Mathematics, 11.10.2020 14:01 ahaug4869

Among all pairs of numbers (x, y) such that 3x+y=15, find the pair for which the sum of squares, x^2+y^2, is minimum. Write your answers as fractions reduced to lowest terms.

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Answer from: melaniegilbreath

(4.5, 1.5)

Step-by-step explanation:

Lets substitute y in the second equation and find its minimum

3x + y = 15y = 15 - 3x

Substitute

x^2 + y^2x^2 + (15 - 3x)^2 =x^2 + 225 - 90 x + 9x^2=10x^2 - 90x + 225

This expression gets minimum value at x = -b/2a as quadratic function's vertex:

x = -(-90)/2*10 = 90/20 = 4.5

Then finding the value of y:

y = 15 -3*4.5 = 15 - 13,5 = 1.5

The answer is (4.5, 1.5)

Answer from: mncampbell01

(9/2, 3/2)

Step-by-step explanation:

We have the equation:

3x+y=15

Which is equivalent to 15 among all pairs of (x, y).

We want to find the pair of solutions (x, y) such that:

x^2+y^2

Is minimum.

Note that our given equation is a line.

And the equation x²+y² is a circle centered on the origin.

In other words, we want to find the radius of the circle such that it is tangent to our line at 3x+y=15.

It must be tangent because this guarantees that it is the smallest value of x²+y².

It's good if we have a visual of this. I've graphed the given linear equation. Please refer to it.

If you remember in geometry, in order for the radius to be tangent to a line, the radius must be perpendicular to our line.

So, let's find the perpendicular equation to our line. Our original equation is:

3x+y=15

Subtract 3x from both sides:

y=-3x+15

So, the slope of our original equation is -3.

This means that the slope of our perpendicular line must be the negative reciprocal of -3. Namely, it is 1/3.

And since our circle is centered on the origin, this line will go through the origin. Therefore, our perpendicular equation is:

$y=\frac{1}{3}x$

Graphing this will yield (please refer to the second graph):

Therefore, the intersection between our old and new line is at (4.5, 1.5).

Therefore, the (x, y) value that grants the minimum sum is 9/2 and 3/2.

We can check this by substituting them into our second equation. This yields:

(9/2)^2+(5/2)^2

Square and add:

=81/4+9/4=22.5

Note that 22.5 is the radius squared.

Graphing this gives us (please refer to the third graph):

We can see that it is indeed tangent to our line. And it is the lowest value of P that does so.

So, our answer is x=9/2 and y=3/2.

Among all pairs of numbers (x,y) such that 3x+y=15, find the pair for which the sum of squares, x^2+

Answer from: Quest

2nd one in the first row is correct

Me on this i’m so tired and i can’t sleep until my progress is at 80 and this is the end of my year

Answer from: Quest

The answer is biodiversity resulting from new ancestors.

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