Find the extremes of: 4x-4y subjected to the following condition x² + 2y² = 1
Answers
The critical points are neither maxima nor minima, but rather saddle points. The function is unbounded above and below on the ellipse x² + 2y² = 1.
How to find the extremes of the function?To find the extremes of the function 4x - 4y subject to the constraint x² + 2y² = 1, we can use the method of Lagrange multipliers.
Let f(x,y) = 4x - 4y, and g(x,y) = x² + 2y² - 1 be the constraint function. Then, we want to find the values of x and y that maximize or minimize f(x,y) subject to g(x,y) = 0.
Using the method of Lagrange multipliers, we set up the following system of equations:
∇f(x,y) = λ∇g(x,y)
g(x,y) = 0
Taking partial derivatives of f and g with respect to x and y, we have:
∂f/∂x = 4
∂f/∂y = -4
∂g/∂x = 2x
∂g/∂y = 4y
Setting these equal to λ times their corresponding partial derivatives, we get:
4 = λ(2x)
-4 = λ(4y)
Dividing these equations gives:
x/y = -1/2
Substituting this into the constraint equation, we get:
(-1/2)² + 2y² = 1
Solving for y, we get:
y = ±√(3/8)
Then, using x/y = -1/2, we can find x:
x = (-1/2)y
So the critical points are:
(√(3)/4, -√(3)/2) and (-√(3)/4, √(3)/2)
To determine whether these are maxima or minima, we need to use the second derivative test or the bordered Hessian method. Since the bordered Hessian is somewhat complicated in this case, we will use the second derivative test.
The second partial derivatives of f are:
∂²f/∂x² = 0
∂²f/∂y² = 0
∂²f/∂x∂y = 0
At the point (√(3)/4, -√(3)/2):
D = ∂²f/∂x² * ∂²f/∂y² - (∂²f/∂x∂y)² = (0)(0) - (0)² = 0
Since D = 0, the second derivative test is inconclusive.
At the point (-√(3)/4, √(3)/2):
D = ∂²f/∂x² * ∂²f/∂y² - (∂²f/∂x∂y)² = (0)(0) - (0)² = 0
Again, the second derivative test is inconclusive.
Therefore, the critical points are neither maxima nor minima, but rather saddle points. The function f(x,y) = 4x - 4y is unbounded above and below on the ellipse x² + 2y² = 1.
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Related Questions
for normally distributed scores, what percentage of scores (or cases) would fall: between the mean and z = -1
Answers
Approximately 84% of scores fall between the mean and z = -1 for normally distributed scores.
In a normal distribution, approximately 68% of the scores fall within one standard deviation of the mean, both above and below. Z-scores are used to describe the location of a score relative to the mean, in terms of standard deviations.
A z-score of -1 indicates that a score is 1 standard deviation below the mean. So, if we consider all the scores that are 1 standard deviation below the mean or closer to the mean, then we would have a total of 68% of the scores, which includes the mean.
And, if we consider only the scores that are between the mean and 1 standard deviation below the mean, then we would have 84% of the scores. Hence, approximately 84% of scores would fall between the mean and z = -1 for normally distributed scores.
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Write an equation in slope-intercept form for the line that is parallel to y=x−6 and that passes through the point (−5,−6).
Answers
Answer:
Equation: y = x - 1
a slope of 1, and a y-intercept of -1Step-by-step explanation:
Given: y = x - 6
Slope-Intercept Form: y = mx + b -
where:
m is the slopeb is the y-intercept (when x = 0)Note: parallel lines have the same slope
1. Determine the slope and y-intercept of the given equation:
⇒ y = x - 6 has a slope of 1, and a y-intercept of -6
2. Find the equation of the line parallel line that passes through the point (-5,-6):
⇒ y = x + b [substitute -5 for x and -6 for y]
⇒ -6 = -5 + b [add 5 to both sides]
⇒ -6 + 5 = -5 + 5 + b
⇒ -1 = b
Equation: y = x - 1
3. Check your answer using point (-5, -6):
⇒ y = x - 1
⇒ -6 = -5 - 1
⇒ -6 = -6 ✔
This statement is correct
Therefore, the equation of the line is y = x - 1
The length of a rectangle is 2cm, and the width is 1 cm. Find the length of a diagonal of this rectangle.
