Question

coordinate values on the map are in kilometers. how much shorter is the path straight to the destination than past the charging station? round your final answer only to the nearest kilometer. km

Explanation:

Step1: Find coordinates

Let the car's position be $(-3, 2)$, the charging - station's position be $(3,-3)$, and the destination's position be $(6,3)$.

Step2: Calculate distance from car to charging - station

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For the car $(-3,2)$ and charging - station $(3,-3)$, we have $d_1=\sqrt{(3 + 3)^2+(-3 - 2)^2}=\sqrt{6^2+(-5)^2}=\sqrt{36 + 25}=\sqrt{61}\approx 7.81$ km.

Step3: Calculate distance from charging - station to destination

For the charging - station $(3,-3)$ and destination $(6,3)$, we have $d_2=\sqrt{(6 - 3)^2+(3 + 3)^2}=\sqrt{3^2+6^2}=\sqrt{9 + 36}=\sqrt{45}\approx 6.71$ km.

Step4: Calculate total distance via charging - station

$D_1=d_1 + d_2\approx7.81+6.71 = 14.52$ km.

Step5: Calculate distance from car to destination

For the car $(-3,2)$ and destination $(6,3)$, we have $D_2=\sqrt{(6 + 3)^2+(3 - 2)^2}=\sqrt{9^2+1^2}=\sqrt{81 + 1}=\sqrt{82}\approx 9.06$ km.

Step6: Calculate the difference

The difference $\Delta D=D_1 - D_2\approx14.52-9.06 = 5.46\approx5$ km.

Answer:

5