Find Two Numbers Whose Difference Is 100 And...

Anonymous

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Find two numbers whose difference is 100 and whose product isa minimum. Please show all steps done to find correct answer.

 

Answer No.1

AlphaGenius81

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You have 2 unknown values so you need two equations relatingthem a - b = 100 (1) product = P = a b (2) since we want to minimize the product, solve equation (1) foreither variable ( I will choose a) and then plug it into equation(2) a = 100 + b (1) so the product in terms of 'b' is... P(b) = (100 + b) b = 100 b + b2 To find critical values take the derivative of P(b), set it to 0and solve for b... P ' (b) = 100 + 2 b = 0 ∴ b =-50 check to make sure it is a minimum... for b < -50 P ' <0 (decreasing) for b > -50 P ' >0 (increasing) therefore b = -50 is a confirmed minimum so plug in to equation (1)to find a... a - (-50) = 100 ∴ a = 50 solutions are 50 and -50

Answer No.2

Anonymous

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A - b = 100 ---> a = 100 + b a * b= M plug in 100 + b for a (100 + b)(b) =M simplify 100b + b2 =M take the derivative 100 + 2b =M' Set M' equals to 0 100 + 2b =0 minus 2b for both side 100 =-2b divide -2 to both side -50 =b a - b =100 plug in -50 for b a + 50 =100 minus 50to both side a =50

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