A coordinate grid is used on the plan of a new housing development. fibre optic cable is being laid with a mico-tunnelling machine. the cable will link points with coordinates A(-18,12), B(-8,1), C(3,4), and D(15,7), in a run beginning at D. if one unit on the grid represents 2.5m, how much cable is required?
Break this into parts by figuring the distance between each point then adding them together to get the total. Then apply unit conversion to get the cable length in meters.
Get the delta x and delta y first (change in x-coordinate, change in y coordinate)
AB: delta x = -8 - (-18) = 10, delta y = (1 - 12) = -11
The use pythagorean's theorem: c = sqrt(a^2 + b^2).
length AB = sqrt(10^2 + (-11)^2) = sqrt(221) = 14.86
BC: delta x = (3 - (-8)) = 11, delta y = 4 - 1 = 3
length BC = sqrt(11^2 + 3^2) = sqrt(130) = 11.40
CD: delta x = 15 - 3 = 12, delta y = 7 - 4 = 3
length CD = sqrt(12^2 + 3^2) = sqrt(153) = 12.37
Sum them to get the total units traversed:
AB + BC + CD = 14.86 + 11.40 + 12.37 = 38.63 units
Apply your unit conversion (one unit = 2.5 meters)
cable length = 38.63 units * 2.5 m/unit = 96.575 m
Answer to nearest tenth is 96.6 m
To calculate the total length of the cable required, we need to find the distance between each pair of consecutive points on the grid.
Given that one unit on the grid represents 2.5 meters, we can use the distance formula to find the distance between two points (x1, y1) and (x2, y2):
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distances step by step:
1. Distance between D(15, 7) and C(3, 4):
Distance = √((3 - 15)^2 + (4 - 7)^2)
= √((-12)^2 + (-3)^2)
= √(144 + 9)
= √153
≈ 12.37 units
2. Distance between C(3, 4) and B(-8, 1):
Distance = √((-8 - 3)^2 + (1 - 4)^2)
= √((-11)^2 + (-3)^2)
= √(121 + 9)
= √130
≈ 11.40 units
3. Distance between B(-8, 1) and A(-18, 12):
Distance = √((-18 - (-8))^2 + (12 - 1)^2)
= √((-18 + 8)^2 + (12 - 1)^2)
= √((-10)^2 + (11)^2)
= √(100 + 121)
= √221
≈ 14.87 units
Now, we can sum up the distances between the consecutive points to find the total length of the cable:
Total Length of Cable = 12.37 + 11.40 + 14.87
≈ 38.64 units
Since each unit on the grid represents 2.5 meters, we can multiply the total number of units by 2.5 to find the cable length in meters:
Total Length of Cable = 38.64 units * 2.5 meters/unit
≈ 96.6 meters
Therefore, approximately 96.6 meters of cable is required for the run.
To determine the amount of cable required, we need to calculate the total distance between the points on the coordinate grid.
The distance between two points in a coordinate grid can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the two other sides of a right triangle.
In this case, we can break down the problem into smaller subproblems: finding the distance between D and C, the distance between C and B, and the distance between B and A.
Let's calculate the distances between the points step by step:
1. Distance between D and C:
- The x-coordinate of point D is 15, and the x-coordinate of point C is 3. Therefore, the horizontal distance between D and C is 15 - 3 = 12 units.
- Similarly, the y-coordinate of point D is 7, and the y-coordinate of point C is 4. Therefore, the vertical distance between D and C is 7 - 4 = 3 units.
- Using the Pythagorean theorem, the distance between D and C can be calculated as follows:
Distance = sqrt((12 * 2.5)^2 + (3 * 2.5)^2) = sqrt(300.25) ≈ 17.32 meters (rounded to two decimal places).
2. Distance between C and B:
- The x-coordinate of point C is 3, and the x-coordinate of point B is -8. Therefore, the horizontal distance between C and B is 3 - (-8) = 11 units.
- The y-coordinate of point C is 4, and the y-coordinate of point B is 1. Therefore, the vertical distance between C and B is 4 - 1 = 3 units.
- Using the Pythagorean theorem, the distance between C and B can be calculated as follows:
Distance = sqrt((11 * 2.5)^2 + (3 * 2.5)^2) = sqrt(92.25) ≈ 9.61 meters (rounded to two decimal places).
3. Distance between B and A:
- The x-coordinate of point B is -8, and the x-coordinate of point A is -18. Therefore, the horizontal distance between B and A is -8 - (-18) = 10 units.
- The y-coordinate of point B is 1, and the y-coordinate of point A is 12. Therefore, the vertical distance between B and A is 12 - 1 = 11 units.
- Using the Pythagorean theorem, the distance between B and A can be calculated as follows:
Distance = sqrt((10 * 2.5)^2 + (11 * 2.5)^2) = sqrt(286.25) ≈ 16.93 meters (rounded to two decimal places).
Finally, to determine the total cable required, we add up the distances between the points:
Total Cable Required = Distance between D and C + Distance between C and B + Distance between B and A
= 17.32 + 9.61 + 16.93
≈ 43.86 meters (rounded to two decimal places).
Therefore, approximately 43.86 meters of cable is required to link points D, C, B, and A on the coordinate grid.
Well, let's take a trip through this housing development with our trusty clown calculator and see how much cable we need!
First, let's calculate the distance from point D to point C. The x-coordinate of D is 15, and the x-coordinate of C is 3. The difference between these two values is 12. Since each unit on the grid represents 2.5m, we multiply 12 by 2.5 to get the horizontal distance: 12 * 2.5 = 30m.
Now, let's calculate the distance from point C to point B. The y-coordinate of C is 4, and the y-coordinate of B is 1. The difference between these two values is 3. We multiply 3 by 2.5 to get the vertical distance: 3 * 2.5 = 7.5m.
Next, let's calculate the distance from point B to point A. The x-coordinate of B is -8, and the x-coordinate of A is -18. Again, the difference between these two values is 10. Multiplying 10 by 2.5 gives us the horizontal distance: 10 * 2.5 = 25m.
Finally, let's calculate the distance from point A back to point D. The y-coordinate of A is 12, and the y-coordinate of D is 7. The difference between these two values is 5. Multiplying 5 by 2.5 gives us the vertical distance: 5 * 2.5 = 12.5m.
Now, we add up all the distances: 30m + 7.5m + 25m + 12.5m = 75m.
So, it looks like we'll need a total of 75 meters of cable to connect all these points in the housing development. Happy tunneling!