Questions: A retailer wants to build a new shop between Jacksonville and Madela. 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 Madela. Estimate the location of the new shop. Near what city will the shop be located? Locate the point where should the shop be built?

Solution Steps

• Jacksonville is located at (-2, 2).
• Madeira is located at (12, 6).

• The retailer wants a 3:1 ratio between the distance from Jacksonville to the shop and the distance from the shop to Madeira.
• Let the shop be at point (x, y).
• The distance formula is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

• Let \( D_1 \) be the distance from Jacksonville to the shop and \( D_2 \) be the distance from the shop to Madeira.
• \( D_1 = 3D_2 \).
• Using the distance formula: \[ \sqrt{(x + 2)^2 + (y - 2)^2} = 3 \sqrt{(12 - x)^2 + (6 - y)^2} \]

• Square both sides to eliminate the square roots: \[ (x + 2)^2 + (y - 2)^2 = 9 \left[ (12 - x)^2 + (6 - y)^2 \right] \]
• Expand and simplify: \[ (x + 2)^2 + (y - 2)^2 = 9 \left[ (12 - x)^2 + (6 - y)^2 \right] \] \[ (x^2 + 4x + 4) + (y^2 - 4y + 4) = 9 \left[ (144 - 24x + x^2) + (36 - 12y + y^2) \right] \] \[ x^2 + 4x + 4 + y^2 - 4y + 4 = 9 \left[ 180 - 24x - 12y + x^2 + y^2 \right] \] \[ x^2 + y^2 + 4x - 4y + 8 = 9x^2 + 9y^2 - 216x - 108y + 1620 \] \[ 0 = 8x^2 + 8y^2 - 220x - 104y + 1612 \]

• This is a system of equations that can be solved using algebraic methods or graphing tools.
• Solving this system, we find the coordinates of the shop.

Final Answer