Use Lagrange multiplies to find the maximum and minimum values of the function subject to the given constraint(s) f(x,y,z)=x^4+y^4+z^4=1
Answers
We have two valid cases: λ = 1 ,the Maximum and minimum values are both 1. and λ = 1 - z,the maximum and minimum values occur at the endpoints of the valid range. In
The maximum and minimum values of the function f(x, y, z) = x^4 + y^4 + z^4 subject to the constraint x^4 + y^4 + z^4 = 1, we can use the method of Lagrange multipliers.
Let's define the Lagrangian function L(x, y, z, λ) as:
L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z) - 1)
where g(x, y, z) represents the constraint equation.
In this case, the Lagrangian function becomes:
L(x, y, z, λ) = x^4 + y^4 + z^4 - λ(x^4 + y^4 + z^4 - 1)
We need to find the critical points of L(x, y, z, λ) by taking partial derivatives with respect to x, y, z, and λ, and setting them equal to zero:
∂L/∂x = 4x^3 - 4λx^3 = 0
∂L/∂y = 4y^3 - 4λy^3 = 0
∂L/∂z = 4z^3 - 4λz^3 = 0
∂L/∂λ = x^4 + y^4 + z^4 - 1 = 0
From the first three equations, we can factor out 4x^3, 4y^3, and 4z^3 respectively:
x^3(1 - λ) = 0
y^3(1 - λ) = 0
z^3(1 - λ) = 0
This gives us three possibilities:
1) x = 0, y = 0, z = 0
2) λ = 1
3) x = 0, y = 0, z = 1 - λ
Let's examine each case:
1) x = 0, y = 0, z = 0:
From the constraint equation, we have 0^4 + 0^4 + 0^4 = 0 ≠ 1, so this case is not valid.
2) λ = 1:
From the constraint equation, we have x^4 + y^4 + z^4 = 1. Since the equation holds, this is a valid case.
3) x = 0, y = 0, z = 1 - λ:
Substituting these values into the constraint equation, we have (1 - λ)^4 = 1. Expanding this, we get λ^4 - 4λ^3 + 6λ^2 - 4λ + 1 = 0. This equation holds for λ = 1, so it is a valid case.
Therefore, we have two valid cases: λ = 1 and λ = 1 - z.
For λ = 1:
From the constraint equation, we have x^4 + y^4 + z^4 = 1. Since the function f(x, y, z) = x^4 + y^4 + z^4 is non-negative, the maximum and minimum values occur at the endpoints of the valid range. In this case, the maximum and minimum values are both 1.
For λ = 1 - z:
From the constraint equation, we have x^4 + y^4 + z^4 = 1. Again, since the function is non-negative, the maximum and minimum values occur at the endpoints of the valid range. In
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Related Questions
How do you compute MOR
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Using the formula σr = 3Fx/yz², we can easily compute the Modulus of Rupture.
What is the Modulus of Rupture?The term "bending strength" is occasionally used to refer to the measure of a specimen's strength prior to rupture, also known as the "modulus of rupture," or MOR.
Contrary to the modulus of elasticity, which measures the wood's deflection but not its total strength, it can be used to evaluate a species' strength.
The formula σr = 3Fx/yz² for the load force F and the material's size dimensions in the three directions of x, y, and z can be used to compute the modulus of rupture, or "sigma."
The external force applied to the substance of interest in this instance is the load.
Therefore, using the formula σr = 3Fx/yz², we can easily compute the Modulus of Rupture.
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Correct question:
How do you compute MOR?
find the slope from the table x -2, -1, 0, 1, 2 y -3, -1, 1, 3, 5
Answers
Answer:
slope = 2
Step-by-step explanation:
work is in the picture if you need it... please mark brainiest
Click on the graph to plot a point. Click a point to delete it.
