Use Lagrange Multipliers to solve the following: Maximize f(x, y, z) = 4x + 2y + z subject to x^2 + y + z^2 = 1.
Answers
To maximize f(x,y,z)=4x+2y+z subject to x2+y+z2=1, we can employ the method of Lagrange multipliers.
The first step is to define the Lagrangian function L(x,y,z,λ)=f(x,y,z)−λ(g(x,y,z)−c),
where λ is a Lagrange multiplier and c is a constant.
We have: L(x,y,z,λ)=4x+2y+z−λ(x2+y+z2−1)
Setting the partial derivatives of L equal to zero gives us a system of equations:
\(∂L∂x=4−2λx=0∂L∂y=2−λ=0∂L∂z=1−2λz=0∂L∂λ=x2+y+z2−1=0\)
Solving for λ from the second equation gives λ=2.
Substituting into the first and third equations yields x=2 and z=1/2, respectively.
Using the fourth equation, we find that y=−15/4.
Thus, the critical point is (2,−15/4,1/2).
To check that this is indeed a maximum,
we need to examine the Hessian matrix of second partial derivatives. This matrix is given by:\(H(f)=⎡⎣⎢02402−2λ02⎤⎦⎥H(f)=(02402−2λ02)H(f)=(02402−22)H(f)=(02−20)H(f)=(0−20)Since λ=2\\\),
we have:H(f)=(02402−4)H(f)=(02−40)
The determinant of this matrix is negative,
so the critical point is a saddle point.
Therefore, the maximum value of f(x,y,z) subject to x2+y+z2=1 is not attained on the surface, but is instead at infinity.
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Related Questions
What is the GCF of 52/65?
Answers
Answer:
The greatest Common factor of 52/65 is 13!
Step-by-step explanation:
Hope this helps :)
Pls make brainliest :p
And have an amazing day <3
Answer:
13
Step-by-step explanation:
The GCF of numbers can be found by first finding the prime factors of the numbers.
52 = 2^2 * 13
65 = 5 * 13
The GCF is the product of common factors with lower exponent.
The only common factor is 13.
GCF = 13
To reduce the fraction, divide the numerator and the denominator by the GCF.
(52/13) / (65/13) = 4/5
Pls help me i need this done tonight
Answers
Answer:
27 yd²
Step-by-step explanation:
Refer to attachment.
Hope it helps :)
Solve 1/2x−5=10−3/4x . Check your solution.
x=___
Answers
Answer
The answer is 12 for sure
Step-by-step explanation:
i just solved it like any other equation
1 i added 3/4 on both sides
2 then i added 5 on both sides
3 after you equation should look like 1 1/4x = 15
4 then i divided and it should be x = 12
pls mark brainliest
The value of x is "12".
Calculating the value of x:\(\to \frac{1}{2}x-5=10-\frac{3}{4}x \\\\\to \frac{1}{2}x+ \frac{3}{4}x=10+5 \\\\\to \frac{2+3}{4}x=15 \\\\\to \frac{5}{4}x=15 \\\\\to x=15 \times \frac{4}{5}\\\\\to x=3 \times 4\\\\\to x=12\)
Therefore, the final value of x is "12".
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Two parallel lines are intersected by a third line so that angles 1 and 5 are congruent.
2 parallel horizontal lines are intersected by a third line. On the first horizontal line where the third line intersects, 4 angles are created. Labeled clockwise, from uppercase left, the angles are 1, 2, 4, 3. On the second horizontal line, where the third line intersects, 4 angles are created. Labeled clockwise, from uppercase left, the angles are 5, 6, blank, blank.
Which statement is true about angles 3 and 5?
They are acute.
They are congruent.
They are complementary.
They are supplementary.
Answers
When two parallel lines are intersected by a third line, eight angles are formed. Vertical angles are pairs of angles formed by two intersecting lines, so angles 1 and 5 must be vertical angles. Since vertical angles are always congruent, angles 1 and 5 are congruent as well.Supplementary angles are two angles whose sum is 180 degrees.
When a transversal intersects two parallel lines, eight angles are formed. Angles 1 and 5 are supplementary angles because they are on opposite sides of the transversal and their measures add up to 180 degrees.
This is because they are consecutive interior angles.So, angles 1 and 5 are congruent and supplementary.
There are a few ways to prove that two lines are parallel, but one way is to use the converse of the Corresponding Angles Postulate.
The postulate states that if two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. Since angles 1 and 5 are congruent, we can use the converse of this postulate to conclude that the two lines are parallel.
Another way to prove that two lines are parallel is to use the Alternate Interior Angles Theorem.
This theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Since angles 1 and 5 are congruent, we can use this theorem to conclude that the two lines are parallel.
