A rectangular field is 1.5 km long and 700 m wide. An asphalt road goes around the outside of the field and a dirt path cuts across the field. A student wants to go from A to B on the field. Path x goes along the road and is shown in red; path y is shown in blue. Which path has the shorter distance? By how much is the path shorter?
How do I calculate this?
I have no idea where the paths are, or points A, B,nor Path x
There is a rectanular field. Point A is in one corner of the field, and Point B is the corner diagonal from Point A. Path x goes along the road from point A (horizontally, and then up vertically to point B.
If I follow it , then one distanceis L+W, the other distance is sqrt(L^2+W^2)
WHich distance is square root? The y distance? I don't get how to calculate the square root thing
To calculate the distance of each path, you would need to use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's start by finding the length of path x, which goes along the road outside the field. Since the road wraps around the entire field, the length of path x would be the perimeter of the field plus the additional distance to get from point A to point B.
The perimeter of the field is equal to the sum of all the sides, which can be calculated using the formula P = 2(L + W), where P is the perimeter, L is the length, and W is the width of the field.
Given that the length of the field is 1.5 km (or 1500 m) and the width is 700 m, we can substitute these values into the formula:
P = 2(1500 + 700)
P = 2(2200)
P = 4400 m
Now, we need to add the additional distance from point A to point B. Looking at the diagram, you can observe that the additional distance is the difference between the width of the field and the dirt path, which is 700 m - 300 m = 400 m.
So, the total length of path x would be 4400 m + 400 m = 4800 m.
Now, let's find the length of path y, which is the straight line distance from point A to point B cutting across the field. Since path y is a straight line, we can use the Pythagorean theorem to calculate its length.
The length of path y can be found using the formula c = √(a^2 + b^2), where c is the length of the hypotenuse (path y), and a and b are the lengths of the two legs of the right-angled triangle formed by path y and the sides of the field.
In this case, one of the legs is the length of the field (1.5 km or 1500 m) and the other leg is the width of the field (700 m). Substituting these values into the formula, we get:
c = √(1500^2 + 700^2)
c = √(2250000 + 490000)
c = √2740000
c = 1655.12 m (rounded to the nearest meter)
Therefore, the length of path y is approximately 1655 meters.
To determine which path is shorter, you need to compare the lengths of path x and path y. In this case, path y has a shorter distance (approximately 1655 m) compared to path x (4800 m).
The difference in distance between the two paths would be the length of path x minus the length of path y:
4800 m - 1655 m = 3145 m.
Therefore, path y is shorter by approximately 3145 meters.