Question

You are in charge of erecting a radio telescope on a newly discovered planet. To minimize interference, you want to place it where the magnetic field of the planet is the weakest. The planet is spherical, with a radius of 6 units. Based on a coordinate system whose origin is the center of the planet, the strength of the magnetic field is given by

M(x,y,z)=

09 + 22 + z - 19

Where should you locate the radio telescope?

Answer 1

solution: of a Given that the equation of the surface sphere of radius is 6 Magnetic field M(x, y, z)= 6x-4² + 2z+60 Let Center = 60,0,0) - د) د ره- ) رہ- ن) ح ہو + كد : ت یہ ہے ا ا ای موجود : (= رل 9 اصل Now, we want to minimize the foncton, | * 2 * 2 - 6 - ( ج ابر) we use Lagrange multipliers, (7 و 79 2 : ( و )Tr Dierentiate perentiate of the quations (6+7, -29 *) : A (2, 29 22) 6 + Z : 24* - 29 : 224 - 227

the second equaton tells that have و طرح حنه می ره تو (اپ 2) - Y=0 on a = -1 IF 2--/ then the other equators are w: -2Z & 6+ 2 = -22 so that 6+ Z: -26-22) G+ Z:42 u:-22 = -2(2) n=-4 - 42-2= 6 32=6 12:2 This tells us that n=-4 & so Got 36-4-16 : Y = 576 = 19=247 i we have two possible poin's (-4, 4, 2) & (-4,-4,2) In the case, where yao we have no information about a and so it is best to ged rid of it 2= 222 6+ z = 2 2 2 = 29(212) 42PZ -

6+2=40²2 6 = 4a?z-z 6 = (42²-,)z Z : - n-- 72 n= 122 42², Note that we can divide by 42²-1 bells us that it cannot equal 2000 putting this into the constraint, and remembering that y=0, we find that 1442² + 36 = 36 (42²-,) 1442 ² + 36 144 2² +36-0 42² (42²+3)=0 then a=0 o 2 = 2 B so the points are 60,0,6) € ( 13 13,0,3)

- 12 03 + 2 (» در به) - ( بهای کے 12 :(2 را به سر من 603+ (= - - - - ( مداد کی 29 = (2 ,4 . --| | 603 + ریه - رده 60 = (0 ,0 ,0) | 6O + 2763 - (3ره 33 ] پر 653+ (3)( + 2- (33اء کے M-3 3, 2, 3) : - 27/3 * 80 ro (43 (3 / 3- +- (3) 3- ا ک وزرات حراره در ک 7 He .اک ( ہے۔ ہے - طه