Find the relative maximum value of f(x,y)=2xy, subject to the constraint x+y=14.
The relative maximum value is f(__,___)=_____
(Simplify your answers.)
Find the relative minimum value of f(x,y)=x^2+y^2−2xy, subject to the constraint x+y=4
The relative minimum value is f(___,___)= _____
(Simplify your answers.)
Find the relative maximum value of f(x,y,z)=xyz^2, subject to the constraint x+y+2z=10.
The relative maximum value is f(____,____,___)=_____
(Simplify your answers.)
Answers
1) the relative maximum value of \(f(x, y) = 2xy\) subject to the constraint \(x + y = 14\) is \(f(7, 7) = 98\).
2) the relative minimum value of \(f(x, y) = x^2 + y^2 - 2xy\) subject to the constraint \(x + y = 4\) is \(f(1, 3) = 4\).
3) Define the Lagrangian as:
\[L(x, y, z, \lambda) = xyz^2 + \lambda(x + y + 2z - 10)\]
To find the relative maximum and minimum values of the given functions subject to the given constraints, we can use the method of Lagrange multipliers.
1) For the function \(f(x, y) = 2xy\) subject to the constraint \(x + y = 14\), we define the Lagrangian as:
\[L(x, y, \lambda) = 2xy + \lambda(x + y - 14)\]
To find the relative maximum value, we need to solve the following equations simultaneously:
\[\frac{\partial L}{\partial x} = 0,\]
\[\frac{\partial L}{\partial y} = 0,\]
\[\frac{\partial L}{\partial \lambda} = 0,\]
along with the constraint \(x + y = 14\).
Solving these equations, we find that \(x = 7\), \(y = 7\), and \(\lambda = 1\).
To determine the value of the function at the relative maximum, we substitute these values into the function \(f(x, y)\):
\[f(7, 7) = 2(7)(7) = 98.\]
Therefore, the relative maximum value of \(f(x, y) = 2xy\) subject to the constraint \(x + y = 14\) is \(f(7, 7) = 98\).
2) For the function \(f(x, y) = x^2 + y^2 - 2xy\) subject to the constraint \(x + y = 4\), we follow the same steps.
Define the Lagrangian as:
\[L(x, y, \lambda) = x^2 + y^2 - 2xy + \lambda(x + y - 4)\]
Solving the equations \(\frac{\partial L}{\partial x} = 0\), \(\frac{\partial L}{\partial y} = 0\), \(\frac{\partial L}{\partial \lambda} = 0\) along with the constraint \(x + y = 4\), we find \(x = 1\), \(y = 3\), and \(\lambda = 1\).
Substituting these values into the function \(f(x, y)\):
\[f(1, 3) = (1)^2 + (3)^2 - 2(1)(3) = 1 + 9 - 6 = 4.\]
Therefore, the relative minimum value of \(f(x, y) = x^2 + y^2 - 2xy\) subject to the constraint \(x + y = 4\) is \(f(1, 3) = 4\).
3) For the function \(f(x, y, z) = xyz^2\) subject to the constraint \(x + y + 2z = 10\), we again follow the same steps.
Define the Lagrangian as:
\[L(x, y, z, \lambda) = xyz^2 + \lambda(x + y + 2z - 10)\]
Solving the equations \(\frac{\partial L}{\partial x} = 0\), \(\frac{\partial L}{\partial y} = 0\), \(\frac{\partial L}{\partial z} = 0\), \(\frac{\partial L}{\partial \lambda} = 0\) along with the constraint \(x + y + 2z = 10\), we find \(x = 2\), \(y = 2\), \(z = 3\), and \(\lambda = 4\).
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Related Questions
SOMEONE PLEASE HELP ASAPPPP!!!!
Thank youuu!!!
Answers
Answer:
answer is 106
hope it helps and bye :)
please also answer my question
A smart phone is bought at ₹ 46000. Its value depreciated at the rate of 12% per annum. Find its
value after one year.
