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?
To walk down Johnson Avenue and turn right on 41st Street, Sylvia would need to walk a total of 80 yards + 60 yards = <<80+60=140>>140 yards.
To walk through the park, Sylvia would need to walk a total of just 80 yards.
Therefore, it is 140 yards - 80 yards = <<140-80=60>>60 yards shorter for Sylvia to walk through the park. Answer: \boxed{60}.
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the distance for both routes and compare them.
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.
- To calculate the total distance for this route, we need to find the hypotenuse of a right-angled triangle, where the two legs are 80 yards and 60 yards.
- We can use the Pythagorean theorem to find the distance:
Distance = √(80^2 + 60^2) = √(6400 + 3600) = √10000 = 100 yards.
2. Walking through the park:
- The distance from 42nd Street to 41st Street is the same as before, which is 80 yards.
- Since Sylvia is walking through the park, we assume she can take a straight line between the two points.
- Therefore, the distance is equal to the hypotenuse of a right-angled triangle with legs of 80 yards and 60 yards.
- Using the Pythagorean theorem again:
Distance = √(80^2 + 60^2) = √(6400 + 3600) = √10000 = 100 yards.
Comparing the distances, we can see that both routes have the same total distance of 100 yards. Therefore, the park route is not shorter; it is the same length as the route along the streets.
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the distance of each route and then compare them.
First, let's calculate the distance if Sylvia walks down Johnson Avenue and turns right on 41st Street. We know that the distance from 42nd Street to 41st Street is 80 yards. Since the streets intersect at a 90° angle, we can use the Pythagorean theorem to calculate the distance from Johnson Avenue to Edison Avenue. The formula for finding the distance using the Pythagorean theorem is:
c^2 = a^2 + b^2
Where c is the hypotenuse and a and b are the other two sides of the right triangle.
In this case, a = 80 yards (distance from 42nd Street to 41st Street) and b = 60 yards (distance from Johnson Avenue to Edison Avenue). Let's solve for c:
c^2 = 80^2 + 60^2
c^2 = 6400 + 3600
c^2 = 10000
c = √10000
c = 100 yards
So the distance from Johnson Avenue to Edison Avenue is 100 yards.
Now let's calculate the distance if Sylvia walks through the park. We can use the Pythagorean theorem again. This time, one side of the triangle is the distance from her house (42nd Street) to the corner of the park (41st Street), which is 80 yards. The other side is the distance from her house (Johnson Avenue) to Edison Avenue, which is 60 yards. Let's solve for the hypotenuse:
c^2 = 80^2 + 60^2
c^2 = 6400 + 3600
c^2 = 10000
c = √10000
c = 100 yards
So the distance through the park is also 100 yards.
Therefore, the distance for both routes is the same – 100 yards. Hence, it is not shorter for Sylvia to walk through the park; both routes have the same distance.