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 calculate the distance Sylvia would walk if she walks down Johnson Avenue and turns right on 41st Street, we need to find the length of the two sides of the right triangle formed by Johnson Avenue and 41st Street.
The length of the side along Johnson Avenue is 80 yards.
The length of the side along 41st Street is 60 yards.
To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's denote the length of the hypotenuse as h.
According to the Pythagorean theorem, we have the equation:
h^2 = 80^2 + 60^2.
Simplifying this equation, we get:
h^2 = 6400 + 3600,
h^2 = 10000.
Taking the square root of both sides, we get:
h = √10000,
h = 100 yards.
Therefore, if Sylvia walks down Johnson Avenue and turns right on 41st Street, she has to walk a total distance of 80 + 60 + 100 = 240 yards.
Now let's calculate the distance Sylvia would walk if she walks through the park.
To find the distance Sylvia would walk through the park, we need to calculate the length of the diagonal connecting the two street intersections. This diagonal represents the shortest path Sylvia can take.
Since the street intersections are at 90° angles, we can use the Pythagorean theorem again to calculate the length of this diagonal.
Let's denote the length of the diagonal as d.
According to the Pythagorean theorem, we have the equation:
d^2 = 80^2 + 60^2.
Simplifying this equation, we get:
d^2 = 6400 + 3600,
d^2 = 10000.
Taking the square root of both sides, we get:
d = √10000,
d = 100 yards.
Therefore, if Sylvia walks through the park, she has to walk a distance of 100 yards.
To find how much shorter it is for Sylvia to walk through the park, we subtract the distance of walking through the park from the distance of walking down Johnson Avenue and turning right on 41st Street:
240 - 100 = 140 yards.
Therefore, it is 140 yards shorter for Sylvia to walk through the park.