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?
To find the distance through the park, we need to calculate the hypotenuse of a right triangle formed by the two legs, which represent the distances from Sylvia's house to 41st Street and from 41st Street to the bank.
The distance from Sylvia's house to 41st Street is 80 yards, and the distance from 41st Street to the bank is 60 yards.
Using the Pythagorean theorem, we can find the length of the hypotenuse:
Hypotenuse^2 = (80^2) + (60^2)
= 6400 + 3600
= 10000
Taking the square root of both sides, we get:
Hypotenuse = √10000
= 100 yards
Therefore, it is shorter for Sylvia to walk through the park because the distance through the park is 100 yards, whereas the distance straight down Johnson Avenue and straight down 41st Street is 80 + 60 = 140 yards.
Therefore, the walk through the park is 140 - 100 = 40 yards shorter.
To find out how much shorter it is for Sylvia to walk through the park instead of going straight down Johnson Avenue and then straight down 41st Street, we need to calculate the total distance for each route and then compare them.
First, let's calculate the distance for the route of going straight down Johnson Avenue and then turning right on 41st Street:
Distance from Sylvia's house to the corner of 42nd Street and Johnson Avenue = 80 yards
Distance from the corner of 42nd Street and Johnson Avenue to the corner of 41st Street and Johnson Avenue = 60 yards
Total distance = 80 yards + 60 yards = 140 yards
Next, let's calculate the distance for the route of walking through the park:
Distance from Sylvia's house to the corner of 42nd Street and Johnson Avenue = 80 yards
Distance from the corner of 42nd Street and Johnson Avenue, through the park, to the corner of 41st Street and Edison Avenue = ?
To calculate the distance through the park, we need to use the Pythagorean Theorem because it forms a right-angled triangle. The two sides of the triangle are 80 yards and 60 yards, with the unknown distance being the hypotenuse.
Using the Pythagorean Theorem:
a² + b² = c²
80² + 60² = c²
6400 + 3600 = c²
10000 = c²
c = √10000
c ≈ 100 yards
Total distance = 80 yards + 100 yards = 180 yards
Therefore, it is 140 - 180 = 40 yards shorter for Sylvia to walk through the park instead of going straight down Johnson Avenue and then straight down 41st Street.
To find out how much shorter it is for Sylvia to walk through the park instead of straight down the streets, we need to calculate the lengths of both paths and compare them.
Let's break it down step by step:
1. Calculate the length of walking down Johnson Avenue (42nd Street to 41st Street):
- Given that the distance from 42nd Street to 41st Street is 80 yards, this path is 80 yards long.
2. Calculate the length of walking along 41st Street (from Johnson Avenue to Edison Avenue):
- Given that the distance from Johnson Avenue to Edison Avenue is 60 yards, this path is 60 yards long.
3. Calculate the length of walking through the park:
- Since the street intersections are 90° angles, we can use the Pythagorean theorem to calculate the diagonal distance from 42nd Street to Edison Avenue.
- The diagonal distance can be calculated as the square root of the sum of the squares of the two legs.
- Leg 1: 80 yards (42nd Street to 41st Street)
- Leg 2: 60 yards (41st Street to Edison Avenue)
- Diagonal distance = √(80^2 + 60^2) = √(6400 + 3600) = √10000 = 100 yards.
4. Compare the lengths of the two paths:
- Length of walking down the streets: 80 yards + 60 yards = 140 yards.
- Length of walking through the park: 100 yards.
Therefore, by walking through the park instead of straight down the streets, Sylvia can save (140 - 100) = 40 yards.