Question

1.12 marks Suppose the measurements of ore lode in a deposit of gold are described by the func- L(x,y,z)-(10 32-y - 3y +0.4ry), where z is the vertical coordinate, with z = 0 at the surface and z < 0 being below ground. The tion (z+3)2 distances are measured in tens of metres (a) Draw contours of the function at a depth of z-3. Use Matlab, Octave or similar. You might like to include the range [x,y] E (-5,5) (b) The edge of the ore body is where the recording shows L 0. Find the equation for this boundary if z =-3. Write this equation in the explicit form y (c) A miner is at the point (2,1,-3). How is the ore lode changing at this point (2,1,-3) if they dig vertically downward? (d) In what direction should they "dig" to head toward the highest concentration of gold? (e) If they dig in the direction b (3,-4,1) what is the rate of increase in gold concentration? (f) Are the results in (d), (e) consistent with your contour plot? Show the vectors on the plot.

Answer 1

Part (a)

Here is the MATLAB code for contour plotting

>>x=linspace(-5,5);
>> y=linspace(-5,5);
>> [X,Y]= meshgrid(x,y);
>> z=-3;
>> L = exp(-(z+3)^2)*(10-3*X.^2-Y.^2-3*Y+0.4*X.*Y);
>> contour(X,Y,L,'ShowText','on')

and the contour plot looks like

for other parts please see this

2 GesturePen 0.9 15 Text (z43) (b) Z -3 15-+ 5d.tiegー2짼-50=。 m, D 70 2 14 -(15-2x)土 ,|ダー 1205-602-296- 10

Gesture Text 15 15 0.9 dot (C) Rat OL 0리 (2,1,-1)

Gesture 0.9 Text w VL at (2,1,-7) - (Zt3) 211,-3) - (2t3) y (4,-3) Fro part c 0

Gesture Text 0.9 26 2 2 6 2,4,-3) 0 2 一3.53 15 of 21
for part (f) kindly look the plot in part (a), and thus the results are consistent.

Thanks.