A. √3cm
B. √5cm
C. 3 cm
D. 25 cm
Answers
Answer:
Ans 2
Step by step explanation :
2 X 1 = 2
Answer:
this is merely the Pythagorean Theorem. so we would do (2^2+1^2)^.5 = 2.236 = square root of 5 = answer B!!!
HELPPPP PLZZZZZZZZZ
WILL MARK BRAINLIEST
Answers
Answer: it is D
Step-by-step explanation:
HELP ASAP PLEASE HELP QUICK!!!!
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Mrs. Berry's class organized the lunch orders for their upcoming party in the table below.
Lunch Orders for the Class Party
Lunch Choice
Number
Vegetarian Sandwich
3
Turkey Sandwich
7
Pepperoni Pizza
8
Cheese Pizza
10
Which ratio represents the number of orders for pepperoni pizza to cheese pizza?
O 4.5
05:4
10
8
18
10
Answers
Answer:
4:5 that's last answer i got
I really extremely need help! I will give you 5 stars and branilest
The roast beef was 48 pounds. After the grinch finished carving there was only 7.5% left. How many pounds of roast beast were left after the feast?
Answers
Answer:
3.6 pounds
Step-by-step explanation:
Write equivalent fractions for 1 3/5 and 1 1/3 using a common denominator
Answers
Answer: 8/15 and 1 1/3
Step-by-step explanation:
The equivalent fractions for 1 3/5 and 1 1/3 using a common denominator are 8/15 and 10/15. This is because 1 3/5 can be written as 8/15 and 1 1/3 can be written as 10/15, and the denominator (bottom number) of both fractions are the same.
Exercise 8-3 Algo A simple random sample of 35 observations is derived from a normally distributed population with a known standard deviation of 6.3. [You may find it useful to reference the z table.]
a. Is the condition that X− is normally distributed satisfied? Yes No
b. Compute the margin of error with 95% confidence. (Round intermediate calculations to at least 4 decimal places. Round "z" value to 3 decimal places and final answer to 2 decimal places.)
c. Compute the margin of error with 90% confidence. (Round intermediate calculations to at least 4 decimal places. Round "z" value to 3 decimal places and final answer to 2 decimal places.)
d. Which of the two margins of error will lead to a wider interval? The margin of error with 90% confidence. The margin of error with 95% confidence.
Answers
a. Yes, the condition that X- is normally distributed is satisfied.
b. The margin of error with 95% confidence is 2.68.
c. The margin of error with 90% confidence is 2.16.
d. The margin of error with 95% confidence will lead to a wider interval than the margin of error with 90% confidence.
a. Yes, the condition that X- is normally distributed is satisfied because the sample size n = 35 is sufficiently large, and the population is normally distributed.
b. For a 95% confidence level, the z-value is 1.96 (from the z-table). The margin of error (ME) can be calculated as
ME = z-value * (standard deviation / √(n))
ME = 1.96 * (6.3 / √(35))
ME ≈ 2.68
Therefore, the margin of error with 95% confidence is 2.68.
c. For a 90% confidence level, the z-value is 1.645 (from the z-table). The margin of error (ME) can be calculated as
ME = z-value * (standard deviation / √(n))
ME = 1.645 * (6.3 / √(35))
ME ≈ 2.16
Therefore, the margin of error with 90% confidence is 2.16.
d. The margin of error with 95% confidence (2.68) will lead to a wider interval than the margin of error with 90% confidence (2.16). This is because a higher confidence level requires a larger margin of error to ensure that the interval contains the true population parameter with a higher probability.
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Brainliest!!!!! Which rule explains why these triangles are congruent
Answers
Answer:
cannot be proven
Step-by-step explanation:
no given length on any side of either triangle to be able to reference
What is the square root rule?
