Find Two Positive Integers Such That The Sum Of Th

Samuel89jack

From: US
Posts: 37
Votes: 74
Find two positive integers such that the sum of the first numberand four times the second number is 1000 and the product of thenumbers is as large as possible.

 

Answer No.1

Anonymous

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QuestionDetails: find two positive integers such that the sum of the first numberand four times the second number is 1000 and the product of thenumbers is as large as possible. LET THE 2 INTEGERS BE X AND Y X+4Y=1000 Z=XY SHOULD BE LARGEST Z=(1000-4Y)Y=-[4Y2-1000Y] = -[(2Y-250)-250*250] = 62500- (2Y-250)2.... HENCE Z WILL BE MAXIMUM WHEN 2Y=250...OR....Y=125 HENCE X=1000-4*125 = 500 SO THE 2 NUMBERS ARE 500 AND 125

Answer No.2

Anonymous

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Let them be x, and y x + 4y = 1000 =>y = (1000-x)/4 let f(x,y) = xy = f(x)=x[(1000-x)/4] = 250x -x2/4 f'(x) = 250 - x/2 = 0 => x = 500 y = (1000-500)/4 = 500/4 = 125 the two number is: 500,125

Answer No.3

WizSam

From: US
Posts: 33
Votes: 99
Let the numbers be x & y. According to first condition: x + 4y = 1000--------------(1) According to second condition : xy = P-------------------(2) Put value of x from eqn (1) to eqn (2) (1000 - 4y) y = P ∴1000y - 4y2 = P Now, dP/dy = 1000 -8y For P to be maximum dP/dy = 0 ∴y = 125 & x =500

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