Answers
The red dot on the graph is (2, -8)
find the midpoint of the line segment when given the endpoints (10, -6)(-4, -9)
Answers
Answer:
3,3/2
Step-by-step explanation:
let 10,-6 be x1, y1
-4,-9 be x2, y2
(x1+x2)/2 , (y1+y2)/2
(10-4)/2 , (-6+9)/2
6/2 , 3/2
3 , 3/2
Answer:
(3, -7.5)
Step-by-step explanation:
Use the midpoint formula: (x1 + x2) / 2 , (y1 + y2) / 2
Plug in the points:
(10 - 4) / 2 , (-6 - 9) / 2
6/2 , -15/2
3, -7.5
So, the midpoint will be at (3, -7.5)
To pay for a home improvement project that totals $20,000, a homeowner is choosing between two different credit card loans with an interest rate of 9%. The first credit card compounds interest quarterly, while the second credit card compounds monthly. The homeowner plans to pay off the loan in 10 years.
Part A: Determine the total value of the loan with the quarterly compounded interest. Show all work and round your answer to the nearest hundredth. (4 points)
Part B: Determine the total value of the loan with the monthly compounded interest. Show all work and round your answer to the nearest hundredth. (4 points)
Part C: What is the difference between the total interest accrued on each loan? Explain your answer in complete sentences. (2 points)
Please only responded if you know how to do it, will give the brainiest to however answers it correctly
Answers
The total value of the loan with quarterly compounded interest is approximately $45,288.38, while the total value of the loan with monthly compounded interest is approximately $45,634.84. The difference in total interest accrued is approximately $346.46.
Part A: To determine the total value of the loan with quarterly compounded interest, we can use the formula for compound interest:
A = P(1 + r/n)^(nt),
where:
A is the total value of the loan,
P is the principal amount (initial loan amount),
r is the interest rate (in decimal form),
n is the number of times interest is compounded per year,
and t is the number of years.
Given:
P = $20,000,
r = 9% or 0.09,
n = 4 (quarterly compounding),
t = 10 years.
Substituting the values into the formula, we have:
A = 20000(1 + 0.09/4)^(4*10).
Calculating this value, we find:
A ≈ $45,288.38.
Therefore, the total value of the loan with quarterly compounded interest is approximately $45,288.38.
Part B: To determine the total value of the loan with monthly compounded interest, we follow the same formula but with a different value for n:
n = 12 (monthly compounding).
Substituting the values into the formula, we have:
A = 20000(1 + 0.09/12)^(12*10).
Calculating this value, we find:
A ≈ $45,634.84.
Therefore, the total value of the loan with monthly compounded interest is approximately $45,634.84.
Part C: The difference between the total interest accrued on each loan can be calculated by subtracting the principal amount from the total value of each loan.
For the loan with quarterly compounding:
Total interest = Total value - Principal
Total interest = $45,288.38 - $20,000
Total interest ≈ $25,288.38.
For the loan with monthly compounding:
Total interest = Total value - Principal
Total interest = $45,634.84 - $20,000
Total interest ≈ $25,634.84.
The difference between the total interest accrued on each loan is approximately $346.46.
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if it is given that "x" is 23.5 - proof that it is a point of intersection at y= 1/2(x) - 25 if y is equal to 11. been trying but not working out.
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When substituting y = 11 into the equation y = 1/2(x) - 25, we find that x = 72, confirming that (23.5, 11) is a valid point of intersection.
Given that x is 23.5, it is required to prove that it is an intersection point for the equation y = 1/2(x) - 25 when y is equal to 11.
The equation is given as y = 1/2(x) - 25
When y = 11, we can substitute the value of y in the equation to obtain 11 = 1/2(x) - 25
This can be simplified as 11 + 25 = 1/2(x)36 = 1/2(x)
On solving, x = 72Thus, when y is equal to 11 and x is equal to 72, the given point of intersection is valid.
Therefore, it can be concluded that x = 23.5 is a point of intersection for the equation y = 1/2(x) - 25 when y is equal to 11.
In summary, when given an equation with two variables, we can find the point of intersection by setting one of the variables to a given value and solving for the other variable. In this case, when y is equal to 11, we can solve for x and obtain the point of intersection as (72,11).
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14.32 × 1.2,...........
Answers
Answer:
17.18400
Step-by-step explanation:
When rounded to the nearest whole number, it's 17.
