Question

Maria is on a hike. If she hikes to the scenic lookout on the following map first, she will have to hike farther than if she went straight to the end of the hike. How much shorter is the path straight to the end of the hike than past the scenic lookout? Round your answer to the near kilometer

Answer 1

In order to find how much shorter is the path straight, we need to find the distance between each pair of points. To find the distance between two points (x1, y1) and (x2, y2), we can use the following formula:

`d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}`

Calculating the distance between "end" (-5, 5) and "Maria" (4, -4), that is, the straight path, we have:

`\begin{gathered} d=\sqrt[]{(4-(-5))^2+(-4-5)^2} \\ d=\sqrt[]{9^2+(-9)^2} \\ d=\sqrt[]{81+81} \\ d=\sqrt[]{162}=9\sqrt[]{2}=12.73 \end{gathered}`

Now, calculating the distance between "end" (-5, 5) and "scenic lookout" (2, 3), we have:

`\begin{gathered} d=\sqrt[]{(2-(-5))^2+(3-5)^2} \\ d=\sqrt[]{7^2+\mleft(-2\mright)^2} \\ d=\sqrt[]{49+4} \\ d=\sqrt[]{53}=7.28 \end{gathered}`

And the distance between "scenic lookout" (2, 3) and "Maria" (4, -4):

`\begin{gathered} d=\sqrt[]{(4-2)^2+(-4-3)^2} \\ d=\sqrt[]{2^2+(-7)^2} \\ d=\sqrt[]{4+49} \\ d=\sqrt[]{53}=7.28 \end{gathered}`

If we sum the distances between "end" and "scenic lookout" and the distance between "scenic lookout" and "Maria", we have:

`7.28+7.28=14.56`

Finally, calculating the difference between this path and the straight path, we have:

`14.56-12.73=1.83`

Rounding to the nearest kilometer, the difference is about 2 kilometers.