Answers
The square root function involves the square root symbol √ which is mostly read as "square root of", the rule says that the square root of a number 'x' is a number 'y' such that y² = x.
We know that the square root of a number can be either positive or negative, but while defining the square root function, we restrict its range to be the set of all positive real numbers and hence making all square roots possible to be positive.
If the range for the square root of a number is set to be positive as well as negative it wont be a function that is why there is a restriction to the range of this set allowing only positive real numbers.
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francisco and meredith are 230 feet apart when they start walking toward one another. they are walking at the same speed, so whenever francisco travels some number of feet, meredith travels the same number of feet. let x x represent the number of feet francisco has traveled since he started walking toward meredith. write an expression in terms of x x that represents the number of feet francisco has walked toward meredith since they started walking. preview write an expression in terms of x x that represents the number of feet meredith has walked toward francisco since they started walking. preview write an expression in terms of x x that represents the total number of feet francisco and meredith have walked toward one another since they started walking. preview write an expression in terms of x x that represents the distance (in feet) between francisco and meredith. preview
Answers
The expressions are:
- Francisco's distance: x
- Meredith's distance: x
- Total distance: 2x
- Distance between Francisco and Meredith: 230 - 2x.
To answer your question, let's break it down into the different expressions:
1. The expression that represents the number of feet Francisco has walked toward Meredith since they started walking can be written as x.
2. The expression that represents the number of feet Meredith has walked toward Francisco since they started walking can also be written as x.
3. The expression that represents the total number of feet Francisco and Meredith have walked toward one another since they started walking can be written as x + x, which simplifies to 2x.
4. The expression that represents the distance between Francisco and Meredith can be written as 230 - (x + x), which simplifies to 230 - 2x.
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What is 10.752 divided by 105? (Enter an exact number.)
Answers
Answer:
0.1024
Step-by-step explanation:
because if you go to any calculator and type in these numbers you would get an answer of 0.1024
Can someone help me out on 2??
Answers
Answer:
x = 7
Step-by-step explanation:
9x+11 = 74 (As they are corresponding angles)
9x = 74-11
9x = 63
x = 63/9
x = 7
The formula for the volume of a cone is given below. Find the rate of change of the volume for each of the radii given below if dr/dt is 5 inches per minute and h= 15r. V=(1/3)πr 2
h (a) r=2 in V=∣π in 3
/min (b) r=16 in V=π in 3
/min
Answers
The rate of change of the volume for the given radii is 1500π cubic inches/min for r = 2 in and 48,000π cubic inches/min for r = 16 in.
Given that the formula for the volume of a cone is V = (1/3)πr²h where h = 15r.
We have to find the rate of change of the volume for each of the radii r = 2 in, r = 16 in, given that dr/dt is 5 inches per minute.
Let's first find the value of h for r = 2 inh = 15r = 15(2) = 30 inches
Now, substitute r = 2 in and h = 30 in in the formula for the volume of the cone.
V = (1/3)π(2)²(30)V = (1/3)π(4)(30)
V = 40π cubic inches
Given that dr/dt = 5 inches/min
Now, differentiate the formula for the volume of the cone V with respect to time t. We get,
dV/dt = (1/3)(2πrh)(dr/dt)
Also, from h = 15r, we get r = h/15
Substitute the values of r, h and dr/dt in the above equation, we get
dV/dt = (1/3)(2πh(h/15))(5) = (π/3)h²
Therefore, for r = 2 in, h = 30 in, we get
dV/dt = (π/3)(30)²(5) = 1500π cubic inches/min
Let's now find the value of h for r = 16 in
h = 15r = 15(16) = 240 inches
Now, substitute r = 16 in and h = 240 in in the formula for the volume of the cone.
