SOLUTION: Among all pairs of numbers (x,y) such that 8x+2y=18, find the pair for which the sum of squares, x2+y2, is minimum. Write your answers as fractions reduced to lowest terms.
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Question 1136263: Among all pairs of numbers (x,y) such that 8x+2y=18, find the pair for which the sum of squares, x2+y2, is minimum. Write your answers as fractions reduced to lowest terms. Found 2 solutions by ankor@dixie-net.com, ikleyn:Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website! Among all pairs of numbers (x,y) such that 8x+2y=18, find the pair for which the sum of squares, x2+y2, is minimum.
Write your answers as fractions reduced to lowest terms.
:
8x + 2y = 18
simplify, divide by 2
4x + y = 9
y = -4x + 9
:
sum = x^2 + y^2
replace y with (-4x+9)
sum = x^2 + (-4x+9)^2
FOIL (-4x+9)(-4x+9)
sum = x^2 + 16x^2 - 36x - 36x + 81
sum = 17x^2 - 72x + 81
min occurs on the axis of symmetry; x = -b/(2a)
x =
x =
x =
:
find y
y = + 81
y = 17( + 81
y = - +
y =
:
Minimum of x^2 + y^2: x = 36/17; y = 81/17
Looks pretty wild, I hope this is right, let me know. ankor@att.net
Really, such a problem can stump unexperienced student.
But it is not so complicated.
The solution is as follows.
From equation 8x + 2y = 18, express y = = -4x + 9 and substitute it into the expression
f(x,y) = = = = . (1)
Now your function is presented as a quadratic function of ONE argument.
You can easily find its minimum by using this very well known shortcut = " ".
In this case a = 17 and b = -72, so = = .
So, you just know the value of " x " where the function has the minimum.
By knowing , you can easily calculate = .
I leave this simple arithmetic for you to complete it on your own.
