Question

you walk along the outside of a park star to answer parts a and b. click the icon to view the graph. a. what is the length of the shortcut in m the length of the shortcut is 2.0 mi. (round to the nearest tenth as needed.) b. what is the total length of the walk in the total length of the walk in the park is (round to the nearest tenth as needed.)

Explanation:

Part a:

Step1: Identify coordinates

The start point (let's say \( P \)) is at \( (0,0) \) and the end point \( Q \) is at \( (1.2, 1.6) \)? Wait, no, wait. Wait, the graph: Wait, the shortcut is the straight line. Wait, maybe the original path is along the sides. Wait, the problem says "the length of the shortcut is 2.0 mi" is given? Wait, no, part a is "What is the length of the shortcut in mi" (maybe the user's image has a typo, but looking at the graph, points: Let's check the coordinates. The point \( R \) is \( (1.2, 0) \), \( Q \) is \( (1.2, 1.6) \)? Wait, no, the x and y axes: x is distance (mi) from 0 to 1.6, y is distance (mi) from 0 to 1.6? Wait, no, the start is at (0,0), then maybe the original path is from (0,0) to (1.2,0) to (1.2,1.6), so the original path length is \( 1.2 + 1.6 = 2.8 \) mi? But the shortcut is the straight line from (0,0) to (1.2,1.6). Wait, using the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). So \( x_1 = 0, y_1 = 0 \), \( x_2 = 1.2, y_2 = 1.6 \). Then \( d = \sqrt{(1.2)^2 + (1.6)^2} = \sqrt{1.44 + 2.56} = \sqrt{4} = 2.0 \) mi. So that's the shortcut length.

Step2: Calculate using distance formula

The distance formula between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \( (x_1,y_1) = (0,0) \) and \( (x_2,y_2) = (1.2,1.6) \). Plugging in: \( d = \sqrt{(1.2 - 0)^2 + (1.6 - 0)^2} = \sqrt{1.44 + 2.56} = \sqrt{4} = 2.0 \) mi.

Step1: Find original path length

The original walk is along the sides: from (0,0) to (1.2,0) (length 1.2 mi) and then from (1.2,0) to (1.2,1.6) (length 1.6 mi). So total original length is \( 1.2 + 1.6 = 2.8 \) mi? Wait, no, wait: Wait, the park's outside walk: maybe the start is at (0,0), then to (1.2,0) (R), then to (1.2,1.6) (Q). So the length of the walk in the park (original path) is \( 1.2 + 1.6 = 2.8 \) mi. Wait, but let's confirm. The x-distance from 0 to 1.2 is 1.2 mi, y-distance from 0 to 1.6 is 1.6 mi. So sum is 1.2 + 1.6 = 2.8 mi.

Step2: Round if needed

The problem says "round to the nearest tenth as needed", but 2.8 is already to the nearest tenth.

Answer:

2.0 mi

Part b: