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?

Assuming the tunnelling machine drills along the shortest distances between the points (a straight line), and not along the coordinate axes, then the total length is the sum of the lengths of the line segments DC, CB and BA.

Given:
A(-18,12), B(-8,1), C(3,4), and D(15,7),
The length of DC can be calculated using pythagoras theorem, namely:
mDC=√((15-3)²+(7-4)²)*2.5m
=√153 * 2.5m
=30.9 m approx.

You can calculate the other two segments in a similar way.

To calculate the amount of cable required, we need to calculate the total distance between the points on the coordinate grid.

First, we need to find the distance between each pair of consecutive points. We can use the distance formula, which is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the distance between each pair of points:

1) Distance between points D(15,7) and C(3,4):
d1 = √((15 - 3)^2 + (7 - 4)^2)
= √(12^2 + 3^2)
= √(144 + 9)
= √153
≈ 12.369

2) Distance between points C(3,4) and B(-8,1):
d2 = √((3 - (-8))^2 + (4 - 1)^2)
= √(11^2 + 3^2)
= √(121 + 9)
= √130
≈ 11.402

3) Distance between points B(-8,1) and A(-18,12):
d3 = √((-8 - (-18))^2 + (1 - 12)^2)
= √(10^2 + (-11)^2)
= √(100 + 121)
= √221
≈ 14.866

Now, let's calculate the total distance by adding these distances:
Total distance = d1 + d2 + d3
≈ 12.369 + 11.402 + 14.866
≈ 38.637

Since one unit on the grid represents 2.5m, we need to convert the total distance to meters:
Total cable required = Total distance * 2.5
≈ 38.637 * 2.5
≈ 96.5925 meters

Therefore, approximately 96.5925 meters of cable is required.

To calculate the length of the cable required, we need to find the distance between each consecutive set of coordinates and add them up.

Let's break down the problem step by step:

Step 1: Calculate the distance between points D(15,7) and C(3,4).
- We'll use the distance formula: distance = √[(x2 - x1)^2 + (y2 - y1)^2]
- Substituting the coordinates, we get: distance = √[(3 - 15)^2 + (4 - 7)^2]
- Simplifying, we get: distance = √[(-12)^2 + (-3)^2]
- Calculating further, we get: distance = √[144 + 9]
- The distance between D and C is √153.

Step 2: Calculate the distance between points C(3,4) and B(-8,1).
- Using the same distance formula, we get: distance = √[(-8 - 3)^2 + (1 - 4)^2]
- Simplifying, we get: distance = √[(-11)^2 + (-3)^2]
- Calculating further, we get: distance = √[121 + 9]
- The distance between C and B is √130.

Step 3: Calculate the distance between points B(-8,1) and A(-18,12).
- Again, using the distance formula, we get: distance = √[(-18 - (-8))^2 + (12 - 1)^2]
- Simplifying, we get: distance = √[(-18 + 8)^2 + (12 - 1)^2]
- Further simplification gives: distance = √[(-10)^2 + (11)^2]
- The distance between B and A is √221.

Step 4: Add the distances together to obtain the total cable length required.
- Adding up the three distances, we get: √153 + √130 + √221

Note: The square root represents the total distance because we are calculating the Euclidean distance in a Cartesian coordinate system.

Finally, to convert the distance to meters, since each unit on the grid represents 2.5m, multiply the result by 2.5 to get the final answer.

Therefore, the total cable length required = (2.5 * (√153 + √130 + √221)).