V = (1/3)π(16)²(240)
V = (1/3)π(256)(240)
V = 2560π cubic inches
Given that dr/dt = 5 inches/min
Now, differentiate the formula for the volume of the cone V with respect to time t. We get,
dV/dt = (1/3)(2πrh)(dr/dt)
Also, from h = 15r, we get r = h/15
Substitute the values of r, h and dr/dt in the above equation, we get dV/dt = (1/3)(2πh(h/15))(5) = (π/3)h²
Therefore, for r = 16 in, h = 240 in, we get dV/dt = (π/3)(240)²(5) = 48,000π cubic inches/min
Therefore, the rate of change of the volume for the given radii is 1500π cubic inches/min for r = 2 in and 48,000π cubic inches/min for r = 16 in.
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Write the equation of the line that passes through (5,-2) and is perpendicular to y = 5/3x-3.
y= -3/5 x +1
y= -3/5x + 19/5
y= 3/5 x -5
y= 3/5 x - 31/3
Answers
Step-by-step explanation:
the slope (a) is always the factor of x in an equation
y = ax + b
it is the ratio (y coordinate difference / x coordinate difference) when going from one point on the line to another.
a perpendicular slope now turns this ratio fraction upside-down and flips the sign.
so, the perpendicular slope of 5/3 is -3/5.
therefore, the last 2 answer options are out.
we have the equation
y = -3/5 x + b
now we use the coordinates of the given point to find b.
-2 = -3/5 × 5 + b = -3 + b
1 = b
so, the correct equation is
y = -3/5 + 1
Someone help me please
Answers
Answer:
(x + 10)(2x + 5)
Step-by-step explanation:
Given
2x² + 25x + 50
Consider the factors of the product of the coefficient of the x² term and the constant term (ac) which sum to give the coefficient of the x- term (b)
ac = 2 × 50 = 100 and b = 25
The factors are + 20 and + 5
Use these factors to split the x- term
2x² + 20x + 5x + 50 ( factor the first/second and third/fourth terms )
= 2x(x + 10) + 5(x + 10) ← factor out (x + 10) from each term
= (x + 10)(2x + 5)
the average 1 bedroom apartment in freeburg is $295 a month with a standard deviation of $25 60% or more of all 1 bedroom apartments would tent for what amount
Answers
To determine the amount that 60% or more of all 1 bedroom apartments in Freeburg would rent for, we can use the concept of standard deviations.
Given that the average rent for a 1 bedroom apartment in Freeburg is $295 and the standard deviation is $25, we can determine the rent threshold for 60% or more of all apartments.
To find the threshold, we need to calculate the Z-score corresponding to the desired percentile. The Z-score measures how many standard deviations a data point is from the mean. The Z-score formula is given by (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
To find the Z-score for the 60th percentile, we look up the corresponding value in the Z-table. The Z-table provides the area under the normal distribution curve up to a certain Z-score. In this case, we want to find the Z-score that corresponds to an area of 0.60.
Assuming a normal distribution, a Z-score of approximately 0.253 corresponds to the 60th percentile.
Using the formula (Z-score * standard deviation) + mean, we can calculate the rent threshold: (0.253 * $25) + $295 = $301.325.
Therefore, 60% or more of all 1 bedroom apartments in Freeburg would rent for approximately $301.325 or higher.
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A car can travel 25 miles per gallon of gas. The gas tank contains
9 gallons. How far can the car travel without refueling?
Answers
Answer:
225 miles
Step-by-step explanation:
The car can travel 225 miles without refuelling.
What is unitary method ?The unitary method is a mathematical approach we first find the value for 1 unit then solve the problem by the given data from the question.
10 balls = 200Rs.
1 Ball = 200/10 = 20Rs.
∴x no. of balls = (20 × x)/10 Rs.
= 20x/10 Rs.
From the given data
In 1 gallon of gas the car can travel 25 miles
∴ In 9 gallons of gas the car can travel (9×25)/1 miles
= 225 miles
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Find the present value of a sequence of annual payments of Rs 25000 each , the first being made at the end of 5th year and the last being paid at the end of 12th year, if money is worth 6%.
Answers
The present value of the sequence of annual payments of Rs 25000 each, the first being made at the end of the 5th year and the last being paid at the end of the 12th year, if money is worth 6% is Rs. 158620.39.
