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A new amusement park is going to be built near 2 major highways. On a cordinate grid of the area, with the scale 1 unit representing 1 km, the park is located at P(3,4). Highway 2 is represented by the equation y=2x+5, and highway 0 is represented by the equation y=-0.5x+2. Determine the coordiantes of the exits that must be built on each highway to result in shortest road to the park.

Guest Mar 1, 2015

Best Answer 

 #1
avatar+88839 
+10

A new amusement park is going to be built near 2 major highways. On a cordinate grid of the area, with the scale 1 unit representing 1 km, the park is located at P(3,4). Highway 2 is represented by the equation y=2x+5, and highway 0 is represented by the equation y=-0.5x+2. Determine the coordiantes of the exits that must be built on each highway to result in shortest road to the park.

 

We first need to find equations of the lines that are perpendicular to these two lines and pass through (3,4)

The equation for the  perpendicular line to Highway 2 is  y - 4 = -.5(x - 3) →  y = .5x + 5.5   (1)

The equation of the perpendicular line to Highway 0 is y - 4 = 2(x -3) →  y = 2x - 2       (2)

We could set the equation for Highway 2 equal to (1) and the equation for Highway 0 equal to (2)  and find the intersection points, but here's a graphical solution...

See the graphs of these lines here..........https://www.desmos.com/calculator/nk5enih0in

The coordinates of the exit on the first line - Highway 2 - should be at (.2, 5.4)

The coordinates of the exit on the second line - Highway 0 - should be at (1.6, 1.2)

 

CPhill  Mar 2, 2015
 #1
avatar+88839 
+10
Best Answer

A new amusement park is going to be built near 2 major highways. On a cordinate grid of the area, with the scale 1 unit representing 1 km, the park is located at P(3,4). Highway 2 is represented by the equation y=2x+5, and highway 0 is represented by the equation y=-0.5x+2. Determine the coordiantes of the exits that must be built on each highway to result in shortest road to the park.

 

We first need to find equations of the lines that are perpendicular to these two lines and pass through (3,4)

The equation for the  perpendicular line to Highway 2 is  y - 4 = -.5(x - 3) →  y = .5x + 5.5   (1)

The equation of the perpendicular line to Highway 0 is y - 4 = 2(x -3) →  y = 2x - 2       (2)

We could set the equation for Highway 2 equal to (1) and the equation for Highway 0 equal to (2)  and find the intersection points, but here's a graphical solution...

See the graphs of these lines here..........https://www.desmos.com/calculator/nk5enih0in

The coordinates of the exit on the first line - Highway 2 - should be at (.2, 5.4)

The coordinates of the exit on the second line - Highway 0 - should be at (1.6, 1.2)

 

CPhill  Mar 2, 2015

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