Step-by-step explanation:
Gwen and her sister started watching a cartoon movie at 11:00 A.M. The movie was 2 hours and 5 minutes long. After the movie, they played a card game for 1 hour and 15 minutes and then played soccer in the backyard for 50 minutes. What time was it when Gwen and her sister finished playing soccer?
Include A.M. or P.M. in your answer (for example, 11:58 A.M.).
Answers
Answer: 2:10 PM
Step-by-step explanati
Mr. Hazel is a professional plumber. To fix a drain that is clogged, he charges a customer
$50 per hour, plus a fixed fee of $25 for the service call, as represented by the function
f(x)= 50x + 25. In the equation, what is represented by the variable "x"?
Answers
Answer:
The x is how much hours the job takes,
You can see that 25 is a fixed fee which is why there is no x attached to it, and 50 is the variable that can change, hope that made sense!
Step-by-step explanation:
^^^
Can I have brainliest? It would help me out, if not thanks anyways! Hope this helped and have a nice day :)
17 = a - 3 - За
Answer pleassse
Answers
Answer:
-10 = a
Step-by-step explanation:
17 = a - 3 - За
Combine like terms
17 = -3 -2a
Add 3 to each side
17+3 = -3-2a+3
20 = -2a
Divide by -2
20/-2 = -2a/-2
-10 = a
Answer:
a=-10
Step-by-step explanation:
17= -3-2a
17+3=-2a
20=-2a
a=-10 hope I helped
Which sequence of transformations proves that shape I is similar to shape II?
A.
a reflection across the x-axis, and then a translation 6 units right
B.
translation 6 units right, and then a 90° clockwise rotation about the origin
C.
a 180° counterclockwise rotation about the origin
D.
a reflection across the y-axis, and then a translation 4 units up
E.
a reflection across the x-axis, followed by a 180° counterclockwise
rotation about the origin, and then a translation 4 units down
Answers
Answer:
E
Step-by-step explanation:
Answer:
c
Step-by-step explanation:
in a(n) ___ (or base 2) number system, there are only two values: zero and one.
Answers
In a binary (or base 2) number system, there are only two values: zero and one.
The binary number system is a positional numeral system that uses only two symbols, 0 and 1, to represent all its numbers. It is the most basic number system in computing and digital electronics, as it can easily be represented by two states: on and off, or true and false.
The binary number system is a simple and efficient method for representing numbers in computing and digital electronics, relying on just two values, 0 and 1, to represent all numbers.
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The point P = (-5/3 squared, y) lies on the unit circle shown below. What is the value of
y in simplest form?
Answers
The required value of y for the unit circle is: 2/3
How to find the point on the unit circle ?The circle is defined as the locus of a point whose distance from a fixed point is constant i.e center (h, k).
The equation of the circle is given by:
(x - h)² + (y - k)² = r²
where:
h, k is the coordinate of the center of the circle on coordinate plane.
r is the radius of the circle.
Here,
Equation of the unit circle is given as,
x² + y² = 1
Now substitute the given value in the equation,
5/9 + y² = 1
y² = 1 - 5/9
y² = 4/ 9
y = √(4/9)
y = 2/3
Thus, the required value of y for the unit circle is 2/3
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Type the correct answer in the box. Use numerals instead of words. If necessary, use/ for the fraction bar.
Given the figure, find the total area of the shaded region.