1: We can use the formula to calculate the present value of the annuity:
P = A x [1 - (1+i)^-n] / i
Where
P = Present Value
A = Annuity
i = Interest Raten = Number of payments
2: Calculate the present value of each payment using the formula:
P1 = 25000 / (1.06)⁵
P2 = 25000 / (1.06)⁶
P3 = 25000 / (1.06)⁷
P4 = 25000 / (1.06)⁸
P5 = 25000 / (1.06)⁹
P6 = 25000 / (1.06)¹⁰
P7 = 25000 / (1.06)¹¹
P8 = 25000 / (1.06)¹²
3: Substitute the values into the formula to find the present value:
P = P1 + P2 + P3 + P4 + P5 + P6 + P7 + P8
P = (25000 / (1.06)⁵) + (25000 / (1.06)⁶) + (25000 / (1.06)⁷) + (25000 / (1.06)⁸) + (25000 / (1.06)⁹) + (25000 / (1.06)¹⁰) + (25000 / (1.06)¹¹) + (25000 / (1.06)¹²)
P = 158620.39
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How do you know if its permutation or combination?
Answers
A permutation is an arrangement of objects in a specific order, whereas a combination is a selection of objects without regard to their order.
To determine if a problem is a permutation or combination, you need to consider whether order matters.
The formula for permutations is nPr = n! / (n - r)!, where n is the total number of items and r is the number of items being chosen at a time. For example, if you had six books and wanted to know how many permutations of three books you could choose, the formula would be 6P3 = 6! / (6 - 3)! = 6 * 5 * 4 / 3 * 2 * 1 = 120.
The formula for combinations is nCr = n! / (r! * (n - r)!), where n is the total number of items and r is the number of items being chosen at a time. For example, if you had six books and wanted to know how many combinations of three books you could choose, the formula would be 6C3 = 6! / (3! * (6 - 3)!) = 6 * 5 * 4 / (3 * 2 * 1 * 3 * 2 * 1) = 20
Therefore, if the order of the objects matters, it is a permutation. If the order does not matter, it is a combination.
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The graphs of y=-4x and y=-4x+2 are shown. Use the drop down menus to complete the statements. The ratios of y/z are the same for the graph of
Answers
Answer:
y = -4x
Step-by-step explanation:
When equations are in the form
y = mx the slope m (is y/x)
help me ill give brainlist i swear i will
Answers
Answer:
36.3
Step-by-step explanation:
If p=180−0.2D, calculate the optimal profit if the total cost is $30,500.
Answers
To calculate the optimal profit, we need to find the value of D when the total cost is \($30,500\) and then substitute it into the given equation.
To find D, we can rearrange the equation to solve for D: Given:\(p = 180 - 0.2D\) and total cost =\($30,500\) Now, substitute the total cost ($30,500) into the equation to find D:
Since D cannot be negative in this context, it means that the given equation is not applicable for a total cost of \($30,500\). it is not possible to calculate the optimal profit using the given equation and the total cost of \($30,500\).
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Martin is using unit cubes to build a tower in the shape of a right rectangular prism. A description of the tower is listed below. Bottom layer is made of 16 unit cubes bottom layer is in the shape of a square prism 9 more equal layers of unit cubes are added on top of the bottom layer What is the total volume, in cubic units, of the completed tower?
Answers
Answer:
160 sq units
Step-by-step explanation: multiple 16 times 10 and you get that
Convert 2676 cm into inches (1 in = 2.54 cm)
Answers
Answer:
1053.543 in
Step-by-step explanation:
I divided the length value by 2.54
Answer:
1053.5 in
Step-by-step explanation:
1 in
2676 cm x ------------ = 1053.5 in
2.54 cm
Amy and Vince want to save $18,000 so that they can take a trip to Europe in five years. How much must they save at the beginning of each month to have the money they need if they can get 5.5% interest in their savings account?