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Answer: D. They are supplementary
Step-by-step explanation: RIGHT ON EDGE2023
HELP PLEASE. A water tank in the shape of a rectangular has a volume of 900 cubic feet. It is 15 feet long, 10 feet wide. What is the height of the water tank?
Options:
A: 6
B: 5
C: 3
D: 9
Answers
Answer:
The Answer is B 5 is the height
Can someone please help me with 1,2 please
Answers
Answer:
1) This HAS a right angle
2) This is an obtuse triangle
Step-by-step explanation:
1)
4² + 3² = 5²
16 + 9 = 25
25 = 25 ✅
2)
5² + 6² = 8²
25 + 36 = 64
61 ≠ 64 ❌
Since c is bigger, this is an obtuse triangle
What's the difference between weather vanes and anemometers?
Answers
Answer:
The second one
Step-by-step explanation:
A snow removal service in Minnesota is deciding to purchase a new snow removal machine. If they dont purchase the machine, they will make $20,000 if the winter is mild, 530,000 il it is typical, and $50,000 the winter is severe. If they purchase the machine, their profits for these conditions will be $30,000, 535,000 and $40,000, respectively. The probability of a mild winter is 0.3. a typical winter is 0.5 and a severe winter is 0.2. What is the EMV for no machine? 32000 34500 35000 31000
Answers
The Europay, Mastercard and Visa (EMV) for not purchase the snow removal machine is $31,000.
The expected monetary value (EMV) for not purchasing the snow removal machine is $31,000. This value is calculated by multiplying the probabilities of each winter condition by the corresponding profits and summing them up. The probabilities for a mild winter, typical winter, and severe winter are 0.3, 0.5, and 0.2, respectively.
The profits for each condition without the machine are $20,000, $530,000, and $50,000. By multiplying each profit by its probability and adding them together, we get the EMV of $31,000 for not purchase the machine.
In detail, the EMV is calculated as follows:
EMV = (Probability of Mild Winter * Profit for Mild Winter) + (Probability of Typical Winter * Profit for Typical Winter) + (Probability of Severe Winter * Profit for Severe Winter)
EMV = (0.3 * $20,000) + (0.5 * $530,000) + (0.2 * $50,000)
EMV = $6,000 + $265,000 + $10,000
EMV = $31,000
Therefore, the EMV for not purchasing the snow removal machine is $31,000.
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A fair six sided dice is rolled twice. What is the theoretical probability that the first number is greater then or equal to the second number. WILL AWARD BRAINLIEST.
Answers
36 total outcomes
Step-by-step explanation:
Since there are a 6/times 6= 36 possible outcomes
For the sequence (5, 8, 11, 14, 17,...), answer the following question.
1.What is the second term of the sequence?
Answers
Answer:8
Step-by-step explanation:5 is 1st term 8 is second term 11 is third term…
-3/7 divide 9/4 as a fraction
Answers
Answer:
-4/21
Step-by-step explanation:
Answer:
-4/21
Step-by-step explanation:
Keep switch flip method. Keep -3/7, switch division to multiplication, and flip 9/4 to become 4/9. then multiply straight through
-3/7 x 4/9 = -12/63
which simplifies to -4/21
Why is it appropriate to use the trigonometric functions on triangle FGJ but not on triangle FHJ?
Answers
Step-by-step explanation:
because it important for the trigonometric
Belinda is jumping on a trampoline. She jumps up with an initial velocity (v0) of 17 feet per second. The function d(t) = v0t-16t^2 gives the height in feet of belinda above the trampoline as a function of time in seconds after her jump. How long after her jump will it take her to return to the trampoline again
Answers
Answer:
1.0625 seconds
Step-by-step explanation:
Given that:
Initial velocity of Belinda, \(v_0=17\ ft/sec\)
Height in feet of Belinda above the trampoline is given as the formula:
\(d(t) = v_0t-16t^2\)
Where \(t\) is the time in seconds.
To find:
Time taken by Belinda to come back to trampoline = ?
Solution:
When Belinda will come back to trampoline, the height will be zero.
Putting \(d(t) =0\)
We have the equation as:
\(0 = 17t-16t^2\\\Rightarrow 16t^2-17t=0\\\because t\neq0\\\Rightarrow 16t=17\\\Rightarrow t=\dfrac{17}{16}\\\Rightarrow t =1.0625\ seconds\)
Therefore, the answer is: 1.0625 seconds
.
a) Find the total surface area of a square-based pyramid in which the area of the base
is 48.5 cm and the area of each triangular face is 24.5 cm.
Answers
Answer:
146.5 sq cm
Step-by-step explanation:
48.5 +24.5x4
24.5 times 4 = 98
98 + 48.5 = 146.5
Determine if the following graphs represent a function. Answer"yes" or "no".
