A music company has a stores in Abby (-3, -2) and Cardenas (3, 6). Find each unit of the coordninate plane represents 1 mile.

a. The company president wants to build a warehouse that is equdistant from the two stores. Write an equation that describes the possible locations.

b. A straight road connects Abby and Cardenas.. The warehouse will be located exactly 4 miles from the road. How many locations are possible?

c. To the nearest tenth of a mile, how far will the warehouse be from each store?

NiteWolf813
Jan 24, 2018

#1**+1 **

**a.**

distance between warehouse and Abby = distance between warehouse and Cardenas

Let (x, y) be the coordinates of the warehouse.

distance between (x, y) and (-3, -2) = distance between (x, y) and (3, 6)

√[ (x + 3)^{2} + (y + 2)^{2} ] = √[ (x - 3)^{2} + (y - 6)^{2} ]

(x + 3)^{2} + (y + 2)^{2} = (x - 3)^{2} + (y - 6)^{2}

x^{2} + 6x + 9 + y^{2} + 4y + 4 = x^{2} - 6x + 9 + y^{2} - 12y + 36

6x + 4y + 4 = -6x - 12y + 36

16y = -12x + 32

y = -3/4x + 2 The warehouse can be any point along this line.

Here's a graph: https://www.desmos.com/calculator/eenwtk2dms

**b.**

The warehouse needs to be on the line y = -3/4x + 2 and needs to be 4 units away from the road. So there are two possible locations.

*edit*

**c.**

I think the answer to this part is 6.4 miles, but I don't have a good explanation.

hectictar
Jan 24, 2018

#1**+1 **

Best Answer

**a.**

distance between warehouse and Abby = distance between warehouse and Cardenas

Let (x, y) be the coordinates of the warehouse.

distance between (x, y) and (-3, -2) = distance between (x, y) and (3, 6)

√[ (x + 3)^{2} + (y + 2)^{2} ] = √[ (x - 3)^{2} + (y - 6)^{2} ]

(x + 3)^{2} + (y + 2)^{2} = (x - 3)^{2} + (y - 6)^{2}

x^{2} + 6x + 9 + y^{2} + 4y + 4 = x^{2} - 6x + 9 + y^{2} - 12y + 36

6x + 4y + 4 = -6x - 12y + 36

16y = -12x + 32

y = -3/4x + 2 The warehouse can be any point along this line.

Here's a graph: https://www.desmos.com/calculator/eenwtk2dms

**b.**

The warehouse needs to be on the line y = -3/4x + 2 and needs to be 4 units away from the road. So there are two possible locations.

*edit*

**c.**

I think the answer to this part is 6.4 miles, but I don't have a good explanation.

hectictar
Jan 24, 2018

#2**+2 **

Here's an explanation for "c"

We have two possible congruent right triangles.....

The midpoint of these two coordinates is (0,2) ......the distance from this point to either store is

sqrt [ (3- 0)^2 + (6 - 2)^2 ] = sqrt (9 + 16) = sqrt (25) = 5 miles

And this is one leg of either right triangle......and the other leg is the distance from the warehouse to the road = 4 miles

So.....the distance from the warehouse to either store is the hypotenuse of these triangles

sqrt ( 5^2 + 4^2) = sqrt ( 41) = 6.4 miles

JUST AS HECTICTAR FOUND....!!!!

CPhill
Jan 24, 2018