(4) use the method of lagrange multipliers to determine the maximum value of f(x, y) = x a y b (the a and b are two fixed positive constants) subject to the constraint x + y = 1. (assume x, y > 0.)
The maximum value of [tex]\(f(x, y)\)[/tex] is given by:
[tex]\(f\left(\frac{b}{a+b}, \frac{a}{a+b}\right) = \left(\frac{b}{a+b}\right)^a \left(\frac{a}{a+b}\right)^b\)[/tex]
To find the maximum value of the function [tex]\(f(x, y) = x^a y^b\)[/tex] subject to the constraint (x + y = 1, use the method of Lagrange multipliers.
Let's set up the Lagrangian function:
[tex]\(L(x, y, \lambda) = x^a y^b + \lambda(x + y - 1)\)[/tex]
Taking the partial derivatives and setting them to zero:
Equation 1: [tex]\(\frac{\partial L}{\partial x} = a x^{a-1} y^b + \lambda = 0\)[/tex]
Equation 2: [tex]\(\frac{\partial L}{\partial y} = b x^a y^{b-1} + \lambda = 0\)[/tex]
Equation 3: [tex]\(\frac{\partial L}{\partial \lambda} = x + y - 1 = 0\)[/tex]
From equations (1) and (2) it can be written as
Equation 4: [tex]\(a x^{a-1} y^b = -\lambda\)[/tex] ... (4)
Equation 5: [tex]\(b x^a y^{b-1} = -\lambda\)[/tex] ... (5)
Dividing equation (4) by equation (5) gives
[tex]\(\frac{a}{b} \frac{x^{a-1}}{x^a} \frac{y^b}{y^{b-1}} = 1\)[/tex]
[tex]\(\frac{a}{b} \frac{y}{x} = 1\)[/tex]
From equation (3) [tex]\(y = 1 - x\)[/tex].
Substituting [tex]\(y = 1 - x\)[/tex] into the equation [tex]\(\frac{a}{b} \frac{y}{x} = 1\)[/tex]
[tex]\(\frac{a}{b} \frac{1-x}{x} = 1\)[/tex]
[tex]\(a x = b (1-x)\)[/tex]
[tex]\((a + b) x = b\)[/tex]
[tex]\(x = \frac{b}{a+b}\)[/tex]
Substituting [tex]\(x = \frac{b}{a+b}\)[/tex] into the constraint equation (3):
[tex]\(y = 1 - x[/tex]
[tex]= 1 - \frac{b}{a+b} = \frac{a}{a+b}\)[/tex]
Therefore, the critical point (x, y) that satisfies the constraint equation is:
[tex]\(x = \frac{b}{a+b}\) and \(y = \frac{a}{a+b}\)[/tex]
Therefore, the critical point gives the maximum value of the function [tex]\(f(x, y) = x^a y^b\)[/tex] subject to the constraint x + y = 1.
Learn more about Lagrangian function here:
https://brainly.com/question/33166274
#SPJ4
Mrs. Carothers is considering reserving a room at the Hotel Indigo for a trip to Fox Theater. Hotel Indigo charges $717 for a 3 night stay.
Mrs. Carothers looked at another hotel. She waited a week before she decided to book nights at that hotel, and now the prices have increased. The original price was $1195. The price for the same room and same number of nights is now $2075. What is the percent increase? Round to the nearest whole percent. Explain and show your work
How many pounds-mass per cubic foot is 40 grams per cubic centimeter?
Question 1 options:
500 lbm/ft^3
3,200 lbm/ft^3.
2,500 lbm/ft^3
1,200 lbm/ft^3
To convert grams per cubic centimeter to pounds-mass per cubic foot, you can use the conversion factors. By applying the conversion factors of 1 lb = 454 g and 1 ft³ = (30.5 cm)³, you can calculate that 40 grams per cubic centimeter is equal to 500 pounds-mass per cubic foot.
Explanation:To convert grams per cubic centimeter to pounds-mass per cubic foot, we need to use conversion factors. Firstly, we need to convert grams to pounds by using the conversion factor 1 lb = 454 g. Then, we convert cubic centimeters to cubic feet by using the conversion factor 1 ft³ = (30.5 cm)³. Finally, we combine the two conversion factors to find the desired unit: pounds-mass per cubic foot.