Answers
2 yes
3 no
hope this helped, please mark brainliest :)
1 yes
2 yes
3 no
helppppp
please?
on a timer so need it ASAP
Answers
Answer:
I think its 2!
Step-by-step explanation:
two nonvertical lines have slopes m 1m 1 and m 2m 2 , respectively. the lines are parallel if and the are unequal; the lines are perpendicular if
Answers
Two non vertical lines have slopes m₁ and m₂ , respectively. The lines are parallel if "m₁ = m₂" and the "y-intercepts" are unequal; the lines are perpendicular if m₁ = 1/m₂.
What is slope of a line?The slope of a line is a measurement of its steepness and direction.
The slope of a line formula computes the "vertical change" to "horizontal change" ratio between two distinct points along a line.
Some key features regarding the slopes of the lines are-
1. Slope of Parallel Lines:
A set of parallel lines always has the same inclination angle. Assume we possess two parallel lines l1 and l2 in the reference plane, inclined at angles θ1 and θ2 with the x-axis, respectively, such that the, θ2 = θ1.
m1 = m2
Therefore, the slopes of the two given parallel lines are equal.
2. Slope of Perpendicular Lines:
A pair of perpendicular lines everytime form a 90 angle. Imagine we have two perpendicular lines l2 and l1 in the coordinate plane, inclined at angles θ1 and θ2 with the x-axis, respectively, so that the provided angles follow the external angle theorem, θ2 = θ1 + 90.
m1 = tan θ1
m2 = tan (θ1 + 90º) = - cot θ1
m1 × m2 = -1
Therefore, the product of slopes of two given perpendicular lines is equal to -1.
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If 58.0 of hydrogen is heated to 140 degrees Celcius and expands to 90.0 L, what was the original temperature?
_____ degree Celcius
(Precision to the nearest whole number)
Answers
The original temperature of the hydrogen gas was approximately 90°C.
To find the original temperature of the hydrogen, we can use the ideal gas law equation, which states that the product of pressure (P), volume (V), and temperature (T) is proportional to the number of moles (n) and the gas constant (R). Mathematically, it can be represented as PV = nRT.
In this scenario, the pressure and volume of the hydrogen remain constant, and we need to find the original temperature (T1). We can set up the following equation:
(P1)(V1) / T1 = (P2)(V2) / T2
Where:
P1 and P2 are the initial and final pressures, respectively.
V1 and V2 are the initial and final volumes, respectively.
T2 is the final temperature (140 degrees Celsius).
V2 is the final volume (90.0 L).
Rearranging the equation to solve for T1, we get:
T1 = (P1)(V1)(T2) / (P2)(V2)
Plugging in the given values:
T1 = (P1)(V1)(T2) / (P2)(V2)
T1 = (1 atm)(58.0 L)(140°C) / (1 atm)(90.0 L)
T1 ≈ 90.44°C
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4p5q3xp2q-4 simplify
Answers
The simplified form of the given expression is 4p⁷/q.
The given expression is 4p⁵q³×p²q⁻⁴.
Using the exponential law, aⁿ×aˣ=aⁿ⁺ˣ, we get
4(p⁵×p²)(q⁻⁴×q³)
= 4p⁵⁺²q⁻⁴⁺³
= 4p⁷q⁻¹
= 4p⁷/q
Therefore, the simplified form of the given expression is 4p⁷/q.
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After a shift in the aggregate demand curve, which variable adjusts to restore general equilibrium? A.real interest rate B.investment spending C.consumption spending D.price level
Answers
In order to restore general equilibrium, price adjustments are undertaken. In the short run, prices increase, and when expectations rise above actual inflation, prices continue to climb until they do. answer is option (d). price level.
What is aggregate demand?The term "aggregate demand" in macroeconomics refers to the overall demand for locally produced commodities, including capital goods, consumer products, and services. Aggregate demand is calculated as the total of spending by consumers, corporate and governmental investment spending, and net imports and exports.
The overall demand of final products and services in an economy at any particular time is known as aggregate demand, often referred to as domestic final demand. Effective demand is a frequent word for it, however occasionally this phrase is used to distinguish between two things. This is the demand for a nation's gross domestic product.
