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)
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the total distance for each route.
Route 1: Walking down Johnson Avenue and turning right on 41st Street:
The distance from 42nd Street to 41st Street is 80 yards.
The distance from Johnson Avenue to Edison Avenue is 60 yards.
Total distance = 80 yards + 60 yards = 140 yards.
Route 2: Walking through the park:
We know the street intersections are 90° angles, which means Sylvia can directly walk from her house to the bank.
The straight-line distance from 42nd Street to Edison Avenue is the hypotenuse of a right-angled triangle with sides 80 yards and 60 yards.
Using the Pythagorean theorem, the straight-line distance is equal to the square root of (80^2 + 60^2) = square root of 6400 + 3600 = square root of 10000 = 100 yards.
Therefore, it is shorter for Sylvia to walk through the park by 140 yards - 100 yards = 40 yards.
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the distance of each option and compare them.
Let's calculate the distance if Sylvia walks down Johnson Avenue and turns right on 41st Street. Since the street intersections are at 90° angles, we can use the Pythagorean theorem to calculate the distance.
The distance from 42nd Street to 41st Street is 80 yards, and the distance from Johnson Avenue to Edison Avenue is 60 yards. So, we have a right-angled triangle with the two sides measuring 80 and 60 yards.
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can calculate the hypotenuse (c) of the triangle, which represents the distance that Sylvia would have to walk.
c^2 = 80^2 + 60^2
c^2 = 6400 + 3600
c^2 = 10000
c = √10000
c = 100 yards
So, if Sylvia walks down Johnson Avenue and turns right on 41st Street, she would have to walk 100 yards.
Now, let's calculate the distance if Sylvia walks through the park. Since the street intersections form a right angle, we can assume that the park is the hypotenuse of another right-angled triangle.
The distance from 42nd Street to 41st Street is still 80 yards, but this time, the other side of the triangle represents the distance of walking through the park. Let's call this side "a."
Using the Pythagorean theorem again, we can solve for a:
a^2 + 80^2 = 100^2
a^2 + 6400 = 10000
a^2 = 10000 - 6400
a^2 = 3600
a = √3600
a = 60 yards
So, if Sylvia walks through the park, she would have to walk 60 yards.
To find out how much shorter it is for Sylvia to walk through the park, we subtract the distance of walking through the park (60 yards) from the distance of walking down Johnson Avenue and turning right on 41st Street (100 yards):
100 - 60 = 40
So, it is 40 yards shorter for Sylvia to walk through the park compared to walking down Johnson Avenue and turning right on 41st Street.
By walking down Johnson Avenue and turning right on 41st Street, Sylvia would need to walk 80 yards + 60 yards = <<80+60=140>>140 yards.
The distance of the hypotenuse walking through the park can be found using the Pythagorean theorem, with one side being 80 yards and the other side being 60 yards.
The square of the hypotenuse is equal to (80 yards)² + (60 yards)² = 6400 yards² + 3600 yards² = 10000 yards².
Therefore, the hypotenuse is sqrt(10000 yards²) = <<sqrt(10000)=100>>100 yards.
Therefore, Sylvia can walk a total of 100 yards - 140 yards = <<100-140=-40>>-40 yards shorter by walking through the park. Answer: \boxed{-40}.