7/8 4/8 how much farther does 7/8 have than 4/8

Lagrange multipliers have a definite meaning in load balancing for electric network problems. Consider the generators that can o

Ivahew [28]

Answer:

The load balance $(x_1,x_2,x_3)=(545.5,272.7,181.8)$ Mw minimizes the total cost

Step-by-step explanation:

Optimizing With Lagrange Multipliers

When a multivariable function f is to be maximized or minimized, the Lagrange multipliers method is a pretty common and easy tool to apply when the restrictions are in the form of equalities.

Consider three generators that can output xi megawatts, with i ranging from 1 to 3. The set of unknown variables is x1, x2, x3.

The cost of each generator is given by the formula

$\displaystyle C_i=3x_i+\frac{i}{40}x_i^2$

It means the cost for each generator is expanded as

$\displaystyle C_1=3x_1+\frac{1}{40}x_1^2$

$\displaystyle C_2=3x_2+\frac{2}{40}x_2^2$

$\displaystyle C_3=3x_3+\frac{3}{40}x_3^2$

The total cost of production is

$\displaystyle C(x_1,x_2,x_3)=3x_1+\frac{1}{40}x_1^2+3x_2+\frac{2}{40}x_2^2+3x_3+\frac{3}{40}x_3^2$

Simplifying and rearranging, we have the objective function to minimize:

$\displaystyle C(x_1,x_2,x_3)=3(x_1+x_2+x_3)+\frac{1}{40}(x_1^2+2x_2^2+3x_3^2)$

The restriction can be modeled as a function g(x)=0:

$g: x_1+x_2+x_3=1000$

Or

$g(x_1,x_2,x_3)= x_1+x_2+x_3-1000$

We now construct the auxiliary function

$f(x_1,x_2,x_3)=C(x_1,x_2,x_3)-\lambda g(x_1,x_2,x_3)$

$\displaystyle f(x_1,x_2,x_3)=3(x_1+x_2+x_3)+\frac{1}{40}(x_1^2+2x_2^2+3x_3^2)-\lambda (x_1+x_2+x_3-1000)$

We find all the partial derivatives of f and equate them to 0

$\displaystyle f_{x1}=3+\frac{2}{40}x_1-\lambda=0$

$\displaystyle f_{x2}=3+\frac{4}{40}x_2-\lambda=0$

$\displaystyle f_{x3}=3+\frac{6}{40}x_3-\lambda=0$

$f_\lambda=x_1+x_2+x_3-1000=0$

Solving for \lambda in the three first equations, we have

$\displaystyle \lambda=3+\frac{2}{40}x_1$

$\displaystyle \lambda=3+\frac{4}{40}x_2$

$\displaystyle \lambda=3+\frac{6}{40}x_3$

Equating them, we find:

$x_1=3x_3$

$\displaystyle x_2=\frac{3}{2}x_3$

Replacing into the restriction (or the fourth derivative)

$x_1+x_2+x_3-1000=0$

$\displaystyle 3x_3+\frac{3}{2}x_3+x_3-1000=0$

$\displaystyle \frac{11}{2}x_3=1000$

$x_3=181.8\ MW$

And also

$x_1=545.5\ MW$

$x_2=272.7\ MW$

The load balance $(x_1,x_2,x_3)=(545.5,272.7,181.8)$ Mw minimizes the total cost

5 0

3 years ago

During his hike, Milt drank 1 liter of water and 1 liter of sports drink. How many milliliters of liquid did he drink?

S_A_V [24]

Answer: 2,000 milliliters.

Step-by-step explanation:

Hi, to answer this question we have to add the amount of water and the amount of sports drink that he drank:

1 liter + 1 liter = 2 liters of liquid.

Finally, we have to convert the result into milliliters.

Since 1 liter = 1000 liters

We have to multiply the liters drank by 1000.

2 x 1000 = 2,000 milliliters.

Feel free to ask for more if needed or if you did not understand something.

5 0

3 years ago

How do I solve calculating prices using discounts and tax rates fruit smoothie: $4.25

kaheart [24]

When your doing tax the tax is 9.25 you divide that by 100

the add that to your price then 12.5 divided by 100 and subtract that from what you ended up with so you should come out with a total of 4.22

8 0

3 years ago

If (x+y , 2x-3) = (5,3) find the value of x and y. plz answer ASAP!!!

Jlenok [28]

Answer:

x=3

y=0

Step-by-step explanation:

2x-3=3

2x=6

x=3

x+y=3

3+y=3

y=0

5 0

3 years ago

Read 2 more answers

Zapisz w jak najprostszej postaci: a) -x-4(1-x) b) 3(5x-2)-5(3-4x) c) √2(3-x)-2(1-x√2)

Sonja [21]

Answer:

Step-by-step explanation:

a) -x-4(1-x) =

− 2 x −4 ( 1 − x ) .

2 x − 4

b) 3(5x-2)-5(3-4x) =

3 ⋅ (5 x − 2 ) − 5 ⋅ ( 5 − 4 x )

35 x − 31

c) √2(3-x)-2(1-x√2) =

√ 2 x ⋅ ( 3 −x ) − 2 ⋅( 1 − x √ 2 ) .

2 x √ 2 + 3 √ 2 x -√ 2 x x − 2

4 0

3 years ago