An aerobatics aircraft is flying in a vertical loop above a hillside. The position of the aircraft at any given time can be found from the set of parametric equations

x(t) = 100 cos(t/2) + 100

z(t) = 100 sin(t/2) + 100

where time t is measured in seconds and the distance in each direction is measured in meters, from a reference point on the hill. The hill has a slope of 3/4 in the xz-plane, show that in this plane, the aircraft is closest to the hill at the point (40,20).

Part 2

If the plane of the hill passes through the origin (0,0,0), and also contains the point (0,2,1), find the distance between the aircraft and the plane of the hill when the aircraft is at the point (40,0,20). Use the slope of the hill in the xz-plane given in part 1.

Part 3

A taught electricity cable runs in a straight line, 20 metres above the ground, from the point p= (0,20,30) to the point q= (80,20,50). Find how close the aircraft comes to the electricity cable. You can assume that this occurs when the aircraft is at the point (40,0,20).