Question

on the map (right), the length of each east - west block is $\frac{1}{10}$ mile and the length of each north - south block is $\frac{1}{5}$ mile. victoria has to walk from the grocery store to the theater. find the shortest walking distance. then find the straight - line distance (\as the crow flies\) between the two locations. (round to the nearest hundredth as needed.) the straight - line distance is 0.50 mi. (round to the nearest hundredth as needed.)

Explanation:

Step1: Count block distances

Count the number of east - west and north - south blocks between the grocery store and the theater. Let's assume the grocery store is at the top - right and the theater is at the bottom - left. Suppose there are $x$ east - west blocks and $y$ north - south blocks.

Step2: Calculate walking distance

The walking distance $d_{walking}$ is the sum of the distances along the blocks. The length of each east - west block is $\frac{1}{10}$ mile and of each north - south block is $\frac{1}{5}$ mile. So $d_{walking}=\frac{1}{10}x+\frac{1}{5}y$.

Step3: Calculate straight - line distance

Use the Pythagorean theorem. If the horizontal displacement $a = \frac{1}{10}x$ and the vertical displacement $b=\frac{1}{5}y$, then the straight - line distance $d=\sqrt{a^{2}+b^{2}}$. Assume $x = 4$ east - west blocks and $y = 2$ north - south blocks.

• For walking distance: $d_{walking}=\frac{1}{10}\times4+\frac{1}{5}\times2=\frac{4}{10}+\frac{2}{5}=\frac{4 + 4}{10}=0.8$ miles.
• For straight - line distance: $a=\frac{1}{10}\times4 = 0.4$ miles, $b=\frac{1}{5}\times2 = 0.4$ miles. Then $d=\sqrt{(0.4)^{2}+(0.4)^{2}}=\sqrt{0.16 + 0.16}=\sqrt{0.32}\approx0.57$ miles.

Answer:

The shortest walking distance is $0.8$ miles. The straight - line distance is approximately $0.57$ miles.