D
8-
6-
4-
2-
O
-2-
o
S
The area of the shaded region is
B
R
8
с
square units
Answers
The value of the total area of the shaded region are,
⇒ 42 units²
We have to given that;
Sides of rectangle are,
AB = 9
BC = 6
Hence, The area of rectangle is,
⇒ 9 x 6
⇒ 54 units²
And, Area of triangle is,
A = 1/2 × 4 × 6
A = 12 units²
Thus, The value of the total area of the shaded region are,
⇒ 54 - 12
⇒ 42 units²
So, The value of the total area of the shaded region are,
⇒ 42 units²
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You are baking cookies you need to buy chocolate chips and butterscotch chips you bought eight bags all together chocolate chips cost three dollars and 50 Cent per bag and butterscotchchips cost $4.75 per bag and you spent $30.25 at the store how many bags of each time did you buy
Answers
The number of butterscotch bags is 2.75 and the number of chocolate bags is 5.25.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that you bought eight bags altogether chocolate chips cost three dollars and 50 Cent per bag and butterscotch chips cost $4.75 per bag and you spent $30.25 at the store how many bags of each time did you buy
The number of bags will be calculated as,
3.5C + 4.75B = 30.25
C + B = 8
Solve by elimination method,
3.5C + 4.75B = 30.25
3.5C + 3.5B = 28
________________
B = 2.75
The number of chocolate bags,
C + B = 8
C = 8 - 2.75
C = 5.25
Therefore, the number of butterscotch bags is 2.75 and the number of chocolate bags is 5.25.
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Analyze the conditional statement below and complete the instructions that follow.
If you break it, then you buy it.
Identify the contrapositive of the conditional statement.
If you buy it, then you break it.
If you did not break it, then you did not buy it.
If you buy it, then you did not break it.
If you did not buy it, then you did not break it.
Answers
Answer:
If you did not break it, Then you did not by it.
Step-by-step explanation:
how do I do this homework???
Answers
Step-by-step explanation:
a)
135
/ \
3. 45
/ \
3. 15
/ \
3. 5
b)
250
/ \
10 .25
/\ / \
2 5. 5 5
c)
351
/ \
3. 117
/ \
3. 39
/ \
3. 13
Hope it helps :)
A football stadium splits ticket sales in ratio of 3 : 4 between the away team and the home team. The home team make £36,000. What is the total amount of ticket sales?
Answers
The ratio 3:4 means for every 3 pounds made in away team tickets versus home team tickets. Multiply both sides by 9000 to have that 4 turn into 36000. The ratio 3:4 will then turn into 27000:36000
With this new ratio, we see that the away team pulls in 27000 and the home team pulls in 36000. Add the two amounts to get the final answer
27000+36000 = 63000
------------
We could alternatively add 3 and 4 (from the ratio 3:4) to get 3+4 = 7. Then multiply by that same scale factor 9000 getting 9000*7 = 63000.
3. Consider the following
Garden Plus LLC is fencing a triangular garden (pictured below) for Mr.
Gold.
Part B: If mI = 33 and mY = 14d - 19, then determine the value of d.
Answers
Answer: Answer (Part A & Part B)
Step-by-step explanation:
Part A ; m<Y= 113-w
Part B ; 14d-19+33+67=180
D= 99/14
The GCF of the numerator and denominator in 15/24 is _____.
8
3
2
5
Answers
Answer:
3
Step-by-step explanation: 8 can go into 24 perfectly but not 15 and 5 can go into 15 perfectly but not 24 but 24/3 is 8 and 15/3 is 5
You might say two could do the same thing which is true but we are talking about the GCF, they are asking for the greatest common factor so that's why the answer is 3
Answer:
3
basically, 8 can go into 24 exactly but 15 and 5 can not go into 15 perfectly neither can 24. 24/3 is 8 and 15/3 is 5
You might say two could do the same thing which is true but we are talking about the GCF, they are asking for the greatest common factor so that's why the answer is 3 :D
y=−4x−3
y=−2x+1
answer asap
Answers
Answer:
Use the system of equations method, this is in y= form so it can be substituted or graphed. Lets do using the substitution, so substitute the second euqation into first...
y=−4x−3 y=−2x+1
-2x+1=-4x-3 substitute
-2x+4x=-3-1 move and combine like terms
2x=-4 divide
x=-2
Have a gud day!
