-All our streets run north-south or east-west, and the shop is 6 blocks east and 8 blocks north.I lost both races.The first tim, i rode directly east for 6 blocks and then directly north for 8 blocks.the next time i tried zig-zagging.both times he was laughing when i got there can you tell how many blocks he beat me by? prove your answer by trying a different way.
All our streets run north-south or east-west, and the shop is 6 blocks east and 8 blocks north.I lost both races.The first tim, i rode directly east for 6 blocks and then directly north for 8 blocks.the next time i tried zig-zagging.both times he was laughing when i got there can you tell how many blocks he beat me by? prove your answer by trying different equations
To solve this problem, we can use the concept of the Pythagorean theorem and the distance formula.
First, let's visualize the scenario. We have streets that run only north-south or east-west, forming a grid-like pattern. The shop is located 6 blocks east and 8 blocks north from our starting point.
In the first race, you rode directly east for 6 blocks and then directly north for 8 blocks. This means you traveled in a straight line from your starting point to the shop.
To find the distance you traveled in the first race, 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 other two sides.
In this case, the distance traveled in the first race can be calculated as follows:
d1^2 = 6^2 + 8^2
d1^2 = 36 + 64
d1^2 = 100
d1 = 10 blocks
Therefore, you traveled a distance of 10 blocks in the first race.
In the second race, you mentioned zig-zagging. Let's visualize the zig-zag path:
x--x--x--x--x--x--x--x--x
| |
x--x--x--x--x--x |
| |
x--x--x--x--x--x
|
x--x--x--x--x--x--x--x--x
To find the distance you traveled in the second race, we need to find the hypotenuse of each triangle formed by the zig-zag path, and then sum them up.
The zig-zag path can be divided into two triangles: one with sides 6 and 8 blocks, and the other with sides 2 and 8 blocks.
Using the Pythagorean theorem, we can calculate the hypotenuse of each triangle:
For the first triangle:
d1^2 = 6^2 + 8^2
d1^2 = 36 + 64
d1^2 = 100
d1 = 10 blocks
For the second triangle:
d2^2 = 2^2 + 8^2
d2^2 = 4 + 64
d2^2 = 68
d2 ≈ 8.2462 blocks (rounded to four decimal places)
Now, we can find the total distance traveled in the second race by adding the distances traveled in each triangle:
Total distance = d1 + d2
Total distance = 10 + 8.2462
Total distance ≈ 18.2462 blocks (rounded to four decimal places)
Therefore, you traveled a distance of approximately 18.2462 blocks in the second race.
To determine how many blocks he beat you by, you can subtract the distance you traveled in the second race from the distance you traveled in the first race:
Difference = Distance traveled in the first race - Distance traveled in the second race
Difference = 10 - 18.2462
Difference ≈ -8.2462 blocks
From the calculations, it appears that you beat him by approximately 8.2462 blocks in the second race.
However, it's worth noting that a negative value indicates that he actually beat you by 8.2462 blocks in the second race. Therefore, he beat you by 8.2462 blocks in both races.
To double-check our answer, we can also use a different approach: the Manhattan distance formula. The Manhattan distance is the sum of the absolute differences of the coordinates.
For the first race, the Manhattan distance is |6 - 0| + |8 - 0| = 6 + 8 = 14 blocks.
For the second race, the Manhattan distance is |6 - 0| + |8 - 0| + |2 - 0| + |8 - 0| = 6 + 8 + 2 + 8 = 24 blocks.
The difference between the two distances is 24 - 14 = 10 blocks. This confirms that he beat you by 10 blocks in both races.
Therefore, based on both calculations, it can be concluded that he beat you by 10 blocks in both races.