Given:
40 grams per cubic centimeterConversion factor:
1 lb = 454 g1 ft³ = (30.5 cm)³Calculation:
40 g/cm³ * (1 lb/454 g) * [(30.5 cm)³/1 ft³] = 40 * 1/(454 * (30.5)³) lb/ft³Simplifying the calculation gives:
40/(454 * (30.5)³) lb/ft³ = 500 lbm/ft³Therefore, 40 grams per cubic centimeter is equal to 500 pounds-mass per cubic foot.
Write the steps for how to use a number line to multiply a 2 digit number by 20
Which number line represents the solution set for the inequality –4(x + 3) ≤ –2 – 2x?
The volume of 10000 drops of liquid is 10 fluid ounces what iscthe volume of 10 drops
The volume of 10 drops of liquid is approximately 0.0001 fluid ounces.
To find the volume of 10 drops of liquid, we can use the given information that the volume of 10000 drops is 10 fluid ounces. Let's set up a proportion to solve for the volume of 10 drops.
Let V be the volume of 10 drops in fluid ounces.
We know that the volume of 10000 drops is 10 fluid ounces, so we can write:
10,000 drops / V = 10 fluid ounces / 10 drops
Now, cross-multiply and solve for V:
10,000 drops * 10 drops = 10 fluid ounces * V
100,000 drops² = 10 fluid ounces * V
Now, divide both sides by 100,000 drops^2 to solve for V:
V = 10 fluid ounces / 100,000 drops^2
V ≈ 0.0001 fluid ounces
Therefore, the volume of 10 drops of liquid is approximately 0.0001 fluid ounces.
Learn about volume here:
brainly.com/question/22886594
#SPJ2
120t-0.4t^4+1000=0
How can i solve this?
The school janitor is going to purchase a vacuum cleaner to use in the classrooms. The regular price of a vacuum cleaner is $70. The janitor can buy a used one for $28. How much money will he save if he buys the used vacuum cleaner?
Graph the line with slope -2 passing through the point (3,-4)
On a map, 12 inch represents 45 mile.
What distance on the map represents 1 mile?
A 2/5 in.
B 5/8 in.
C 1 3/5 in.
D 2 1/2 in.
5/8 in.
i just took a test.
A combination lock uses three distinctive numbers between 0 and 49 inclusive. how many different ways can a sequence of three numbers be selected? (show work)
write the rule that applies to the following number:0÷7=0
Raphael graphed the functions g(x) = x + 2 and f(x) = x – 1. How many units below the y-intercept of g(x) is the y-intercept of f(x)? The y-intercept of f(x) is units below the y-intercept of g(x).Raphael graphed the functions g(x) = x + 2 and f(x) = x – 1. How many units below the y-intercept of g(x) is the y-intercept of f(x)? The y-intercept of f(x) is units below the y-intercept of g(
Answer:
The y-intercept of f(x) is 3 units below the y-intercept of g(x).
Step-by-step explanation:
We are given,
The functions [tex]f(x)=x-1[/tex] and [tex]g(x)=x+2[/tex].
Now, y-intercept is the point where the graph of the function crosses y-axis.
So, y-intercept is obtained when x= 0.Substituting x= 0 in the functions, we get,
[tex]f(0)=0-1\\f(0)=-1[/tex] and [tex]g(0)=0+2\\g(0)=2[/tex].
Thus, we get that,
The y-intercept of the the function f(x) is -1.
The y-intercept of the function g(x) is at 2.
Thus, the y-intercept of f(x) is 2-(-1)= 3 units below the y-intercept of g(x) as seen below.
Which reflection rule, if any, can be used to prove that rectangle A(-8, -3), B(-2, -3), C(-2, -6), D(-8, -6) and rectangle A'(8, -3), B'(2, -3), C'(2, -6), D'(8, -6) are congruent?
A) (x, y) → (-x, y)
B) (x, y) → (x, -y)
C) (x, y) → (-x, -y)
D) The rectangles are not congruent.
