At the rectangular intersection of two streets, the crosswalks are 60 feet from north to south, and 80 feet from west to east. If a diagonal crosswalk was installed, how many fewer feet would a person have to walk to get from the NW corner to the SE corner?
you should recognize a scaled-up 3-4-5 right triangle. So, the diagonal is 100 ft.
The rest is a cinch, right?
Why did the pedestrian go to the circus?
To take a shortcut from the NW corner to the SE corner! With a diagonal crosswalk, you could save them some steps. Let's calculate how much.
Using the Pythagorean theorem (a² + b² = c²), where a = 60 (north-south) and b = 80 (west-east), we can find the length of the diagonal crosswalk (c):
c² = 60² + 80²
c² = 3600 + 6400
c² = 10000
c = √10000
c ≈ 100 feet
So, with the diagonal crosswalk, a person would only have to walk about 100 feet to get from the NW corner to the SE corner. Since the original distance would be 60 + 80 = 140 feet, you'd be saving them a whopping 40 feet!
To find out how many fewer feet a person would have to walk from the NW corner to the SE corner with the diagonal crosswalk, we need to compare the distance of the original route with the distance of the new route.
1. Calculate the distance of the original route:
The original route is along the sides of a rectangle. The NW corner connects to the NE corner, and then continues to the SE corner.
- The distance from the NW corner to the NE corner is the width of the rectangle, which is 80 feet.
- The distance from the NE corner to the SE corner is the length of the rectangle, which is 60 feet.
So, the distance of the original route is 80 feet + 60 feet = 140 feet.
2. Calculate the distance of the new route:
The new route uses the diagonal crosswalk, connecting the NW corner to the SE corner.
- The diagonal of a rectangle forms a right triangle with the sides of the rectangle.
- The width and length of the rectangle are the legs of the right triangle.
- Using the Pythagorean theorem, we can find the length of the diagonal.
The diagonal (d) can be calculated using the following equation: d = √(width^2 + length^2)
- The width (w) of the rectangle is 80 feet.
- The length (l) of the rectangle is 60 feet.
Plugging the values into the formula:
d = √(80^2 + 60^2) = √(6400 + 3600) = √10000 = 100 feet.
So, the distance of the new route is 100 feet.
3. Calculate the difference in distance:
To find the difference in distance, subtract the distance of the new route from the distance of the original route.
140 feet - 100 feet = 40 feet.
Therefore, a person would have to walk 40 fewer feet to get from the NW corner to the SE corner with the diagonal crosswalk.
To find out how many fewer feet a person would have to walk to get from the NW corner to the SE corner using a diagonal crosswalk, we need to compare the distances of the two routes: one using the two perpendicular crosswalks and the other using the diagonal crosswalk.
Let's break down the problem:
1. Without the diagonal crosswalk:
The person would have to walk the 60 feet along the north-south crosswalk, then turn and walk the 80 feet along the west-east crosswalk. So, the total distance traveled would be 60 + 80 = 140 feet.
2. With the diagonal crosswalk:
The person would have to walk diagonally from the NW corner to the SE corner. Since this forms a right-angled triangle, we can use the Pythagorean Theorem to determine the length of the diagonal.
The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the two sides are the 60 feet and 80 feet.
Applying the theorem, we have:
diagonal^2 = 60^2 + 80^2
Therefore, diagonal^2 = 3600 + 6400 = 10000
Taking the square root of both sides, we get:
diagonal = √10000
Simplifying, we find that the diagonal = 100 feet.
Thus, with the diagonal crosswalk, the person would only have to walk 100 feet to get from the NW corner to the SE corner.
To determine how many fewer feet the person would have to walk, we subtract the distance with the diagonal crosswalk from the distance without the diagonal crosswalk:
140 feet - 100 feet = 40 feet
Therefore, a person would have to walk 40 fewer feet to get from the NW corner to the SE corner if a diagonal crosswalk was installed.