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3 years ago

2 ways to tell if there is a proportional relationship

Mathematics

1 answer:

Rus_ich [418]3 years ago

5 0

Answer:

straight line and line goes through the origin.

Step-by-step explanation:

without a graph, you will need to divide y/x to seeing there is a constant rate or a constant number every time you divide.

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A mix of factoring by grouping. Factor the exressions. 17ax+17a-x-1 Please explain/show steps on how to do this!

mario62 [17]

Answer:

$(x+1)(17a-1)$

Step-by-step explanation:

We are given that

$17ax+17a-x-1$

We have to factorize the given expression.

Pair up the terms

$(17ax+17a)+(-x-1)$

Now, taking common factor

$17a(x+1)-1(x+1)$

Taking common (x+1)

$(x+1)(17a-1)$

Now, we get

$(x+1)(17a-1)$

This is required factors of given expression.

4 0

2 years ago

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:

<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:

$g: x_1+x_2+x_3=1000$

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

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

What decimal number is represented by the shaded part

Anna11 [10]

Answer:

The decimal that represents the shaded portion of the figure is 0.75.

Step-by-step explanation:

4 0

2 years ago

Which of the following below is the equation of the quadratic function f(x) with x-intercepts of 0 and -3 for which f(-1)=-2. A.

Paladinen [302]

✿————✦————✿

Answer: D

✿————✦————✿

7 0

2 years ago

Find an equation of the line containing (-3,4) and having slope -2. If this line contains the points (a,8) and (5,b), find a and

larisa [96]

Answer:

Step-by-step explanation:

Remark

The slope intercept line is written as y = mx + b in it's general form.

You can get the slope, m, immediately because it is given as -2

So far what you have is y = -2x + b

y intercept

To get the y intercept, use the point (-3 , 4)

x = - 3

y = 4

4 = -2*(-3) + b

4 = 6 + b Subtract 6 from both sides

4 - 6 = 6-6 + b

b = - 2

Line

y = -2x - 2

(a,8)

y = -2a - 2

8 = -2a - 2 Add 2 to both sides

8 +2 = -2a -2+2 Combine

10 = - 2a Divide by - 2

10/-2 = a

a = -5

(5,b)

y = -2x - 2

x = 5

y = b

b = -2*5 - 2

b = -10 - 2

b = - 12

6 0

2 years ago