find 2 positive numbers so that the sum of first and twice the second is 100 and the product is a maximum.
2007-01-15 15:52:51 · 6 answers · asked by fathotra 1 in Science & Mathematics ➔ Mathematics
x + 2y = 100
x*y is a maximum.
so x = 100-2y.
(100-2y)*y ... distribute
100y - 2y^2 ... differentiate
100 - 4y ... solve for y.
100 -4y = 0 ... add 4y to both sides
100 = 4y ... divide both sides by 4
25 = y
plug back into first equation and you get x = 50
(x,y) => (50,25)
2007-01-15 15:57:43 · answer #1 · answered by David W 3 · 0⤊ 0⤋
The first number I can come up with is 50 and 25.
50+2*25 = 100
50*25 = 1250
You are trying to find the numbers, one of which would be closer to the median (or 50), if you come too close to 100, the number of additions would be less (x=96, y=2 --> 96*2 = 96+96), however making one 25 and the other 50, we add "25" 50 times. When you figure these problems, you need to go only to the middle, or 50% - there's no need to be thinking about first number to be greater than 50, if you tried it between 0 and 50: the numbers show the same pattern: 1 and 98, 2 and 96 .... 50 and 25...96 and 2, 98 and 1. Anyways, the answer should be that one of the numbers is in the middle.
Eugene
2007-01-16 00:09:37 · answer #2 · answered by eugene89us 2 · 0⤊ 0⤋
David W is correct.
Further, if you differentiate line 6 (the first derivative with respect to y) again with respect to y you get -4, which confirms that the critical value y=25 does indeed induce a maximum for the product function.
2007-01-16 00:11:44 · answer #3 · answered by answerING 6 · 0⤊ 0⤋
x= 1st number
y = 2nd number
Sum = x+2y =100
product = P = xy = x(100-x)/2 = 50x - x^2/2
dP/dx = 50 -x
0 = 50 -x
x = 50
y =25
2007-01-16 00:22:43 · answer #4 · answered by ironduke8159 7 · 0⤊ 0⤋
33 and 34
33 + 33 +34 = 100
33 * 34 = 1122
2007-01-15 23:58:32 · answer #5 · answered by Big.Al.B 2 · 1⤊ 1⤋
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2007-01-16 00:02:49 · answer #6 · answered by Anonymous · 0⤊ 2⤋