Question

the anderson family needs gas. if they visit the historic landmark on the following map first, they will have to drive farther than if they drive straight to the gas station.
coordinate values on the map are in kilometers.
how much shorter is the path straight to the gas station than past the landmark?
round your final answer only to the nearest kilometer.
\boxed{} km

Explanation:

Step1: Identify Coordinates

First, we determine the coordinates of each location. Let's assume:

• Gas station (GS): Let's say its coordinates are \((-5, -6)\) (from the map, estimating the grid).
• Landmark (L): Coordinates \((-5, 4)\) (since it's on the same x - coordinate as GS, x=-5, y = 4).
• Current location (C): Coordinates \((7, 3)\) (from the map, x = 7, y=3).

Step2: Calculate Distance via Landmark

First, distance from C to L: Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For C\((7,3)\) and L\((-5,4)\):
\(d_{CL}=\sqrt{( - 5-7)^2+(4 - 3)^2}=\sqrt{(-12)^2+1^2}=\sqrt{144 + 1}=\sqrt{145}\approx12.04\)
Then, distance from L to GS: For L\((-5,4)\) and GS\((-5,-6)\), since x - coordinates are the same, distance is \(|4-(-6)| = 10\)
Total distance via landmark: \(d_{via\ landmark}=12.04 + 10=22.04\)

Step3: Calculate Straight - line Distance to GS

For C\((7,3)\) and GS\((-5,-6)\), using distance formula:
\(d_{straight}=\sqrt{(-5 - 7)^2+(-6 - 3)^2}=\sqrt{(-12)^2+(-9)^2}=\sqrt{144 + 81}=\sqrt{225}=15\)

Step4: Find the Difference

Difference in distances: \(d_{via\ landmark}-d_{straight}=22.04 - 15 = 7.04\approx7\) (Rounding to nearest kilometer)

Answer:

7