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 to Johnson Avenue and turn right on 41st street, or she can walk through the park. The distance from 42nd 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?
To walk through the park, Sylvia would only need to walk along one side of the right triangle formed by the streets. The two legs of the triangle are 80 yards and 60 yards. To find the hypotenuse, we can use the Pythagorean theorem:
c^2 = a^2 + b^2
where c is the hypotenuse and a and b are the legs of the triangle.
c^2 = 80^2 + 60^2
c^2 = 6400 + 3600
c^2 = 10000
Taking the square root of both sides, we find:
c = 100 yards
Therefore, the distance Sylvia would need to walk through the park is 100 yards.
To walk through the streets, Sylvia would need to walk 80 yards and then 60 yards, for a total of 80 + 60 = <<80+60=140>>140 yards.
Therefore, by walking through the park, Sylvia would walk 140 - 100 = <<140-100=40>>40 yards less. Answer: \boxed{40}.
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the distance of both routes and compare them.
Let's start by calculating the distance of the first route, where Sylvia walks down to Johnson Avenue and turns right on 41st street.
From Sylvia's house to Johnson Avenue is 80 yards, and from Johnson Avenue to the bank (Edison Avenue) is 60 yards. Since these distances form the legs of a right-angled triangle, we can use the Pythagorean theorem to find the distance:
Distance = √(80² + 60²)
Distance ≈ √(6400 + 3600)
Distance ≈ √10000
Distance ≈ 100 yards
Now, let's calculate the distance of the second route, where Sylvia walks through the park.
The distance from Sylvia's house (42nd street) to the bank (41st street and Edison Avenue) can be calculated directly as the hypotenuse of a right-angled triangle, using the distances provided.
Distance = √(80² + 60²)
Distance ≈ √(6400 + 3600)
Distance ≈ √10000
Distance ≈ 100 yards
Comparing the two distances, we can see that both routes have the same distance of 100 yards. Therefore, walking through the park does not provide a shorter route for Sylvia.
To determine how much shorter it is for Sylvia to walk through the park, we need to calculate the distance for both routes: walking through Johnson Avenue and turning right on 41st street, and walking through the park.
1. Walking through Johnson Avenue:
- Distance from 42nd street to the corner of Johnson Avenue and Edison Avenue is 80 yards.
- Distance from Johnson Avenue to Edison Avenue is 60 yards.
- Total distance = 80 + 60 = 140 yards.
2. Walking through the park:
- Distance from 42nd street to the corner of 41st street and Edison Avenue can be calculated using the Pythagorean theorem.
- The distance between 42nd street and 41st street is 1 street unit, which is equivalent to the side of a right-angled triangle.
- The distance between Johnson Avenue and Edison Avenue is also 1 street unit.
- Using the Pythagorean theorem, the diagonal distance between these two points is √(1^2 + 1^2) = √2.
Now, to calculate the actual distance through the park:
- From the corner of 42nd street and Johnson Avenue, Sylvia walks a distance of 80 yards.
- She then turns onto 41st street, where the diagonal distance (√2) is equivalent to 1 street unit.
- Multiplying the diagonal distance (√2) by the street unit length (60 yards) gives: √2 * 60 = 84.85 yards (rounded to two decimal places).
Therefore, the total distance of walking through the park is approximately 80 + 84.85 = 164.85 yards.
Now, to find the difference between the two routes, we subtract the distance through the park from the distance via Johnson Avenue:
140 - 164.85 ≈ -24.85 yards.
This negative value implies that it is 24.85 yards longer to walk through the park than to walk through Johnson Avenue.