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The sum of a number and three times another number is 18. find the numbers if their product is a maximum.Please show all your work and explain the steps thank you! Write as quadratic eqaution plzz

Answer :

kvnmurty
Let the two numbers be X  and Y respectively.

   X + 3 Y = 18
   X Y = K = their product.
   X (18 - X) / 3 = K
   18 X - X^2 = 3 K

We have to maximize the product K.   So its differential is zero wrt to the variable X or Y.  
   d K / d X =  6 - 2 X / 3    = 0            =>        X = 9
                   Then Y = 3    and  K = 27

   X = 8,  Y = 10/3    => K = 80/3 = 26.67
   X = 10,  Y = 8/3    =>  K = 80/3 = 26.67


  d K/ dX is positive for X < 9  and  is  negative for X > 9.    So  Their product is maximum when X = 9.

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