Mathematics, 08.04.2020 19:55 anna8352

Find two numbers x and y such that a) 2x+y=100 and A=2x+2xy+y is maximized b) 2x+4y-15=0 and B= √x2+y2is minimized. Note that in this case, minimizing B is equivalent to minimizing B2.

Answers: 3

Show answers

Answers

Answer from: Lizzy527663

a) x = 25, y = 50

b) x = 1.5, y = 3

Step-by-step explanation:

We have to use Lagrange Multipliers to solve this problem. The maximum of a differentiable function f with the constraint g(x,y) = b, then we have that there exists a constant $\lambda$ such that

$\nabla f(x,y) = \lambda \, \nabla g(x,y)$

Or, in other words,

$f_x(x,y) = \lambda \, g_x(x,y) \\ f_y(x,y) = \lambda \, g_y(x,y)$

a) Lets compute the partial derivates of f(x,y) = 2x+2xy+y. Recall that, for example, the partial derivate of f respect to the variable x is obtained from derivating f thinking the variable y as a constant.

f_x(x,y) = 2 + 2y

On the other hand,

f_y(x,y) = 2x+1

The restriction is g(x,y) = 100, with g(x,y) = 2x+y. The partial derivates of g are

g_x(x,y) = 2; g_y(x,y) = 1

This means that the Lagrange equations are

$2y + 2 = 2 \, \lambda$ $2x +1 = \lambda$ 2x + y = 100

(this is the restriction, in other words, g(x,y) = 100)

Note that 2y + 2, which is $2 \, \lambda$ is the double of 2x+1, which is $\lambda$ . Therefore, we can forget $\lambda$ for now and focus on x and y with this relation:

2y+2 = 2 (2x+1) = 4x+2

2y = 4x

y = 2x

If y is equal to 2x, then

g(x,y) = 2x+y = 2x+2x = 4x

Since g(x,y) = 100, we have that

4x = 100

x = 100/4 = 25

And, therefore y = 25*2 = 50

Therefore, x = 25, Y = 50.

b) We will use the suggestion and find the minumum of f(x,y) = B² = x²+y², under the constraing g(x,y) = 0, with g(x,y) = 2x+4y-15. The suggestion is based on the fact that B is positive fon any x and y; and if 2 numbers a, b are positive, and a < b, then a² < b². In other words, if (x,y) is the minimum of B, then (x,y) is also the minimum of B² = f.

Lets apply Lagrange multipliers again. First, we need to compute the partial derivates of f:

$f_x(x,y) = 2x \\f_y(x,y) = 2y$

And now, the partial derivates of g:

$g_x(x,y) = 2 \\ g_y(x,y) = 4$

This gives us the following equations:

$2x = 2 \, \lambda \\ 2y = 4 \, \lambda \\ 2x+4y-15 = 0$

If we compare 2x with 2y, we will find that 2y is the double of 2x, because 2y is equal to $4 \, \lambda$ , while on the other hand, $2x = 2 \, \lambda$ . As a consequence, we have

2y = 2*2x

y = 2x

Now, we replace y with 2x in the equation of g:

0 = g(x,y) = 2x+4y-15 = 2x+4*2x -1x = 10x-15

10 x = 15

x = 15/10 = 1.5

y = 1x5*2 = 3

Then, B is minimized for x 0 1.5, y = 3.

Answer from: Quest

l.ln.lln.k.nkl

step-by-step explanation:

Answer from: Quest

r and t.

both are 64 degrees.

17. given triangle rst as shown below. which two sides of the triangle are congruent to each other?

Another question on Mathematics

Mathematics, 21.06.2019 17:30

If jaime wants to find 60% of 320, which table should he use?

Answers: 1

Answer

Mathematics, 21.06.2019 23:20

Point r divides in the ratio 1 : 5. if the coordinates of e and f are (4, 8) and (11, 4), respectively, what are the coordinates of r to two decimal places?

Answers: 2

Answer

Mathematics, 22.06.2019 00:10

Given: p: 2x = 16 q: 3x – 4= 20 which is the converse of p - q? if 2x + 16, then 3x - 4 20. if 3x –4 +20, then 2x # 16. lf 2x = 16, then 3x - 4 = 20. lf 3x – 4= 20, then 2x = 16. o o

Answers: 1

Answer

Mathematics, 22.06.2019 00:20

What is the value for this expression? 2e-5

Answers: 1

Answer

You know the right answer?

Find two numbers x and y such that a) 2x+y=100 and A=2x+2xy+y is maximized b) 2x+4y-15=0 and B= √x2+...

Questions

Mathematics, 20.11.2020 02:20

PLEASEE HELPP Match the correct graph to the story below. You get two movies free, but then you get charged at a fixed rate per movie.x =...