Question

a retailer wants to put 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... locate the point where should the shop be built? options: (0,3), (4,4), (8,5). chart with grid, points including jacksonville (-4,2), madeira, and other locations like ames, kent, lynn, bend

Explanation:

Step1: Identify Coordinates

Jacksonville: \((-4, 2)\), Madeira: \((11, 7)\) (assuming from graph). Ratio \(3:1\), so section formula: \(x=\frac{3\times11 + 1\times(-4)}{3 + 1}\), \(y=\frac{3\times7 + 1\times2}{3 + 1}\)

Step2: Calculate x-coordinate

\(x=\frac{33 - 4}{4}=\frac{29}{4}=7.25\approx8\)

Step3: Calculate y-coordinate

\(y=\frac{21 + 2}{4}=\frac{23}{4}=5.75\approx6\) (close to \((8,5)\) or check options. Wait, maybe Madeira's y is 7? Wait, maybe the graph has Madeira at \((11,7)\) and Jacksonville at \((-4,2)\). The ratio is 3:1 (Jacksonville to shop : shop to Madeira). So section formula for internal division: \(x=\frac{m x_2 + n x_1}{m + n}\), \(y=\frac{m y_2 + n y_1}{m + n}\), where \(m = 3\), \(n = 1\), \((x_1,y_1)=(-4,2)\), \((x_2,y_2)=(11,7)\)
\(x=\frac{3\times11 + 1\times(-4)}{4}=\frac{33 - 4}{4}=\frac{29}{4}=7.25\approx8\)
\(y=\frac{3\times7 + 1\times2}{4}=\frac{21 + 2}{4}=\frac{23}{4}=5.75\approx6\), but option is \((8,5)\). Maybe Madeira's y is 7? Wait, maybe the graph's Madeira is at \((11,7)\) and Jacksonville at \((-4,2)\). The point \((8,5)\) is near Ames? Wait, the options for location: Ames. So the shop should be near Ames, at \((8,5)\)? Wait, the question is to locate the point, and the options for coordinates: \((8,5)\), and location: Ames.

Answer:

The shop should be located at \((8, 5)\) near Ames.