Find the point on the plane 2x-y+2z=20 nearest the origin.

korporasidn

korporasidn

Answered question

2021-12-07

Find the point on the plane 2x-y+2z=20 nearest the origin.

Answer & Explanation

Befoodly

Befoodly

Beginner2021-12-08Added 17 answers

d=x2+(2x+2z20)2+z2 
f(x,z)=x2+4(x+z10)2+z2 
which will have the minimum point. 
Locate the critical points for f from equations 
0=fx=2x+8(x+z10)=10x+8z80...(1) 
0=fy=2z+8(x+z10)=10z+8x80...(2) 
From (1) we have that x=88z10. Substituting in (2) we have that z=409 and from x=88z10 we obtain that x=409. The only critical point for the function f is (409,409). Since the plane has no boundaries, it is impossible to locate a point on it such that its distance from the origin is greater than or equal to that of any other point on the plane. If not, geometry provides us with the possibility of a point whose distance from the origin is as minimal as feasible.

 

The critical points i minima for the function f and the point we need to find is (409,209,409)

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