To determine how much shorter it is for Sylvia to walk through the park compared to walking down Johnson Avenue and then turning right on 41st Street, we first need to visualize her two routes.
Route 1 (Walking down Johnson Avenue and then turning right):
- Sylvia walks down Johnson Avenue from 42nd Street to 41st Street (1 block).
- Then she turns right on 41st Street and walks to Edison Avenue (1 block).
Thus, the total distance for Route 1 is:
- 1 block (Johnson Avenue) + 1 block (41st Street) = 2 blocks.
Route 2 (Walking through the park):
- Sylvia walks diagonally through the park, which forms a right triangle with her walking distances as the legs of the triangle. The vertical distance between 42nd Street and 41st Street is 1 block, and the horizontal distance from Johnson Avenue to Edison Avenue is also 1 block.
Using the Pythagorean theorem, we determine the hypotenuse (the distance through the park) as follows:
\[ \text{Distance} = \sqrt{(\text{leg1})^2 + (\text{leg2})^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}. \]
The distance through the park is \(\sqrt{2}\) blocks.
Comparison:
- Route 1 (via streets): 2 blocks
- Route 2 (through the park): \(\sqrt{2}\) blocks
To find out how much shorter it is to walk through the park compared to walking along the streets:
\[ \text{Difference} = 2 - \sqrt{2}. \]
Calculating this difference:
First, we approximate \(\sqrt{2} \approx 1.414\):
\[ 2 - \sqrt{2} \approx 2 - 1.414 = 0.586. \]
Thus, walking through the park is approximately 0.586 blocks shorter than walking the street route.
In conclusion, Sylvia's route through the park is \(\boxed{2 - \sqrt{2}}\) blocks shorter than walking down Johnson Avenue and then turning onto 41st Street, and this value is approximately 0.586 blocks shorter.