8 = - 1 ⇒ vertex ( 3, -1).

Now you have plenty information to sketch the curve.

The graph of the two curves is in the diagram attached. It was drawn witha graphical tool but you can do it in a sheet of paper using the features of the curves found.

The intersection points on the graph are (0,8) and (4,0), meaning tha the solutions to the paths of the two boats cross at those coordinates.

2. Algebraic solution.

a) equal the two equations:

• 
  `x^2-6x+8=-2x+8`

b) Solve:

`x^2-6x+8+2x-8=0\\ \\ x^2-4x=0\\ \\ x(x-4)=0\\ \\ x=0;x-4=0\\ \\ x=0;x=4`

Meaning that the two paths cross at x = 0 and x = 4.

Thus, you can find the y-coordinates substituting those x-coordinates in either equation of the paths. I will use y = -2x +8

• y = -2(0) + 8 = 8

• y = -2(4) + 8 = - 8 + 8 = 0

Thus, the paths of the two boats cross at (0,8) and (4,0), such as we found with the graphical method.