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If a categorical variable that can take the values from the set {Red, Blue, Green, Yellow} is included as an independent variable in a linear regression, the number of dummy variables that are created is: 2
Answers
If a categorical variable that can take the values from the set {Red, Blue, Green, Yellow} is included as an independent variable in a linear regression, the number of dummy variables that are created is two
When a categorical variable that has n categories is to be included as an independent variable in a linear regression analysis, it must be converted to n - 1 dummy variables. The reason for this is that including all n categories as dummy variables would cause perfect multicollinearity in the regression analysis, making it impossible to estimate the effect of each variable.In this case, the set of categories {Red, Blue, Green, Yellow} has four categories. As a result, n - 1 = 3 dummy variables are required to represent this variable in a linear regression. This is true since each category is exclusive of the others, and we cannot assume that there is an inherent order to the categories.The dummy variable for the first category is included in the regression model by default, and the remaining n - 1 categories are represented by n - 1 dummy variables. As a result, the number of dummy variables that are required to represent the categorical variable in the regression model is n - 1.
Thus, if a categorical variable that can take the values from the set {Red, Blue, Green, Yellow} is included as an independent variable in a linear regression, the number of dummy variables that are created is two .
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HELP ME WITH THISSS PLZZZZ.
Answers
Scale factor is 3.5
Write down two properties of parallelogram.
Can somebody help me? Pls
Answers
Answer: Opposite sides are congruent (AB = DC).
Opposite angels are congruent (D = B).
Consecutive angles are supplementary (A + D = 180°).
If one angle is right, then all angles are right.
The diagonals of a parallelogram bisect each other.
Each diagonal of a parallelogram separates it into two congruent triangles.
these are all the choices lol
Step-by-step explanation: Brainliest pls
Answer:
Step-by-step explanation:
Properties of parallelogram:
1) opposite sides are equal and parallel.
2) opposite angles are equal.
3) diagonals bisect each other.
-10 plus -9 ??????????
Answers
Answer: -19
Step-by-step explanation:
(-10) + (-9)
-10-9
-19
what was the most inaccurate version of pi? explain who, when and what the value was.
Answers
The value of pi is known to over 31 trillion decimal places, thanks to the use of powerful computers and sophisticated algorithms.
Describe about the history of pi?The history of pi dates back thousands of years, and over time, various civilizations have attempted to calculate its value with varying degrees of accuracy. One of the most inaccurate versions of pi was recorded by the ancient Babylonians around 2000 BC.
The Babylonians calculated the value of pi as 3.125, which is off by more than 6% from the actual value. It is believed that the Babylonians arrived at this value by using a rough approximation of a circle as a hexagon. They measured the perimeter of the hexagon and divided it by the diameter to get their approximation of pi.
This value was later refined by the ancient Egyptians and Greeks, who were able to calculate pi with greater accuracy. The Greek mathematician Archimedes, for instance, was able to calculate pi to within 1% accuracy by using a method of exhaustion.
It wasn't until the development of calculus in the 17th century that mathematicians were able to derive an exact formula for pi. Today, the value of pi is known to over 31 trillion decimal places, thanks to the use of powerful computers and sophisticated algorithms.
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Gina walked 0.8 miles before she started jogging at an average pace of 6 miles per hour. The equation d = 0.8 + 6t can be used to relate the total distance, d, in miles to the time, t, that Gina spent jogging. What are the independent and dependent variables?
Answers
Answer:
independent: tdependent: dStep-by-step explanation:
The way the equation is written, d is the dependent variable; t is the independent variable (on which d depends).
Answer:
the independent variable is t (time) and the dependent variable is d (distance)
Step-by-step explanation:
Time is called independent variable because time determines the distance and it stands alone, while distance is dependent variable because the length of distance is determined with the time, meaning when time (t) changes, the distance changes too.
for ex. if t=1, d= 0.8+6 (1) = 6.8
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order √2, 2π, 0.8
Please help ASAP
Answers
Answer:
0.8,√2,2π
Step-by-step explanation:
Answer:
the second one, then the first, then the last one
Step-by-step explanation:
the first one is = to 1
the second one is = to 6.3
the third one is = to .8
Mrs. Wright is planning her best friend's baby shower. She calculates that the food will cost $9 per guest, and she has already spent $43 on decorations. Mrs. Wright can afford to spend a total of $160 on the baby shower. How many guests can there be?
Answers
Answer:
Mrs. Wright is planning her best friend's baby shower. She calculates that the food will cost $9 per guest, and she has already spent $43 on decorations.
9x + 43
Which is a correct first step in solving 5 – 2x < 8x – 3?
5 < 6x – 3
3x < 8x – 3
5 < 10x – 3
2 – 2x < 8
Answers
Answer:
5 < 10x - 3
Step-by-step explanation:
5 - 2x < 8x - 3
Add '2x' to both the sides
5 - 2x + 2x < 8x + 2x - 3
5 < 10x - 3
Solve for the value of s
Answers
The two angles are vertical angles, which mean they are the same.
6s-2 = 94
Add 2 to both sides:
6s = 96
Divide both sides by 6:
S = 96/6
S = 16