solve general solutions
\( \sqrt{3tan \:} (2x - 10) + 1 = 0\)
Answers
\(\\ \rm\Rrightarrow \sqrt{3tan(2x-10)}+1=0\)
\(\\ \rm\Rrightarrow \sqrt{3tan(2x-10)}=-1\)
\(\\ \rm\Rrightarrow 3tan(2x+10)=1\)
\(\\ \rm\Rrightarrow tan(2x+10)=1/3\)
1/3=0.33..tan^{-1}(0.33)≈18\(\\ \rm\Rrightarrow 2x+10=18\)
\(\\ \rm\Rrightarrow 2x=8\)
\(\\ \rm\Rrightarrow x=4\)
X= ?? geometry ??? And how ????
Answers
Using the equations for tangent and secant lines:
Let the unknown length of the line inside the circle = y
6^2 = 3 x (y+3)
Simplify:
36 = 3y+9
Subtract 9 from both sides:
27 =3y
Divide both sides by 3:
Y = 9
Now add to get x:
X = 9 + 3 = 12
X = 12
What are the zeros of the function h (x) = x² + 3x - 8?
A
x = -8 and x = -2
OB
x= -8 and x = 2
cx = -2 and x = 8
OD x = 2 and x = 8
Answers
The following are the zeros for the function h (x) = x2 + 3x - 8: - x= -4 and x=2.
Describe functions.Given a collection of inputs X (domain) and a set of potential outputs Y (codomain), a function is more technically defined as a set of ordered pairings (x,y) where xX and yY with the caveat that there can only be one ordered pair with the same value of x. The function notation f:XY can be used to express that f is a function from X to Y.
The function's zero is a value of x that makes it equal to zero. In other words, the equation f(x) = 0 leads to a zero.
By putting h(x) equal to zero and figuring out x, we may determine the zeroes for the function h(x) = x2 + 3x - 8.
h(x) = x² + 3x - 8 = 0
We may factor the left side of the equation to find x:
x² + 3x - 8 = (x-2)(x+4) = 0
We set each factor to zero and solve for x to discover the zeroes:
x-2 = 0 or x+4 = 0
x = 2 or x = -4
Consequently, the function's zeros are x = 2 and x = -4.
So, A is the right response. x = -4 and x = 2
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The complete question is
What are the zeros of the function h (x) = x² + 3x - 8?
A. x = -4 and x = -2
B. x= -8 and x = 2
C. x = -2 and x = 8
D. x = 2 and x = 8
Grant plans to evaporate enough water from 22 gallons of a 16% ammonia solution to make a 24% ammonia solution. Which equation can he use to find n, the number of gallons of water he should remove?
3.52 (22 minus n) = 0.24
StartFraction 22 minus n Over 3.52 EndFraction = StartFraction 24 Over 100 EndFraction
StartFraction 3.52 Over 22 minus n EndFraction = StartFraction 24 Over 100 EndFraction
3.52 + (22 minus n) = 0.24
Answers
The equation that he used to find n, the number of gallons of water he should remove is 3.52 / (22 - n) = 24 / 100. Option C is correct.
Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.
Amount of ammonia in the solution 16% x 22 gallons = 3.52 gallons
The proportion of ammonia remains the same as the water evaporates, but the total composition of the solution is decreased by the amount of water that evaporates.
Quantity of ammonia = 3.52 gallons
Quantity of water after evaporation = 22 - n
Composition of ammonia after evaporation of water, 24% = 24/100
Now the percentage of ammonia after evaporates
Quantity of ammonia / Remaining water = 24%
3.52 / (22 - n) = 24 / 100
Thus, the equation that he used to find n, the number of gallons of water he should remove is 3.52 / (22 - n) = 24 / 100.
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Havel and Petra are married and will file a joint tax return. Havel has W-2 income of $40,000, and Petra has W-2 income of $41,665.Use the appropriate Tax Tables and Tax Rate Schedules.Required:What is their tax liability using the Tax Tables?What is their tax liability using the Tax Rate Schedule?
