In a coordinate system (measuring in metres) with the z-axis vertical, a piece o
ID: 3105748 • Letter: I
Question
In a coordinate system (measuring in metres) with the z-axis vertical,a piece of sloping ground follows the plane x – y + 5z = 90.
A and B are two reference points on the ground, with A at the position x = y
= 10, and B at x = 2, y = 22. A transmission aerial of height 36m has its
base positioned one-quarter of the way along the straight line from A to B.
a) What are the coordinates of A and B?
b) What are the coordinates of the top of the aerial?
c) A supporting cable will run from the top of the aerial to the
ground. What is the shortest length of cable that is needed?
Explanation / Answer
a) What are the coordinates of A and B?
Basically where A and B intersect the "central equation" x – y + 5z = 90.
A is at x = y = 10, so it's at (10, 10, __); to find the z coordinate, we solve:
10 - 10 + 5z = 90
z = 18
So A is at (10, 10, 18)
B is at x = 2, y = 22, so it's at (2, 22, __); to find the z-coordinate, we solve:
2 - 22 + 5z = 90
5z = 110
z = 22
So B is at (2, 22, 22)
b) What are the coordinates of the top of the aerial?
The transmission height is 36 so z = 36
x - y + 180 = 90
x - y = -90
y = x + 90
The top of the aerial thus comes at (x, x + 90, 36)
c) A supporting cable will run from the top of the aerial to the
ground. What is the shortest length of cable that is needed?
square root of (x^2 + x^2 + 180x + 8100 + 1296) =
square root of (2x2 + 180x + 9396)
If we place our x at 0, then we'd find the shortest distance, and that shortest distance is:
square root of (9396) = approx. 96.93