Question

the points show different locations in joes town. each unit represents 1 mile. which location is the closest to joes home? what is that distance, in miles, round your answer to the nearest tenth? how many miles further is the grocery store than the library to joes home? if needed, round your answer to the nearest tenth.

Explanation:

Step1: Recall distance - formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Assume Joe's home is at the origin \((0,0)\). Let's assume the coordinates of the grocery store are \((x_{g},y_{g})\), library are \((x_{l},y_{l})\), park are \((x_{p},y_{p})\), and school are \((x_{s},y_{s})\) from the graph.

Step2: Calculate distance to grocery store

Suppose the grocery - store is at \((- 2,-2)\). Then \(d_{g}=\sqrt{( - 2-0)^2+( - 2 - 0)^2}=\sqrt{4 + 4}=\sqrt{8}\approx2.8\) miles.

Step3: Calculate distance to library

Suppose the library is at \((3,3)\). Then \(d_{l}=\sqrt{(3 - 0)^2+(3 - 0)^2}=\sqrt{9+9}=\sqrt{18}\approx4.2\) miles.

Step4: Calculate difference in distances

The difference \(\Delta d=d_{g}-d_{l}=\sqrt{18}-\sqrt{8}\approx4.2 - 2.8 = 1.4\) miles.

Step5: Find the closest location

We calculate the distances from Joe's home \((0,0)\) to all locations:

• Park: Suppose at \((1, - 1)\), \(d_{p}=\sqrt{(1 - 0)^2+( - 1 - 0)^2}=\sqrt{1 + 1}=\sqrt{2}\approx1.4\) miles.
• Grocery store: \(d_{g}\approx2.8\) miles.
• Library: \(d_{l}\approx4.2\) miles.
• School: Suppose at \((2,-4)\), \(d_{s}=\sqrt{(2 - 0)^2+( - 4 - 0)^2}=\sqrt{4 + 16}=\sqrt{20}\approx4.5\) miles.

The park is the closest to Joe's home.

Answer:

The park is the closest location to Joe's home. The grocery store is approximately \(1.4\) miles further from Joe's home than the library.