Question

a retailer wants to build a new shop between jacksonville and madeira. the retailer wants there to be a 3:1 ratio between the distance from jacksonville to the shop and the distance from the shop to madeira. estimate the location of the new shop. near what city will the shop be located? locate the point. where should the shop be built? options: ames, bend, kent, lynn (image shows a grid with jacksonville at (-4,2), madeira at (12,6), and cities bend, kent, lynn, ames marked on the grid.)

Explanation:

Step1: Recall section formula

The section formula for a point \((x,y)\) dividing the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \((\frac{mx_2 + nx_1}{m + n},\frac{my_2 + ny_1}{m + n})\). Here, Jacksonville is \((-4,2)\), Madeira is \((12,6)\), and the ratio \(m:n = 3:1\) (Jacksonville to shop : shop to Madeira).

Step2: Calculate x - coordinate

Substitute \(x_1=-4\), \(x_2 = 12\), \(m = 3\), \(n = 1\) into the x - coordinate formula: \(\frac{3\times12+1\times(-4)}{3 + 1}=\frac{36-4}{4}=\frac{32}{4}=8\).

Step3: Calculate y - coordinate

Substitute \(y_1 = 2\), \(y_2=6\), \(m = 3\), \(n = 1\) into the y - coordinate formula: \(\frac{3\times6+1\times2}{3 + 1}=\frac{18 + 2}{4}=\frac{20}{4}=5\). So the point is \((8,5)\), which is near Ames? Wait, no, looking at the graph, the point \((8,5)\) is near Ames? Wait, the cities: Kent, Lynn, Ames, Bend. Wait, the coordinates: Jacksonville \((-4,2)\), Madeira \((12,6)\). The ratio 3:1, so the point is closer to Madeira? Wait, no, ratio \(m:n = 3:1\) means from Jacksonville to shop is 3 parts, shop to Madeira is 1 part. So the x - coordinate: \(\frac{3\times12+1\times(-4)}{4}=\frac{36 - 4}{4}=8\), y - coordinate: \(\frac{3\times6+1\times2}{4}=\frac{20}{4}=5\). Looking at the graph, Ames is at (8,7)? Wait, maybe I misread. Wait, the graph: Jacksonville is \((-4,2)\), Bend is at (2,3)? Kent at (4,3)? Lynn at (6,4)? Ames at (8,7)? Madeira at (12,6). Wait, maybe the ratio is 1:3? Wait, the problem says "3:1 ratio between the distance from Jacksonville to the shop and the distance from the shop to Madeira". So Jacksonville --- shop --- Madeira, with J to S : S to M = 3:1. So using section formula, the coordinates of S are \((\frac{3\times12+1\times(-4)}{3 + 1},\frac{3\times6+1\times2}{3 + 1})=(8,5)\). Now, looking at the cities: Lynn is at (6,4), Ames at (8,7), Kent at (4,3), Bend at (2,3). The point (8,5) is near Ames? Wait, maybe the graph's Ames is at (8,6) or something. Alternatively, maybe I made a mistake. Wait, let's re - check. The x - difference between Jacksonville \((-4,2)\) and Madeira \((12,6)\) is \(12-(-4)=16\). The ratio 3:1, so the distance from J to S is \(\frac{3}{4}\) of 16, which is 12. So x - coordinate of S is \(-4 + 12 = 8\). The y - difference is \(6 - 2 = 4\), so distance from J to S in y is \(\frac{3}{4}\times4 = 3\), so y - coordinate is \(2+3 = 5\). So (8,5). Now, looking at the options: Ames is the closest? Wait, the options are Ames, Bend, Kent, Lynn. So the answer should be Ames? Wait, maybe the graph's Ames is at (8,6) or (8,5) - adjacent. So the shop is near Ames.

Answer:

Ames