Bennett tried to solve exercise 1 a few different ways. Which of his methods are correct? Of the other methods, which makes the most sense to you? Explain
A. 5% sales tax means that for every dollar you spend, you need to pay a nickel in tax. If you buy a nickel tax. If you buy something for $21, you need to pay 21 nickels in tax.
B. You can set up a promotion and solve for the missing value.
$.05 = x <-- (These are fractions)
___ __
$1.00 $21.00
C. If you know that 10% of $21 is $2.10, so 5% should be half of $2.10
D. 5% is equal to 1/20. To find the amount of tax on $21, find $21 ÷ 20
E. 1% of $21.00 is $0.21 so 5% of $21.00 is 5 x $ 0.2
STVU is an isosceles trapezoid. If SV = 6x - 5 and TU = 2x + 7, find the value of x.
x=3
What is the product
3/8 * (-3/6)
Enter your answer as a fraction, in simplified form, please
Solve 5x + 7 > 17. {x | x < 2} {x | x > 2} {x | x < -2} {x | x > -2}
Answer:
The solution set for the provided inequity is {x | x > 2}.
Step-by-step explanation:
Consider the provided inequity.
[tex]5x+7>17[/tex]
Subtract both the side by 7.
[tex]5x+7-7>17-7[/tex]
[tex]5x>10[/tex]
Now divide both the side by 5.
[tex]\frac{5x}{5}>\frac{10}{5}[/tex]
[tex]x>2[/tex]
Hence, the solution set for the provided inequity is {x | x > 2}.
Combine like terms: –10.9p + 3.9 = –9.18 Apply the next steps to solve the equation. What is the solution? p =
Answer:
P=1.2
Step-by-step explanation:
Please mark as brainlest
Rate and give me thanks
If you know that a < b, and both a and b are positive numbers, then what must be true about the relationship between the opposites of these numbers? Explain.
Answer: just copy and paste
We want to find the relationship between the opposites of a and b, given that both are positive and a < b.
That relation is:
-a > -b.
Remeber that the opposite of a real number is just -1 times that number.
The opposite of a is: -a
The opposite of b is: -b
Remember that for negative numbers, the closer the number is to zero, the larger is the number.
Because we know that a and b are positive, and a < b, then we know that a is closer to zero.
So for -a and -b, we know that -a is also closer to zero (then -a is larger than -b), so the relationship between these two is:
-a > -b.
When a < b and both a and b are positive numbers, the opposites of these numbers will have opposite signs.
Explanation:In mathematical terms, when two positive numbers add, the answer has a positive sign. Similarly, when two negative numbers add, the answer has a negative sign. However, when a negative number is added to a positive number, the answer will depend on the magnitudes of the numbers:
If the positive number is greater than the negative number, the answer will have a positive sign.If the positive number is smaller than the negative number, the answer will have a negative sign.So, in the given scenario where a < b, if both a and b are positive numbers, the opposites of these numbers will have opposite signs.
Learn more about positive and negative numbers here:https://brainly.com/question/30287693
#SPJ3
Henry invest 4500 at a compound interest rate of 5% per annum. At the end of n complete years the investment has grown to 5469.78. Find the value of n
The value of n is 4. This means that it took 4 years for the investment to grow to £5469.78.
How to calculate the value of nWe can use the compound interest formula to solve this problem:
A = P(1 + r/n)nt
where:
A is the future value of the investment
P is the principal amount invested
r is the annual interest rate
n is the number of times per year the interest is compounded
t is the number of years
In this problem, we have the following information:
A = 5469.78
P = 4500
r = 0.05 (5%)
n = ?
t = n
We can plug these values into the formula and solve for n:
5469.78 = 4500(1 + 0.05/n)n
1.2155066 = (1.05)n
n = 4
Therefore, the value of n is 4. This means that it took 4 years for the investment to grow to £5469.78.
Learn more about interest on
https://brainly.com/question/29451175
#SPJ2
How fast (in rpm) must a centrifuge rotate if a particle 6.00 cm from the axis of rotation is to experience an acceleration of 125,000 g's?
The speed of the particle in rpm is 43170.32 rpm.
