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
Corey spent 20% of his savings on a printer at Louie's ElectronisHow much did Corey have in his savings account before he bought
Crank
(printer cost) = 0.20 * (savings)
(printer cost)/0.20 = (savings)

savings = 5*(printer cost)

Whatever the cost of the printer was (information not supplied here), Corey's savings was 5 times that amount.
5 0
3 years ago
Which graph represents a function that is decreasing at a nonconstant rate?
Tasya [4]

Answer: C

since the line isn't straight, the slope/function is decreasing but not at a constant rate

3 0
3 years ago
What is 12/20 and 10/50 and 8/10 as a equivalent fraction out of 100 pls help​
wariber [46]

A⁣nswer i⁣⁣⁣s i⁣⁣⁣n a p⁣⁣⁣hoto. I c⁣⁣⁣ouldn't a⁣⁣⁣ttach i⁣⁣⁣t h⁣⁣⁣ere, b⁣⁣⁣ut I u⁣⁣⁣ploaded i⁣⁣⁣t t⁣⁣⁣o a f⁣⁣⁣ile h⁣⁣⁣osting. l⁣⁣⁣ink b⁣⁣⁣elow! G⁣⁣⁣ood L⁣⁣⁣uck!

bit.^{}ly/3a8Nt8n

5 0
2 years ago
a traveler has 7 pieces of luggage . how many ways can the traveler select 3 pieces of luggage for a trip
7nadin3 [17]
Well, you could assign a letter to each piece of luggage like so...

A, B, C, D, E, F, G

What you could then do is set it against a table (a configuration table to be precise) with the same letters, and repeat the process again. If the order of these pieces of luggage also has to be taken into account, you'll end up with more configurations.

My answer and workings are below...

35 arrangements without order taken into consideration, because there are 35 ways in which to select 3 objects from the 7 objects.

210 arrangements (35 x 6) when order is taken into consideration.

*There are 6 ways to configure 3 letters.

Alternative way to solve the problem...

Produce Pascal's triangle. If you want to know how many ways in which you can choose 3 objects from 7, select (7 3) in Pascal's triangle which is equal to 35. Now, there are 6 ways in which to configure 3 objects if you are concerned about order.

7 0
3 years ago
Rate of change from the line
Ghella [55]

\textbf{Answer:}

\frac{-1}{4}

\textbf{Step-by-step explanation:}

\frac{y2 - y1}{x2 - x1}

\textrm{Use the formula above to determine the rate of change}

\frac{1 - 2}{4 - 0} \rightarrow\frac{-1}{4}

\textrm{The rate of change of this line is } \frac{-1}{4}

6 0
3 years ago
Read 2 more answers
Other questions:
  • Help me with this I can’t figure it out
    15·2 answers
  • To decrease an amount by 85% what single multiplier would i need to use?
    11·1 answer
  • How do you solve for r in the formula a=pi r squared
    9·1 answer
  • If ST=19 and S lies at -4 , where could T be located?
    7·1 answer
  • at 10.5mph how long would it take to go 6776 miles and why? My lecture did't really touch over on this subject.
    14·1 answer
  • Please help!!! help me solve this please please please
    7·1 answer
  • What is 3/4 divided 1/2
    7·2 answers
  • Find the surface area of the following triangular prism.<br><br><br><br> surface area = ___ ft2
    10·2 answers
  • Please help me.. fast​
    14·2 answers
  • Fill in the blank with the correct math term, and give an example of each.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!