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°

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?

(1 point)
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 determine how much shorter Sylvia's walk through the park is compared to walking straight down Johnson Avenue and turning right on 41st Street, we can analyze both routes.

Route down Johnson Avenue and turning right on 41st Street:
- The distance from Sylvia's house at 42nd Street to 41st Street is 80 yards (vertical distance).
- The distance from Johnson Avenue to Edison Avenue is 60 yards (horizontal distance).
- Therefore, the total distance for this route is: \[ 80 \text{ yards} + 60 \text{ yards} = 140 \text{ yards} \]
Route through the park:
- This route can be visualized as the hypotenuse of a right triangle where:
  - One side (vertical leg) is 80 yards (from 42nd Street to 41st Street).
  - The other side (horizontal leg) is 60 yards (from Johnson Avenue to Edison Avenue).
- We can use the Pythagorean theorem to find the distance: \[ \text{Distance} = \sqrt{(80 \text{ yards})^2 + (60 \text{ yards})^2} \] \[ = \sqrt{6400 + 3600} = \sqrt{10000} = 100 \text{ yards} \]
Difference in distances:
- Now, we calculate how much shorter the walk through the park is compared to walking straight down the streets: \[ \text{Difference} = \text{Distance via streets} - \text{Distance through park} \] \[ = 140 \text{ yards} - 100 \text{ yards} = 40 \text{ yards} \]

Thus, if Sylvia walks through the park instead of straight down Johnson Avenue and down 41st Street, the walk will be 40 yards shorter.