Given:
The distance of particle (radius) from the axis is [tex]r=6\rm \: cm=0.06\; m[/tex].
The acceleration (centripetal) experienced by the particle is,
[tex]a=125000\rm \; g's\\a=125000\times 9.81 \rm \; m/s^2\\a=1226250\rm \; m/s^2[/tex]
Let ω be the angular velocity of the particle and N be the speed in rpm.
So, ω will be [tex]\omega=\dfrac{2\pi N}{60}[/tex]
Now, the acceleration of the particle can be written as,
[tex]a=\omega^2r\\a=\left(\dfrac{2\pi N}{60}\right)^2\times 0.06[/tex]
So, the speed of the particle in rpm will be,
[tex]a=\left(\dfrac{2\pi N}{60}\right)^2\times 0.06\\12262550=0.01095N^2\times 0.06\\N=43170.32\rm \; rpm[/tex]
Therefore, the speed of the particle in rpm is 43170.32 rpm.
For more details, refer the link:
https://brainly.com/question/8825608?referrer=searchResults
Which function can be used to find the nth term of the arithmetic sequence?
Find the equation of the tangent plane to (a) z = 4x 2 − y 2 + 2y at (−1, 2, 4)
Δ ABC is isosceles. Angles B and C are congruent. If m∠B = (5x + 5)° and m∠C = (7x - 25)°, Find m∠A. A) 10° B) 15° C) 20° D) 80°
Answer:
option C
[tex]20\°[/tex]
Step-by-step explanation:
we know that
An isosceles triangle has two equal angles and two equal sides. The two equal angles are the base angles and the third angle is the vertex angle
In this problem the base angles are angle B and angle C
so
m∠B=m∠C
substitute
[tex](5x+5)\°=(7x-25)\°[/tex]
[tex](7x-5x)\°=(5+25)\°[/tex]
[tex]2x=30\°[/tex]
[tex]x=15\°[/tex]
m∠B=[tex](5x+5)\°=5*15+5=80\°[/tex]
Remember that the sum of the internal angles of the triangle is equal to [tex]180\°[/tex]
Find m∠A
m∠A=[tex]180\°-2*80\°=20\°[/tex]
The reported unemployment is 5.5% of the population. what measurement scale is used to measure unemployment? nominal ordinal interval or ratio descriptive
The answer is interval or ratio. In interval level of measurement, the distances between have attributes does have a meaning. The best example to be used here is temperature. The distance from 40-50 is like the distance from 70-80. While in ratio level of measurement, there is constantly an absolute zero that is significant. Meaning, that you can build a meaningful fraction (or ratio) with a ratio variable.
In a chemical compound there are three parts zinc for every 16 parts copper by mass. A piece of the compound contains 320 g of copper. Write and solve an equation to determine the amount of zinc in the chemical compound.
Write and solve an equation for the situation. Eight and one-half years ago, Steven was 7 years old. How old is he now? Make sure you explain and know your answer. I am confused and frustrated, also, I have been working of this for a while and I do not fully understand. Thanks, :). Answer Fast. This means a lot.
Eight and one-half years ago the age of Steven was 7 years, then the present age of Steven is 15.5 years.
What is an equation?Mathematical equations with two algebraic symbols on either side of the equal (=) sign are called equation. This connection is illustrated by the left and right expression being equal to one another. The left-hand side equals the right-hand side is a basic, straightforward equation.
Assume the current age of Steven is x years.
From the given information in the question,
8.5 years ago from now,
x - 8.5 = 7 years
x = 15.5 years.
So, the current age is 15.5 years.
To know more about Equation:
https://brainly.com/question/10413253
#SPJ3
Which fraction is equal to 0.20%? 1/20 1/40 1/50 1/400 1/500
Writing a check on an account with insufficient funds is allowed under certain conditions.
True
False
Answer:
This is FALSE.
Step-by-step explanation:
Writing a check on an account with insufficient funds is never allowed by any bank. This will only lead to check bounce and sometimes can land you up in legal trouble. When a check bounces, you will have to pay your bank the bounce fee also. Hence, we should be careful while writing any check.