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

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.

Answer 1

To find out how much shorter it is for Sylvia to walk through the park instead of down Johnson Avenue and then to 41st Street, we can calculate the distances for both routes.

1. Walking straight down Johnson Avenue and then turning right on 41st Street:

   • The distance from 42nd Street to 41st Street is 80 yards (this is the distance along Johnson Avenue).
   • Then, she walks 60 yards along 41st Street to reach Edison Avenue.

   Total distance via Johnson Avenue: \[ 80 \text{ yards} + 60 \text{ yards} = 140 \text{ yards} \]

2. Walking through the park:

   • This forms a right triangle where one leg is 80 yards (the distance from 42nd Street to 41st Street) and the other leg is 60 yards (the distance from Johnson Avenue to Edison Avenue).
   • We can use the Pythagorean theorem to calculate the direct distance through the park: \[ d = \sqrt{(80)^2 + (60)^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100 \text{ yards} \]

3. Comparing the distances:

   • The distance if Sylvia walks through the park: 100 yards.
   • The distance along Johnson Avenue and 41st Street: 140 yards.

4. Calculating how much shorter the park route is: \[ 140 \text{ yards} - 100 \text{ yards} = 40 \text{ yards} \]

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