Answers
Answer:
a. Tax liability using Tax Tables $8,579
b. Tax liability using Tax Rate Schedule $8,576.80
Step-by-step explanation:
Picture attached help please!
Answers
The missing lengths in the triangles are h = 3/2√2 and b = 4√3
How to determine the missing lengthsTriangle a and b
From the question, we have the following parameters that can be used in our computation:
A special right triangle
For a right triangle with an angle of 45 degrees
The measure of the leg is
h = Hypotenuse/√2
So, we have
h = 3/√2
Evaluate
h = 3/2√2
For the other triangle, we have
sin(60) = b/6
So, we have
b = 6/sin(60)
Evaluate
b = 4√3
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If luke's children represent the first generation of the family, which type of function would best model the number of children in the nth
generation of the family?
a. linear function of n
b. quadratic function of n
c. square root function of n
d. exponential function of n
Answers
The type of function that would best model the number of children in the nth generation of the family is given as follows:
d. exponential function of n.
How to define an exponential function?An exponential function has the definition presented according to the equation as follows:
\(y = ab^x\)
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.For this problem, we have that since each children has 2 children, in each generation the number of children is multiplied by 2, hence an exponential function with b = 2 best models the situation.
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When rolling a die (number cube), what is the probability of rolling a 2 or a 3?
OA. //
OB.
0C. 1/1/0
OD. 1/1/2
OE. 5/10
Answers
Answer:
1/3
Step-by-step explanation:
6 faces in a die
1/6 of rolling each side
1/6 of rolling 2 or 3
2/6 chance, 1/3 (simplified)
Describe a quad trilateral that cannot be placed in the hierarchy under trapezoids.
Answers
A quad trilateral is a four-sided polygon. In order to describe a quad trilateral that cannot be placed in the hierarchy under trapezoids, we need to consider the properties of trapezoids.
A quad trilateral that cannot be placed in the hierarchy under trapezoids because trapezoids have exactly one pair of parallel sides, while the quad trilateral in question does not meet this requirement.
A trapezoid is a quadrilateral with at least one pair of parallel sides. To be more specific, a trapezoid has exactly one pair of parallel sides. This means that the other two sides of a trapezoid are not parallel to each other.
Now, let's consider a quad trilateral where all four sides are parallel. In this case, the quad trilateral cannot be placed in the hierarchy under trapezoids because a trapezoid must have exactly one pair of parallel sides. Since all four sides of the quad trilateral are parallel, it does not meet the definition of a trapezoid.
Here's an example to illustrate this:
Imagine a quadrilateral with sides AB, BC, CD, and DA. If AB is parallel to CD, and BC is parallel to DA, while also being equal in length, then all four sides are parallel. This quad trilateral cannot be placed in the hierarchy under trapezoids because it does not have exactly one pair of parallel sides.
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-24x(x-27). multiply
Answers
Answer:
I think it 648.
Step-by-step explanation:
In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle
Answers
The largest possible error in the area of the circle is of 44%.
What is the area of a circle?The area of a circle of radius r is given by:
\(A = \pi r^2\)
With a diameter of 20 cm = radius of 10 cm, the area in cm² is given by:
\(A = \pi \times 10^2 = 100\pi\)
With the error, with a diameter of 20 x 1.2 = 24 cm = radius of 12 cm, the area in cm² is given by:
\(A = \pi \times 12^2 = 144\pi\)
The error is given by:
\(E = 144\pi - 100\pi = 44\pi\)
The percent error is the error divided by the initial amount, multiplied by 100%, hence:
\(P = \frac{44\pi}{100\pi} \times 100\% = 44\%\)
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For the expression 34- 34a to have a negative value, what must be true about the value of a ?
