Question

meg walks to school and work each day and wants to track how far she walks each day. in the morning, meg walks 7 blocks due east to school. after school, she walks 2 blocks north and then 4 blocks west to reach work. she walks straight home from work. how far does she walk in all? enter the correct number in the box. round to the nearest tenth.

Explanation:

Step1: Determine horizontal and vertical distances

Meg walks 7 blocks east to school, then 4 blocks west to work. So the horizontal distance from work to home is \( 7 - 4 = 3 \) blocks (east - west, so net east - west distance is 3 blocks east? Wait, no: from school, she walks 2 north and 4 west to work. Wait, let's re - parse the problem.

Morning: Meg walks 7 blocks due east to school.

After school: she walks 2 blocks north and then 4 blocks west to work.

Then she walks straight home from work.

So, to find the distance from work to home, we can use the Pythagorean theorem. The horizontal (east - west) displacement from home to work: from home to school is 7 east, from school to work is 4 west. So the net horizontal displacement from home to work is \( 7-4 = 3 \) blocks (east direction? Wait, no: home to school is 7 east, school to work is 4 west. So the horizontal distance between home and work is \( 7 - 4=3 \) blocks (in the east - west direction, with home being 7 east from... Wait, maybe better: Let's consider home as the origin \((0,0)\).

- School is at \((7,0)\) (7 blocks east, 0 north).
- After school, she walks 2 blocks north: so from school \((7,0)\) to \((7,2)\), then 4 blocks west: from \((7,2)\) to \((7 - 4,2)=(3,2)\). So work is at \((3,2)\).

Now, the distance from work \((3,2)\) to home \((0,0)\) is given by the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \), where \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(3,2)\).

Step2: Apply the distance formula

Substitute into the formula: \( d=\sqrt{(3 - 0)^2+(2 - 0)^2}=\sqrt{3^2 + 2^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.6 \) (rounded to the nearest tenth). Wait, but wait, the problem says "how far does she walk in all"? Wait, maybe I misread. Wait, the problem: "Meg walks to school and work each day and wants to track how far she walks each day. In the morning, Meg walks 7 blocks due east to school. After school, she walks 2 blocks north and then 4 blocks west to work. She walks straight home from work. How far does she walk in all?"

Oh! I misread. We need to calculate the total distance: morning (school) + after school (to work) + from work to home.

Morning: 7 blocks.

After school: 2 (north) + 4 (west) = 6 blocks.

From work to home: we need to calculate that distance. As above, work is at (3,2) if home is (0,0), school is (7,0). So distance from work to home: \( \sqrt{(3)^2+(2)^2}=\sqrt{13}\approx3.6 \)

Total distance: \( 7+6+\sqrt{13}=13+\sqrt{13}\approx13 + 3.6055=16.6055\approx16.6 \)? Wait, no, wait: Wait, morning: home to school: 7 blocks.

After school: school to work: 2 north + 4 west: 6 blocks.

Work to home: let's recast the coordinates correctly.

Let home be (0,0).

- Home to school: 7 blocks east: school is (7,0).

- School to work: 2 blocks north (so y - coordinate increases by 2: (7,0) to (7,2)) then 4 blocks west (x - coordinate decreases by 4: (7,2) to (3,2)). So work is (3,2).

- Work to home: distance between (3,2) and (0,0) is \( \sqrt{(3 - 0)^2+(2 - 0)^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.6055 \)

Now, total distance: home to school (7) + school to work (2 + 4 = 6) + work to home (\( \sqrt{13}\approx3.6055 \))

Total = \( 7+6+\sqrt{13}=13+\sqrt{13}\approx13 + 3.6055 = 16.6055\approx16.6 \)

Wait, but maybe I made a mistake in the coordinate system. Let's check again.

Alternative approach: The path from home to school to work to home forms a triangle? Wait, home to school is 7 east. School to work is 2 north and 4 west. So the horizontal leg from home to work is \( 7-4 = 3 \) (east - west), and the vertical leg is 2 (north - south). Then the distance from work to home is the hypotenuse of a right triangle with legs 3 and 2. Then total distance is 7 (home - school) + 2 + 4 (school - work) + \( \sqrt{3^2 + 2^2} \) (work - home).

Calculating:

7+2 + 4=13

\( \sqrt{9 + 4}=\sqrt{13}\approx3.6055 \)

Total: \( 13+\sqrt{13}\approx16.6 \)

Answer:

\( 16.6 \)