Question

joes commute to work can be represented in the coordinate plane as follows: his house is at h (0,0). his favorite coffee - shop is at c (7,6) where he stops every morning. his office is at w (4,13). he goes back home from his office every day without stopping. part a. assume that joe lives in a city where the roads are parallel to the coordinate axes and each intersection occurs at integer coordinates. represent his route on the coordinate plane where each city block is one coordinate unit by one coordinate unit, which measures 175 yards by 175 yards. part b. what is the total distance, in yards, that joe commutes every day, if he stays on the roads? part c. if joe could take the most direct route (cutting across city blocks) for his commute, what would be his total distance, in yards, that he commutes every day? check for understanding: find the perimeter and area of the figure.

Explanation:

Step1: Calculate distance from home (0,0) to coffee - shop (7,6) on grid

For moving along grid - lines from (0,0) to (7,6), we move 7 units horizontally and 6 units vertically. The distance $d_1$ in grid units is $7 + 6=13$ units. Since each unit is 175 yards, the distance in yards is $d_{1y}=13\times175 = 2275$ yards.

Step2: Calculate distance from coffee - shop (7,6) to office (4,13) on grid

We move $|7 - 4|=3$ units horizontally and $|13 - 6| = 7$ units vertically. The distance $d_2$ in grid units is $3+7 = 10$ units. In yards, $d_{2y}=10\times175=1750$ yards.

Step3: Calculate distance from office (4,13) to home (0,0) on grid

We move $|4 - 0| = 4$ units horizontally and $|13 - 0|=13$ units vertically. The distance $d_3$ in grid units is $4 + 13=17$ units. In yards, $d_{3y}=17\times175 = 2975$ yards.

Step4: Find total distance for part B

The total distance $D_B$ when staying on roads is $D_B=d_{1y}+d_{2y}+d_{3y}=2275 + 1750+2975=7000$ yards.

Step5: Calculate straight - line distance from home (0,0) to coffee - shop (7,6)

Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, for points $(0,0)$ and $(7,6)$, $d_{s1}=\sqrt{(7 - 0)^2+(6 - 0)^2}=\sqrt{49 + 36}=\sqrt{85}$ units. In yards, $d_{s1y}=\sqrt{85}\times175$ yards.

Step6: Calculate straight - line distance from coffee - shop (7,6) to office (4,13)

For points $(7,6)$ and $(4,13)$, $d_{s2}=\sqrt{(4 - 7)^2+(13 - 6)^2}=\sqrt{(-3)^2+7^2}=\sqrt{9 + 49}=\sqrt{58}$ units. In yards, $d_{s2y}=\sqrt{58}\times175$ yards.

Step7: Calculate straight - line distance from office (4,13) to home (0,0)

For points $(4,13)$ and $(0,0)$, $d_{s3}=\sqrt{(4 - 0)^2+(13 - 0)^2}=\sqrt{16 + 169}=\sqrt{185}$ units. In yards, $d_{s3y}=\sqrt{185}\times175$ yards.

Step8: Find total distance for part C

$D_C=( \sqrt{85}+\sqrt{58}+\sqrt{185})\times175\approx(9.22+7.62+13.60)\times175=(30.44)\times175 = 5327$ yards.

Answer:

Part B: 7000 yards
Part C: 5327 yards