a city block 500ft by 500ft is an unobstructed paved lot, except for a small office building 100ft by 100ft, centered in the middle of the block. What's the shortest distance from the SW corner to the NE corner, going through the paved lot and along (but not through) the building (to the nearest foot)?
The shortest distance from the SW corner of the lot to the NE corner of the lot is along two straight lines which pass through the SE or NW corner of the building.
If the path passes through the SE corner of the building, the x-displacement is 250+50=300 ft, and the y-displacement is 250-50=200 ft. Find the distance by Pythagoras theorem. Since by symmetry, the other leg has the same length, the total distance is 2√(200²+300²)
The diagonal (c) across the lot:
500^2 + 500^2 = c^2
250,000 + 250,000 = 500,000^2
c = 707.11 feet
Less the diagonal through the building:
100^2 + 100^2 = c^2
10,000 + 10,000 = 20,000^2
c = 141.43 feet
707.11 - 141.42 = 565.69 feet
565.69 + 200 = 765.69 = 766 feet
To find the shortest distance from the southwest (SW) corner to the northeast (NE) corner, we can break down the problem into two parts:
1. The distance from the SW corner to the nearest corner of the building.
2. The distance from the NW corner of the building to the NE corner.
For the first part, we need to find the length of the diagonal from the SW corner of the paved lot to the corner of the building. Since the lot is a square with sides measuring 500ft, the diagonal can be calculated using the Pythagorean theorem:
diagonal = sqrt(500^2 + 500^2) = sqrt(250,000 + 250,000) = sqrt(500,000) ≈ 707.1ft.
So, the distance from the SW corner to the nearest corner of the building is approximately 707.1ft.
For the second part, the building is located at the center of the block, which means the distance from the NW corner of the building to the NE corner will be half the length of one side of the building (100ft), since it is centered. Therefore, this distance is 100/2 = 50ft.
Now, we can find the total shortest distance by summing up the two parts:
total distance = 707.1ft + 50ft = 757.1ft.
Therefore, the shortest distance from the SW corner to the NE corner, going through the paved lot but not through the building, is approximately 757.1ft (to the nearest foot).