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
rewona [7]
2 years ago
7

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.
Mathematics
1 answer:
zaharov [31]2 years ago
4 0

Answer:

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.

You might be interested in
To make a greeting card, Bryce used 1/8 sheet of red paper, 3/8 sheet of green paper, and 7/8 sheet of white paper. How many she
Mashutka [201]
11/8 or 1 3/8 or 1.375
6 0
2 years ago
How many integers between 14 and 3,452 can be divided by 42 with a remainder of zero?
soldi70 [24.7K]
      _______
42 I 3,452

82 with a remainder

There are <u>82</u> integers between 14 and 2452 that can be divided with a remainder of 0
4 0
3 years ago
IV
nikitadnepr [17]
The answer is all real numbers
5 0
3 years ago
A local bottler in Hawaii wishes to ensure that an average of 22 ounces of passion fruit juice is used to fill each bottle. In o
Mandarinka [93]

Answer:

(a) Null Hypothesis, H_0 : \mu = 22 ounces  

    Alternate Hypothesis, H_A : \mu\neq 22 ounces

(b) The value of the test statistic is -2.687.

(c) The critical values are -1.96 and 1.96.

(d) We conclude that Reject H_0 since the value of the test statistic is less than the negative critical value.

Step-by-step explanation:

We are given that a local bottler in Hawaii wishes to ensure that an average of 22 ounces of passion fruit juice is used to fill each bottle.

He takes a random sample of 65 bottles. The mean weight of the passion fruit juice in the sample is 21.54 ounces. Assume that the population standard deviation is 1.38 ounce.

<em>Let </em>\mu<em> = average ounces of passion fruit juice used to fill each bottle.</em>

(a) So, Null Hypothesis, H_0 : \mu = 22 ounces     {means that the average of 22 ounces of passion fruit juice is used to fill each bottle}

Alternate Hypothesis, H_A : \mu\neq 22 ounces     {means that the average different from 22 ounces of passion fruit juice is used to fill each bottle}

The test statistics that would be used here <u>One-sample z test statistics</u> as we know about the population standard deviation;

                        T.S. =  \frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }  ~ N(0,1)

where, \bar X = sample mean weight of the passion fruit juice = 21.54 ounces

            \sigma = population standard deviation = 1.38 ounce

            n = sample of bottles = 65

So, <u><em>test statistics</em></u>  =  \frac{21.54-22}{\frac{1.38}{\sqrt{65} } }  

                               =  -2.687

(b) The value of z test statistics is -2.687.

(c) Since, in the question we are not given with the level of significance so we assume it to be 5%. Now, at 5% significance level the z table gives critical values of -1.96 and 1.96 for two-tailed test.

<em>Since our test statistics does not lie within the range of critical values of z, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which </em><u><em>we reject our null hypothesis</em></u><em>.</em>

Therefore, we conclude that Reject H_0 since the value of the test statistic is less than the negative critical value which means that the average different from 22 ounces of passion fruit juice is used to fill each bottle.

6 0
3 years ago
Which choice shows the coordinates of C' if the trapezoid is reflected across the y-axis?
nirvana33 [79]

Answer:

(5,3)

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Other questions:
  • Solve for x 2x=6 plz help mee
    9·1 answer
  • Subtract (5y3 + 6y + 3y4) from (4y4 - 8y3).
    9·2 answers
  • A tank can hold 50 liters of water. At present, it is only 30% full. How many liters of water shall I put in the tank, so that i
    11·1 answer
  • ❗️ASKED ONCE AND NO ONE ANSWERED, WILL MARK BRAINLIEST,,,,,,,,,,❗️. Five men and eight women work in a firm’s PR office. Their e
    10·2 answers
  • What is the maximum or minimum of the quadratic function y = 16x^2 +40x +25
    14·1 answer
  • System of Linear Equations that have no solution? I need a system.
    10·1 answer
  • The small rectangle was enlarged to create the big rectangle. A smaller rectangle has a length of 12 feet and a width of 2 feet.
    10·2 answers
  • Please help!! Will give brainliest
    10·1 answer
  • Explain why 2 raised to the third power is not equal to 6.​
    14·1 answer
  • Seven less than twice a number is 13
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!