1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Maslowich
3 years ago
15

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

utput xi megawatts, i = 1, 2, 3 each generator cost
Ci = 3xi + ( i /40) xi^2 i .

If the total power needed is 1000MW. What load balance (x1, x2, x3) minimizes cost?
Mathematics
1 answer:
Ivahew [28]3 years ago
5 0

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:

<u>Optimizing With Lagrange Multipliers</u>

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

You might be interested in
How many tens are in 60,000
algol [13]
600000 is the answer to
5 0
3 years ago
Read 2 more answers
What is the common factor of 10 15and 35​
anygoal [31]

Answer:

the common factor of all the numbers is 5

7 0
3 years ago
Which of the following is not a triangle congruence theorem?<br><br> A) ASA<br> B) AAS<br> C) SSA
serg [7]
Don't listen to them asa is a very real thing! It's the angle side angle!! It would be C! Ssa! That doesn't exist!
3 0
3 years ago
Read 2 more answers
Find the value of each of the following: exponents
Nataly [62]

Answer:

7) is 625

8) 169

9) 1.44

10) 180??

11) 1000000

12) 3430

hopefully these are right

Step-by-step explanation:

7 0
2 years ago
How many times does 6 go into 64
Bumek [7]
6 goes into 64 ten times with a remainder of 4
you know that
3 0
3 years ago
Read 2 more answers
Other questions:
  • Look at the picture below and answer if you can.
    7·1 answer
  • If the sum of a number and six is tripled. the result is five less than twice the number
    7·2 answers
  • Georgia bought her car with 15,000 miles already on it. She drives an additional 735 miles each month for work. If she now has 2
    14·1 answer
  • A scale of a room in a blueprint is 3 in.:5 ft. A wall in the same blueprint is 18 in.
    11·1 answer
  • A bag contains 8 blue marbles, 6 green marbles, 12 yellow marbles, and 10 orange marbles. A marble is drawn at random from the b
    13·1 answer
  • Which is greater three hours or 300 minutes
    7·2 answers
  • 5 ^ 3 + 1/2 ^ 2. <br> I need help.
    15·1 answer
  • A multivitamin tablet contains 0.13g of vitamin C. How much vitamin C does a bottle of 20 tablets contain? Write your answer in
    9·1 answer
  • Help please and thank youuuu need it to path 8th grade and go to HS
    5·2 answers
  • Find the slope of ( 3,7 ) ( 4,10 )
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!