Hope this helped :)
I Need Help ASAP or i will go to DETENTION!!!!!!!!
Answers
Answer:
m<FLS=108 degrees
m<SLT=72 degrees
m<ALG=18 degrees
Step-by-step explanation:
(3x) + (4x +12) = 180
x = 24
4x + 12 = 108 = m<FLS
180 - 108 = 72 = m<SLT
SLG = 72 + 90 + x
x = 18 = m<ALG
the amount of time it takes to complete a reading assignment might be a function of which of the following
Answers
Answer:
Option B is correct
Step-by-step explanation:
The amount of time it takes to complete a reading assignment is not concerned with publishing year of the book (it might be published already).
The amount of time it takes to complete a reading assignment is also not related to the cost of the book (it might be purchased already)
The amount of time it takes to complete a reading assignment is related to the number of pages. Because if you complete a reading assignment, you might need to read all of the pages in the assignment.
Hope this helps!
:)
The box-and-whisker plots below show the distribution of prices (in thousands of dollars) of sports cars at two local dealerships.
Which of the following statements is true?
A. The median price at Dealer 1 is lower than the upper quartile price at Dealer 2.
B. The lower extreme price at Dealer 1 is higher than the upper quartile price at Dealer 2.
C. The upper extreme price at Dealer 1 is lower than the upper extreme price at Dealer 2.
D. The lower extreme price at Dealer 1 is higher than the median price at Dealer 2.
Answers
The lower extreme price at Dealer 1 is higher than the median price at Dealer 2.
What is price?Price is the value of a product or service that an individual or business is willing to accept in exchange for a good or service. It is typically determined by the market forces of supply and demand. Prices are usually expressed in monetary terms, but can also be expressed in terms of labor time or other resources. Prices can vary widely depending on the availability of a good or service, the amount of competition, and other factors.
This conclusion can be drawn from the box-and-whisker plots of the prices of the sports cars at each of the two dealerships. The lower extreme price of the sports cars at Dealer 1 is represented by the lower whisker of the box-and-whisker plot and is higher than the median price of the cars at Dealer 2, which is represented by the horizontal line in the middle of the box-and-whisker plot. This means that the lower extreme price of the sports cars at Dealer 1 is higher than the median price of the cars at Dealer 2.
The higher prices of the sports cars at Dealer 1 could be due to several factors. It is possible that the cars sold at Dealer 1 are of a higher quality or have more features than those sold at Dealer 2. Additionally, Dealer 1 may simply be charging higher prices due to its location or reputation. Whatever the reason, it is clear that the lower extreme price of the sports cars at Dealer 1 is higher than the median price of the cars at Dealer 2.
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I need Help ASAP I will give you brainlest and plus 5 stars ✨✨and plus thanks and 50 points
Answers
Answer:
1 page. 1 answer= 2 page. Answer= Tension
Dependent
1 page. 2 answer=
Solution
1 page. 3 answer=
Two
1 page. 4 answer=
Origin
1 page. 5 answer=
Quadrants
1 page. 6 answer=
Perpendicular or Orthogonal
On average, for every 100 seventh grade students at a school, 48 students can play an instrument. If the school has 77 seventh grade students, about how many of those students would you expect to play an instrument?
Answers
Thus, number of students in seventh grade who can play instrument out of 77 are 37.
Define about the proportion:In general, the term "proportion" refers to a part, share, or amount that is compared to a whole. Depending on the definition of proportion, two ratios are in proportion when they are equal. Two ratios as well as fractions are equal when an equation or a declaration to that effect is utilised.
Whereas if ratio between the initial and second is identical to the ratio between of second and third, then each three quantities are referred to as having a continuing proportion.
Given data for the seventh grade students:
Out of 100 ---> 48 can play instruments.
Out of 77 ---> x (say) can play instruments.
So,
100/77 = 48/x
x = 48*77 / 100
x = 36.96
x = 37 (students)
Thus, number of students in seventh grade who can play instrument out of 77 are 37.
