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.

coordinate plane with points labeled: maria at (3, -4), scenic lookout at (2, 3), end at (-4, 5).

coordinate values on the map are in kilometers.
how much shorter is the path straight to the end of the hike than past the scenic lookout?
round your final answer only to the nearest kilometer.
km

Explanation:

Step1: Identify Coordinates

Maria's coordinates: \((3, -4)\)
Scenic lookout: \((2, 3)\)
End: \((-4, 5)\)

Step2: Distance Formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Distance Maria to Scenic Lookout:

\(x_1 = 3, y_1 = -4\); \(x_2 = 2, y_2 = 3\)
\(d_1 = \sqrt{(2 - 3)^2 + (3 - (-4))^2} = \sqrt{(-1)^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} \approx 7.07\)

Distance Scenic Lookout to End:

\(x_1 = 2, y_1 = 3\); \(x_2 = -4, y_2 = 5\)
\(d_2 = \sqrt{(-4 - 2)^2 + (5 - 3)^2} = \sqrt{(-6)^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32\)

Total via Scenic Lookout:

\(d_{\text{total}} = d_1 + d_2 \approx 7.07 + 6.32 = 13.39\)

Distance Maria to End (Straight):

\(x_1 = 3, y_1 = -4\); \(x_2 = -4, y_2 = 5\)
\(d_{\text{straight}} = \sqrt{(-4 - 3)^2 + (5 - (-4))^2} = \sqrt{(-7)^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.40\)

Step3: Difference

\(\text{Difference} = d_{\text{total}} - d_{\text{straight}} \approx 13.39 - 11.40 \approx 2\)

Answer:

2