A coordinate grid is used on the plan of a new housing development. fibre optic cable is being laid with a micro-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 A and ending at D . if one unit on the grid represents 2.5m, how much cable is required?
To find out how much cable is required, we need to calculate the distance between each consecutive pair of points and then sum them up.
Let's calculate the distance between A and B first:
- The change in x-coordinate is -8 - (-18) = 10 units.
- The change in y-coordinate is 1 - 12 = -11 units.
- Using the Pythagorean theorem, we can find the distance:
distance_AB = sqrt((10 * 2.5)^2 + (-11 * 2.5)^2)
Next, let's calculate the distance between B and C:
- The change in x-coordinate is 3 - (-8) = 11 units.
- The change in y-coordinate is 4 - 1 = 3 units.
- Using the Pythagorean theorem, we can find the distance:
distance_BC = sqrt((11 * 2.5)^2 + (3 * 2.5)^2)
Lastly, let's calculate the distance between C and D:
- The change in x-coordinate is 15 - 3 = 12 units.
- The change in y-coordinate is 7 - 4 = 3 units.
- Using the Pythagorean theorem, we can find the distance:
distance_CD = sqrt((12 * 2.5)^2 + (3 * 2.5)^2)
Finally, we can sum up the distances to find the total cable required:
total_cable = distance_AB + distance_BC + distance_CD
Now, let's calculate the distances step-by-step:
1. Calculate the distance between A and B:
- distance_AB = sqrt((10 * 2.5)^2 + (-11 * 2.5)^2)
- distance_AB = sqrt((25^2 + 11^2) * 2.5^2)
- distance_AB = sqrt((625 + 121) * 2.5^2)
- distance_AB = sqrt(746 * 2.5^2)
- distance_AB = sqrt(746) * 2.5
- distance_AB ≈ 31.446 units
2. Calculate the distance between B and C:
- distance_BC = sqrt((11 * 2.5)^2 + (3 * 2.5)^2)
- distance_BC = sqrt((27.5^2 + 7.5^2) * 2.5^2)
- distance_BC = sqrt((755.625 + 56.25) * 2.5^2)
- distance_BC = sqrt(811.875 * 2.5^2)
- distance_BC = sqrt(811.875) * 2.5
- distance_BC ≈ 39.447 units
3. Calculate the distance between C and D:
- distance_CD = sqrt((12 * 2.5)^2 + (3 * 2.5)^2)
- distance_CD = sqrt((30^2 + 7.5^2) * 2.5^2)
- distance_CD = sqrt((900 + 56.25) * 2.5^2)
- distance_CD = sqrt(956.25 * 2.5^2)
- distance_CD = sqrt(956.25) * 2.5
- distance_CD ≈ 49.913 units
Finally, we can calculate the total cable required:
total_cable = distance_AB + distance_BC + distance_CD
total_cable ≈ 31.446 + 39.447 + 49.913
total_cable ≈ 120.806 units
Therefore, approximately 120.806 units of cable, each representing 2.5m, will be required for laying the fiber optic cable in the housing development.
To calculate the cable length required for the run from point A to D, we need to find the distance between each consecutive pair of points (A to B, B to C, and C to D) and sum them up.
Here's how you can find the distance between two points on a coordinate grid:
1. Determine the difference in the x-coordinates (Δx) and the difference in the y-coordinates (Δy) between the two points.
Δx = x2 - x1
Δy = y2 - y1
2. Use the Pythagorean theorem to calculate the straight-line distance (d) between the two points.
d = √(Δx² + Δy²)
Let's calculate the cable length:
1. Distance between A and B:
Δx_AB = -8 - (-18) = 10
Δy_AB = 1 - 12 = -11
d_AB = √(10² + (-11)²) = √(100 + 121) = √221
2. Distance between B and C:
Δx_BC = 3 - (-8) = 11
Δy_BC = 4 - 1 = 3
d_BC = √(11² + 3²) = √(121 + 9) = √130
3. Distance between C and D:
Δx_CD = 15 - 3 = 12
Δy_CD = 7 - 4 = 3
d_CD = √(12² + 3²) = √(144 + 9) = √153
4. Add up the distances:
total distance = d_AB + d_BC + d_CD
total distance = √221 + √130 + √153
Finally, you mentioned that one unit on the grid represents 2.5m. So to convert the total distance to meters, you need to multiply it by the scaling factor:
cable length in meters = total distance * 2.5
By following these steps, you can determine the amount of cable required for the run from point A to D.