Answer:
Since we want a positive number, x = 24. y = 192/x = 8
Explanation:
Let x and y be the two positive numbers.
- Their product is 192: x*y = 192
- the sum of the first plus 3 times the second is a minimum: x + 3y
From the first equation, y = 192/x. Substitute that into the second equation:
x + 3y = x + 3(192/x) = x + 576/x
Now take the first derivative, set it to zero, and solve for x:
d(x + 576/x)/dx = 1 - 576/x2
0 = 1 - 576/x2
0 = x2 - 576 [Multiplied both sides by x2]
576 = x2
√576 = √x2
±24 = x
Since we want a positive number, x = 24. y = 192/x = 8
As a check, the factors of 192 are 192*1, 96*2, 48*4, 24*8, 16*12
192 + 3*1 = 195
96 + 3*2 = 102
48 + 3*4 = 60
24 + 3*8 = 48
16 + 3*12 = 52
24 + 3*8 is indeed the minimum