Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41st Street and Edison Avenue. Her house is at the corner of 42nd Street and Johnson Avenue. She can walk down Johnson Avenue and turn right on 41st Street, or she can walk through the park. The distance from 42nd Street to 41st Street is 80 yards, and it is 60 yards from Johnson Avenue to Edison Avenue. Assuming the street intersections are 90° angles, how much shorter is it for Sylvia to walk through the park?

Question

Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41st Street and Edison Avenue. Her house is at the corner of 42nd Street and Johnson Avenue. She can walk down Johnson Avenue and turn right on 41st Street, or she can walk through the park. The distance from 42nd Street to 41st Street is 80 yards, and it is 60 yards from Johnson Avenue to Edison Avenue. Assuming the street intersections are 90°  angles, how much shorter is it for Sylvia to walk through the park?   
If Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be  yards shorter.

GPT-4o mini · Answer

To find out how much shorter it is for Sylvia to walk through the park rather than taking the streets, we need to calculate the two different distances.

Walking down Johnson Avenue and then turning onto 41st Street:
- The distance from her house (corner of 42nd Street and Johnson Avenue) to the corner of 42nd Street and 41st Street is 80 yards (distance between the two streets).
- The distance from 41st Street and Johnson Avenue to the bank (corner of 41st Street and Edison Avenue) is 60 yards (distance between the two avenues).
Therefore, the total distance using this route is: \[ 80 \text{ yards} + 60 \text{ yards} = 140 \text{ yards} \]
Walking through the park:
- The path she takes through the park forms a right triangle, where the legs of the triangle are 80 yards (the distance between 42nd Street and 41st Street) and 60 yards (the distance between Johnson Avenue and Edison Avenue).
- We can use the Pythagorean theorem to find the distance through the park: \[ d = \sqrt{(80 \text{ yards})^2 + (60 \text{ yards})^2} \] \[ d = \sqrt{6400 + 3600} = \sqrt{10000} = 100 \text{ yards} \]
Calculating the difference:
- Now, to find out how much shorter the park walk is compared to the street route: \[ \text{Difference} = \text{Distance on streets} - \text{Distance through the park} \] \[ \text{Difference} = 140 \text{ yards} - 100 \text{ yards} = 40 \text{ yards} \]

Thus, walking through the park is 40 yards shorter than taking the street route.