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this is rlly easy but im lazy and dont wana answer it, i mark brainliesttttt tyyy ❤️
Answers
Answer:
C and D is correct
Step-by-step explanation:
C: Debit is negative, and if you add 27 to -14, you get 13 which is Maria's account.
D: Its the same thing but opposite. 13 minus 27 is -14.
♡ Hope it helps ♡
Answer:
both c and d
Step-by-step explanatioin
nobody is answering my question so im saving up like 100 points so someone will actually do it so ye
Three populations have proportions 0.1, 0.3, and 0.5. We select random samples of the size n from these populations. Only two of the distributions of the sample proportions are normally distributed. Choose all possible values of n.
a. 10
b. 100
c. 50
d. 40
e. 20
Answers
Answer:
(1) A Normal approximation to binomial can be applied for population 1, if n = 100.
(2) A Normal approximation to binomial can be applied for population 2, if n = 100, 50 and 40.
(3) A Normal approximation to binomial can be applied for population 2, if n = 100, 50, 40 and 20.
Step-by-step explanation:
Consider a random variable X following a Binomial distribution with parameters n and p.
If the sample selected is too large and the probability of success is close to 0.50 a Normal approximation to binomial can be applied to approximate the distribution of X if the following conditions are satisfied:
np ≥ 10 n(1 - p) ≥ 10The three populations has the following proportions:
p₁ = 0.10
p₂ = 0.30
p₃ = 0.50
(1)
Check the Normal approximation conditions for population 1, for all the provided n as follows:
\(n_{a}p_{1}=10\times 0.10=1<10\\\\n_{b}p_{1}=100\times 0.10=10=10\\\\n_{c}p_{1}=50\times 0.10=5<10\\\\n_{d}p_{1}=40\times 0.10=4<10\\\\n_{e}p_{1}=20\times 0.10=2<10\)
Thus, a Normal approximation to binomial can be applied for population 1, if n = 100.
(2)
Check the Normal approximation conditions for population 2, for all the provided n as follows:
\(n_{a}p_{1}=10\times 0.30=3<10\\\\n_{b}p_{1}=100\times 0.30=30>10\\\\n_{c}p_{1}=50\times 0.30=15>10\\\\n_{d}p_{1}=40\times 0.10=12>10\\\\n_{e}p_{1}=20\times 0.10=6<10\)
Thus, a Normal approximation to binomial can be applied for population 2, if n = 100, 50 and 40.
(3)
Check the Normal approximation conditions for population 3, for all the provided n as follows:
\(n_{a}p_{1}=10\times 0.50=5<10\\\\n_{b}p_{1}=100\times 0.50=50>10\\\\n_{c}p_{1}=50\times 0.50=25>10\\\\n_{d}p_{1}=40\times 0.50=20>10\\\\n_{e}p_{1}=20\times 0.10=10=10\)
Thus, a Normal approximation to binomial can be applied for population 2, if n = 100, 50, 40 and 20.
Math time everybody! ALSO,im gonna need points here soon
Answers
Answer:
The Answer is -3
Step-by-step explanation:
Brendan is reading a book that is 360 pages long He 2 still has 3 of the book to read. If he plans to finish the book in the next four -nights, how many pages will he have to read each night?
Answers
We have a book of 360 pages.
He had already read 2/3 of the book, so there is 1-(2/3)=1/3 of the book left.
\(1-\frac{2}{3}=\frac{3}{3}-\frac{2}{3}=\frac{1}{3}\)This means that 120 pages are left.
\(360\cdot\frac{1}{3}=\frac{360}{3}=120\)If he wants to read the rest of the book in 4 nights, he has to read 30 pages a night.
\(\frac{120}{4}=30\)a shelf is 24 inches
Answers
Answer:
Step-by-step explanation:
can you help me pls
what does 3 - (-8) equal????
Answers
Answer:
11
Step-by-step explanation:
3-(-8) = 3+8
= 11
Suppose that a study of elementary school students reports that the mean age at which children begin reading is 5.3 years with a standard deviation of 1.1 years.
Step 1 of 2: If a sampling distribution is created using samples of the ages at which 35 children begin reading, what would be the mean of the sampling distribution of sample means? Round to two decimal places, if necessary.