Group of answer choices
$435.39
$260.13
$323.45
$224.22
Answers
Amy and Vince must save approximately $323.45 at the beginning of each month to have the $18,000 they need for their trip to Europe in five years if they can get 5.5% interest in their savings account.
We know that Amy and Vince want to save $18,000 in five years and can earn 5.5% interest in their savings account, we need to adjust the interest rate to a monthly rate.
First, let's calculate the monthly interest rate (r) from the annual interest rate:
r = (1 + annual interest rate)^(1/12) - 1
= (1 + 0.055)^(1/12) - 1
= 0.0045107
Next, let's calculate the number of periods (n) in months:
n = 5 * 12
= 60 months
Using the future value of an ordinary annuity formula, we can solve for the monthly savings (PMT):
PMT = FV / [((1 + r)^n - 1) / r]
where:
FV = Future Value (desired savings amount)
r = Monthly interest rate
n = Number of periods
FV = $18,000
PMT = 18,000 / [((1 + 0.0045107)^60 - 1) / 0.0045107]
≈ $323.45
Therefore, Amy and Vince must save approximately $323.45 at the beginning of each month to have the $18,000 they need for their trip to Europe in five years.
Therefore, the correct option is $323.45.
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the graph of a linear function is given below
Answers
Answer:
A
Step-by-step explanation:
The zero of the function is where the y-value is zero.
then it's -5
The dance committee has $75 to spend on centerpieces. They spent $30 on flowers.
If each vase costs $3, write and solve an inequality to find how many vases they
can buy. (Example 3)
Plzzz help I only have till 11 to finish these
Answers
Answer:
135
Step-by-step explanation:
75 x - 30 = 45
Inequality
45 x 3 =
135 (Inequality)
suppose ax=b has a solution. explain why the solution is unique precisely when ax=0 has only the trivial solution.
Answers
Translation of the Ax=0 solution set yields the Ax-b solution set since Ax-b is consistent. The Ax = b solution set is therefore a single vector if and only if the Ax = 0 solution set is a single vector, which occurs if and only if Ax 0 has only the trivial solution.
what is solution ?Finding the solution to an equation is like finding the solution to a puzzle. A mathematical equation proves the equality of two algebraic expressions. Determine the values of the variables that make the equation a true statement in order to solve an equation. An equation's solution is any value that causes the equation to be true.
given
Let's suppose that there is a solution to the equation ax = b.
Now, we want to demonstrate that the trivial solution exists for the equation ax = b when ax = 0.
Ax = 0 is homogeneous.
In the event that this equation was true for b, we define
To be a set of vectors with the form ax = b
w = m + gh
The subscript "h"
A solution to ax = 0 is gh.
Ax=b is in the form of w= m+gh based on the information we know had.
with
m = ax=solution b's
Soulution of gh = ax=0
Only a simple solution, ax = 0, exists.
gh = 0
where gh = 0
The relationship between ax=b and w=m
Ax = b thus has no alternative solutions.
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BC=6, EF=12
Based on the given information, choose the similarity statement that you would use to say ABC~DEF. If you could NOT conclude the triangles similar, then choose NOT
A: AA
B: SAS
C: SSS
D: NOT
Answers
Answer:
C: SSS
Step-by-step explanation:
Two triangle are said to be similar if they have the same shape and proportion. This means that their angles are the same or their sides are in proportion. They are similar if:
i) All three corresponding angles are the same (AAA)
ii) All three corresponding sides are the same proportion (SSS)
iii) Two corresponding angles are equal and a side is in the same proportion (ASA)
iv) Two sides are in the same proportion and a corresponding angles are the same (SAS)
Given that:
AB = 12, BC = 6, AC = 3, DE = 24, EF = 12 and DF = 6
\(\frac{DE}{AB} =\frac{DF}{AC} =\frac{EF}{BC}\\\\\frac{24}{12}=\frac{6}{3}=\frac{12}{6}=2\)
Since All the sides are in the same proportion, it supports the SSS rule.
Answer: SSS
I have done this assignment before