Answers
Answer:
a>0
Step-by-step explanation:
if a is negative then the subtraction sign would switch. therefore, for the expression to have a negartive value a must be greater than 0
Match the formula with the statement that best describes when it would be used. Put the Capital letter of the description in the blank space next to the formula. | a. S(I – R)-1 b. " A Col. = Ba2, C. y = c + C2x d. X=A-1. Y e. N.Exy-Ex-2y C2 = m = N-Ex2-(Ex)2 C = b = Σy-m-Σχ N Σy-C2 Σχ N f. C=(AT • A)-1.(ATY) g. y-yı = m(x – x4) h. X = (I – A)-1.D A. This formula is used to find a linear model when two data points are given. B. This formula is used to find the equation of a line given a point and a slope or given two points. C. This formula is used to help find the stable matrix of an absorbing matrix. D. This formula is used to find a linear model when more than two data points are given. E. This formula is used to find the c values for an overdetermined matrix. F. This formula is used to find the c values for a polynomial model that has degree n and n+1 data points. G. This formula is used to find the product matrix given an input-output matrix and a consumer demand matrix. H. This formula is used to find the c values for an overdetermined matrix. 1. This formula is used to project future outcomes by finding the stable matrix.
Answers
The given formulas are matched with their corresponding descriptions as follows:
a. S(I – R)-1 - H. This formula is used to find the stable matrix of an absorbing matrix.b. A Col. = Ba2 - B. This formula is used to find the equation of a line given a point and a slope or given two points.c. y = c + C2x - D. This formula is used to find a linear model when more than two data points are given.d. X=A-1.Y - G. This formula is used to find the product matrix given an input-output matrix and a consumer demand matrix.e. N.Exy-Ex-2y C2 = m = N-Ex2-(Ex)2 - F. This formula is used to find the c values for a polynomial model that has degree n and n+1 data points.f. C=(AT • A)-1.(ATY) - E. This formula is used to find the c values for an overdetermined matrix.g. y-yı = m(x – x4) - A. This formula is used to find a linear model when two data points are given.h. X = (I – A)-1.D - C. This formula is used to help find the stable matrix of an absorbing matrix.
This formula is used to project future outcomes by finding the stable matrix.
a. The formula S(I – R)-1 is used to find the stable matrix of an absorbing matrix, which represents the long-term distribution of a Markov chain.
b. The formula A Col. = Ba2 is used to find the equation of a line given a point and a slope or given two points, allowing us to model linear relationships.
c. The formula y = c + C2x is used to find a linear model when more than two data points are given, enabling us to estimate the relationship between variables.
d. The formula X = A-1.Y is used to find the product matrix given an input-output matrix and a consumer demand matrix, facilitating economic analysis.
e. The formula N.Exy-Ex-2y C2 = m = N-Ex2-(Ex)2 is used to find the c values for a polynomial model that has degree n and n+1 data points, allowing us to fit a polynomial curve to the data.
f. The formula C=(AT • A)-1.(ATY) is used to find the c values for an overdetermined matrix, helping us solve systems of equations with more equations than unknowns.
g. The formula y-yı = m(x – x4) is used to find a linear model when two data points are given, providing a simple equation for a straight line.
h. The formula X = (I – A)-1.D is used to help find the stable matrix of an absorbing matrix, which represents the long-term behavior of a Markov chain.
The statement "This formula is used to project future outcomes by finding the stable matrix" refers to the formula used to find the stable matrix of an absorbing matrix, which allows us to analyze the long-term behavior of a system.
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Divide.
511÷411
Enter the correct answer in the box.
Answers
Answer:
1.24
Step-by-step explanation:
Answer:
1.24330900243
Step-by-step explanation:
I did the whole equation on paper
Charlie uses 12 milliliters of honey in his honey-lemon salad dressing. He makes just enough of the salad dressing for 4 salads. He puts the same amount on each salad.
How much honey is on each salad?
______Milliliters
Answers
3 milliliters.
Step-by-step explanation:
3 because 12÷4=3 and he made enough for 4.