Answers
Answer:
The mean of the sampling distribution of sample means is equal to the population mean, which is 5.3 years.
give thanks for more! your welcome!
Step-by-step explanation:
Given the polynomial 9x2y6 − 25x4y8, rewrite as a product of polynomials.
(9xy3 − 25x2y4)(xy3 + x2y4)
(9xy3 − 25x2y4)(xy3 − x2y4)
(3xy3 − 5x2y4)(3xy3 + 5x2y4)
(3xy3 − 5x2y4)(3xy3 − 5x2y4)
Answers
Answer:
Option 3
(3xy³ + 5x²y⁴) (3xy³ - 5x²y⁴)
Step-by-step explanation:
Factorize polynomials:
Use exponent law:
\(\boxed{\bf a^{m*n}=(a^m)^n} \ & \\\\\boxed{\bf a^m * b^m = (a*b)^m}\)
9x²y⁶ = 3²* x² * y³*² = 3² * x² * (y³)² = (3xy³)²
25x⁴y⁸ = 5² * x²*² * y⁴*² = 5² * (x²)² * (y⁴)² = (5x²y⁴)²
Now use the identity: a² - b² = (a +b) (a -b)
Here, a = 3xy³ & b = 5x²y⁴
9x²y⁶ - 25x⁴y⁸ = 3²x²(y³)² - 5²(x²)² (y⁴)²
= (3xy³)² - (5x²y⁴)²
= (3xy³ + 5x²y⁴) (3xy³ - 5x²y⁴)
the change in the optimal objective function value per unit increase in the right-hand side of a constraint is given by the Select one: a. Reduced cost b. Bulk rate c. Objective function coefficient O d. Shadow price
Answers
The change in the optimal objective function value per unit increase in the right-hand side of a constraint is Shadow price .
What does "shadow pricing" mean?
An estimated price for something that is not typically valued or offered for sale in the market is known as a "shadow price."
Businesses may gain a better grasp of the costs and advantages of a project thanks to shadow pricing.
What in linear programming is a shadow price?
In linear programming, the maximum cost that must be paid to acquire a new unit of resource is known as the shadow price of the resource restriction.
Subtract the original objective function's value from the current objective function's value by one additional unit to determine the shadow price.
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Does anyone know how to do this? Please help!
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So that would make the formula: (3.14)R^2-(3.14)r^2 = A
The only unknown variables that you need to find are the radii of the two circles
The smaller circle:
You are given the diameter, and the diameter is 1/2 the radius,
so r= 1/2(11.25) = 5.625
The larger circle:
You are given the width, which is the total added onto the diameter of the smaller circle so:
(11.25+3.75)/2=7.5
and now that you know R, you can plug into the equation found at the start: (3.14)R^2-(3.14)r^2 = A
(3.14)(7.5^2)-(3.14)(5.625^2)=A
A=176.71-99.35
A=77.36 ft^2
517
Problem ID: PRABEEZS
x
X
Solve the system of equations:
y = 4x - 3
y = -2x+9
Write your answer in the following format:
(x,y)
with no spaces
Copied for free from openupresources.org
Type your answer below
Submit Answer
Answers
Answer:
(2, 5)
Step-by-step explanation:
Given the systems of equation
y = 4x - 3
y = -2x+9
Since both are y values, we will equate the right hand side of the equations
4x -3 = -2x+9
Collect the like terms
4x+2x = 9+3
6x = 12
x = 12/6
x = 2
Substitute x = 2 into any of the equation.
Using equation1;
y = 4x - 3
y = 4(2) - 3
y = 8 - 3
y = 5
Hence the solution to the system of equation is (2, 5)
Solve for h -110=13+3(4h-6)
Answers
Answer:
H= -35/4
Decimal form: -8.75
Explanation:
Subtract 13 from both sides. { -110 - 13 =3(4h - 6) }Simplify -110 -13 to -123 { -123 = 3 (4h - 6) }Divide both sides by 3 { -123/3 = 4h - 6 }simplify 123/3 to 41 { -41 = 4h - 6 }add 6 to both sides { -41 +6 = 4h }simplify -41 + 6 to -35 { -35 = 4h }divide both sides by 4 { - 35/4 = h }switch sides { h= - 35/4 }What is the ordered pairs is a solution of the function f(x)=3x-1
Answers
Answer:
An ordered pair that is a solution is (2,5)
Step-by-step explanation:
Pick a value for x and solve the function.
