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.

1 answer

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

  1. Distance along the streets (Johnson Ave and 41st St):

    • The distance from her house (42nd Street and Johnson Avenue) to 41st Street is 80 yards (walking south).
    • Then, she walks 60 yards east along 41st Street to get to Edison Avenue.
    • Therefore, the total distance when walking along the streets is: \[ \text{Total distance along streets} = 80 \text{ yards} + 60 \text{ yards} = 140 \text{ yards} \]
  2. Distance through the park:

    • Since the problem suggests walking through the park, we use the diagonal distance from her house (42nd Street and Johnson Avenue) to the bank (41st Street and Edison Avenue).
    • The distance is a right triangle where:
      • One leg (vertical) is the distance from 42nd Street to 41st Street: 80 yards.
      • The other leg (horizontal) is the distance from Johnson Avenue to Edison Avenue: 60 yards.
    • We can use the Pythagorean theorem to find the direct distance through the park: \[ \text{Distance through the park} = \sqrt{(80 \text{ yards})^2 + (60 \text{ yards})^2} \] \[ = \sqrt{6400 + 3600} = \sqrt{10000} = 100 \text{ yards} \]
  3. Calculating the difference:

    • Now we calculate the difference in distance between the street route and the park route: \[ \text{Difference} = \text{Distance along streets} - \text{Distance through the park} \] \[ = 140 \text{ yards} - 100 \text{ yards} = 40 \text{ yards} \]

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