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I NEED HELP I WILL GIVE YOU BRAINLES you just have to tell me how to do it

I NEED HELP I WILL GIVE YOU BRAINLES you just have to tell me how to do it-example-1
User Avishay
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1 Answer

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8 votes

Answer:

  1. Hwy 2: (0.2, 5.4); Hwy 10: (1.6, 1.2)
  2. x+2y = 4, x+2y = 11; 2x-y = 2, 2x-y = -5

Explanation:

Scenario 1

In this scenario, you are given a point and the equations of two lines, and you are asked for the point on each line nearest to the given point.

There are several ways you could approach this. Perhaps the most straightforward is to write the equation of the perpendicular line through the given point, then solve the system of equations for the point of intersection between the given line and its perpendicular.

We note that you are asked to use the desmos graphing tool. That tool simplifies a lot of this effort. It easily graphs the equations, and it tells you the points at which the lines intersect.

Exit from Hwy 2

The equation for Hwy 2 is y=2x+5. This is in slope-intercept form. The slope is the coefficient of x, which is 2. The perpendicular line will have a slope that is the opposite reciprocal of this: -1/2. That means we can write the point-slope equation of the line from the Hwy 2 exit to the park in point-slope form as ...

y -k = m(x -h) . . . . . . line with slope m through point (h, k)

y -4 = -1/2(x -3) . . . . line with slope -1/2 through point (4, 3), the park

Desmos will plot this equation and show us the point of intersection with Hwy 2. The exit from Hwy 2 is located at (0.2, 5.4).

Exit from Hwy 10

The equation for Hwy 10 is y=-0.5x+2. This has slope -0.5, so its perpendicular has slope -1/-0.5 = 2. The equation of the line from the Hwy 10 exit to the park is ...

y -4 = 2(x -3)

Desmos shows the Hwy 10 exit location to be (1.6, 1.2).

_____

Scenario 2

The two highways in Scenario 1 are perpendicular to each other. The equations for the exit routes are perpendicular to those. This means the four equations of Scenario 1 are four equations that create lines that form a figure with right angles.

As it happens, the distance from each exit to the park is the same, so the rectangle formed is a square. None of these lines is aligned with the grid.

Effectively, the equations of Scenario 1 are a solution to Scenario 2.

Just for consistency, we can write each equation in standard form.

Hwy 2: y = 2x+5 ⇒ 2x -y = -5

Exit from Hwy 2: y -4 = -1/2(x -3) ⇒ 2y -8 = -x +3 ⇒ x +2y = 11

Hwy 10: y = -0.5x +2 ⇒ 2y = -x +4 ⇒ x +2y = 4

Exit from Hwy 10: y -4 = 2(x -3) ⇒ y -4 = 2x -6 ⇒ 2x -y = 2

__

The distance from Exit 2 to the park is ...

d = √((x2 -x1)² +(y2 -y1)²) = √((0.2 -3)² +(5.4 -4)²) = √((-2.8)² +1.4²) = √9.8

The distance from Exit 10 to the park is ...

d = √((1.6 -3)² +(1.2 -4)²) = √((-1.4)² +(-2.8)²) = √9.8

Adjacent sides of the rectangle are the same length, so the rectangle is a square.

I NEED HELP I WILL GIVE YOU BRAINLES you just have to tell me how to do it-example-1
User Chaser
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