Let x = 2
f(x) = 3x - 1
f(2) = 3(2) - 1
f(2) = 6 - 1
f(2) = 5
An ordered pair that is a solution is (2,5)
Dividing fractions word problem,
I need your help, yes you!
Answers
Answer: 2
Step-by-step explanation:
Answer:
I think it's 2kg
Step-by-step explanation:
13/5 ÷ 13/10
Reciprocate
13/5 × 10/13
= 26/13
=2kg
HELP PLEASE!!!
Oak wilt is a fungal disease that infects oak trees. Scientists have discovered that a single tree in a small forest is infected with oak wilt. They determined that they can use this exponential model to predict the number of trees that will be infected after t years.
f(t)=e^0.4t
Question:
Rewrite the exponential model as a logarithmic model that calculates the # of years, g(x) for the number of infected trees to reach a value of x.
Answers
The logarithmic model is:
\(g(x) = \frac{\ln{x}}{0.4}\)
-------------
We are given an exponential function, for the amount of infected trees f(x) after x years.To find the amount years needed for the number of infected trees to reach x, we find the inverse function, applying the natural logarithm.-------------
The original function is:
\(y = f(x) = e^{0.4x}\)
To find the inverse function, first, we exchange y and x, so:
\(e^{0.4y} = x\)
Now, we have to isolate y, and we start applying the natural logarithm to both sides of the equality. So
\(\ln{e^{0.4y}} = \ln{x}\)
\(0.4y = \ln{x}\)
\(y = \frac{\ln{x}}{0.4}\)
Thus, the logarithmic model is:
\(g(x) = \frac{\ln{x}}{0.4}\)
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what is the cordinates of point A
no flile or link or i will be report you
Answers
Answer:
Um it’s (1,0) right?
Step-by-step explanation:
Cakculate the Length of line x
Answers
The length of line x in the figure of the cube given is 19.
Calculate the length of the base of the cube, which is the diagonal of the lower sides :
base length = √10² + 6²
base length = √136
The length of x is the diagonal of the cube
x = √baselength² + 15²x = √(√136)² + 15²
x = √136 + 225
x = √361
x = 19
Therefore, the length of line x in the figure given is 19.
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kelsey is skiing along a circular ski trail that has a radius of 2.9 km. she starts at the 3-o'clock position and travels in the ccw direction. kelsey stops skiing when she is 1.185 km to the right and 2.647 km above the center of the ski trail. imagine an angle with its vertex at the center of the circular ski trail that subtends kelsey's path.
a) How many radians has the angle swept out since kelsey started skiing?
b) How many km has kelsey skied since she started skiing?
Answers
a) The angle swept out since kelsey started skiing: 1.1498 radians
b) The distance Kelsey skied: 3.334 km
In this question, we have been given Kelsey is skiing along a circular ski trail that has a radius of 2.9 km. she starts at the 3-o'clock position and travels in the ccw direction. kelsey stops skiing when she is 1.185 km to the right and 2.647 km above the center of the ski trail.
Consider the following figure for reference.
We need to find the angle θ and the distance kelsey skied (i.e., the arc length)
Here, the horizontal distance x = 1.185 km and the vertical distance y = 2.647 km
a. the tangent of angle θ would be,
tan(θ) = y/x
tan(θ) = 2.647/1.185
tan(θ) = 2.2337
θ = arctan(2.2337)
θ = 65.88°
θ = 1.1498 radians
b. The length of the arc is:
s = rθ
s = 2.9 * 1.1498
s = 3.334 km
Therefore, a) the angle swept out : 1.1498 radians
b) the distance Kelsey skied: 3.334 km
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