This extreme value problem has a solution with both a maximum value and a minimum value. use lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = 8x + 8y + 4z; 4x2 + 4y2 + 4z2 = 36
The extreme max value is "[tex]f(x,y,z) = 36[/tex]" and min value is "[tex]f(x,y,z) = -36[/tex]".
According to the question,
[tex]f(x, y, z) = 8x + 8y + 4z[/tex]let,
[tex]g(x,y,z) = 4x^2 + 4y^2 + 4z^2 - 36[/tex]By using Lagrange multipliers,
→ [tex]\bigtriangledown f= \lambda \bigtriangledown g[/tex]...(equation 1)
[tex]\bigtriangledown f = <8,8,4>[/tex][tex]\bigtriangledown g = <8x, 8y,8z>[/tex]∴ [tex]<8,8,4>=<8\lambda x, 8 \lambda y, 8 \lambda z >[/tex]
then,
[tex]8 \lambda x = 8[/tex]...(equation 2)[tex]8 \lambda y =8[/tex]...(equation 3)[tex]8 \lambda z = 4[/tex]...(equation 4)→ [tex](x = \frac{1}{\lambda}, y=\frac{1}{\lambda}, z =\frac{1}{2 \lambda} )[/tex]
As we know,
→ [tex]4x^2 +4y^2+4z^2=36[/tex]
By substituting the values, we get
→ [tex]4(\frac{1}{\lambda} )^2+ 4(\frac{1}{\lambda} )^2+4(\frac{1}{2\lambda} )^2 =36[/tex]
→ [tex]\frac{4}{\lambda^2} +\frac{4}{\lambda^2} +\frac{1}{\lambda^2} =36[/tex]
→ [tex]\frac{9}{\lambda^2}=36[/tex]
[tex]\frac{1}{\lambda}= \pm \frac{6}{3}[/tex]
[tex]\frac{1}{\lambda} = \pm 2[/tex]
∴ x = 2, y = 2, z = 1
then,
→ [tex]f(x,y,z) = 36[/tex] (Extreme max value)
and,
∴ x = -2, y = -2, z = -1
then,
→ [tex]f(x,y,z) = -36[/tex] (min value)
Thus the above answer is correct.
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Chris plans to mulch the Border surrounding a rectangular Garden in his backyard if the length of L of the garden (exculding its border) is 3 more than twice its width w and the width of the border is 1/4w which function repersents tje area A to be mulched in terms of w?
19,050,000 in scientific notation
Using sum or difference formulas, find the exact value of sin(165∘)
Express your answer in the form
sin(165∘)=√a(√b−1)/4 for some numbers a and b
We have a = 2 and b = 3.
To find the exact value of sin(165°) using sum or difference formulas, we notice that 165° can be expressed as the sum of two angles whose sine and cosine values we know exactly, for example, 120° and 45°. The sum formula for sine is sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
Therefore, sin(165°) = sin(120° + 45°) = sin(120°)cos(45°) + cos(120°)sin(45°).
From here, we use the known values:
sin(120°) = sin(180° - 60°) = sin(60°) = √3/2
cos(45°) = √2/2
cos(120°) = cos(180° - 60°) = -cos(60°) = -1/2
sin(45°) = √2/2
Plugging these values into our formula gives:
sin(165°) = ( √3/2)(√2/2) + (-1/2)(√2/2) = (√6 - √2)/4
This fraction can be rewritten to match the given form by factoring out √2 in the numerator:
sin(165°) = √2(√3 - 1)/4
Hence, in the form sin(165°) = √a(√b - 1)/4, we have a = 2 and b = 3.
a diamond's density is 3.2 g/cm3
determine the volume of a diamond that has a mass of 1 g
Brenda can deliver 644 newspapers in 7 hours. How many newspapers can Brenda deliver in 9 hours?
A child types the letters q, w, e, r, t, y, randomly producing 1000 letters in all. what is the expected number of times that the sequence qqqq appears, counting overlaps
The expected number of times the sequence 'qqqq' appears in 1000 random typings of 6 different letters is approximately 0.769 times.
Explanation:
This problem falls under the category of discrete probability in mathematics. The child randomly types one of 6 different letters.
Therefore, the probability of typing any one specific letter, such as 'q', is 1/6. The sequence 'qqqq' is a sequence of 4 specific letters. Thus, the probability of typing this sequence at any given point is (1/6)^4 or 1/1296. To find the expected number of times the sequence 'qqqq' will appear in 1000 typed letters, we multiply the probability of the event by the number of attempts. However, because 'qqqq' is a sequence of 4 letters, our attempts are based on 4-letter sequences and not individual letters. So, instead of 1000 letters, we have 1000 - 4 + 1 = 997 possible 4-letter sequences in 1000 letters.The expected number of times the sequence 'qqqq' will appear is then 997 * (1/1296) which is approximately 0.769. In simple words, we would expect the sequence 'qqqq' to occur around 0.769 times or less than once in 1000 random typings of the 6 different letters.Learn more about Expected Value here:https://brainly.com/question/35639289
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It is important to re-evaluate financial goals periodically. In which of the following situations would it be necessary to change an existing financial goal?
a. You fell sharply behind your expected schedule with regard to saving.
b. You recovered from an unexpected expense and are rattled that you did not see it coming.
c. You married, and your spouse has a similar financial goal.
d. You came across unexpected income.
Please select the best answer from the choices provided
A
B
C
D
i have 3 1/8 pounds of macaroni and want to make 10 equal serving of macaroni salad. how many pounds of macaroni do i have for each serving
describe how you would Use the distributive property to simplify 39 * 5
URGENT! PLEASE ANSWER!
Layla is using the model to solve a division problem. Which question would the model be most helpful in solving?
How many 1/6s are in 4?
How many 1/6s are in 2/3?
How many 2/3s are in 4?
How many 2/3s are in 1/6?
Answer:
the answer is B
Step-by-step explanation:
The model would be most helpful in solving the problem 'How many 1/6s are in 2/3?'. This is found by dividing 2/3 by 1/6 through fractional division, giving a result of 4.
Explanation:The model would be most helpful in solving the question 'How many 1/6s are in 2/3?'. Let me explain why:
To solve this, you simply need to divide 2/3 by 1/6. When we say 'how many 1/6s are in 2/3', it means how many times can 1/6 fit into 2/3. Division is the process of finding out how many times one number (the divisor) is contained within another (the dividend).
So we divide 2/3 by 1/6. But rather than directly divide, in fractional division we multiply the first fraction by the reciprocal of the second. The reciprocal of 1/6 is 6/1.
So, 2/3 * 6/1 = 4.
That means, there are 4 1/6s in 2/3.
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Write the fact family for the numbers 32,8,and 4
Someone please help me!
The sum of 3 numbers is -4. The second number decreased by the third is equal to the first. The sum of the first and second numbers is -5. Write the system of equations and find the numbers.
The parent function, f(x)=5^x has been vertically compressed by a factor of one-half shifted to the right three units and down two units.
Choose the correct function to represent the transformation
Function transformation involves changing the form of a function
The function that represents the transformation is: g(x) = 1/2(5^(x -3))
The parent function is given as:
f(x) = 5^x
When the function is vertically compressed by a factor of one-half, the function becomes
f'(x) = 1/2(5^x)
When the function is shifted 3 units right, it becomes
f"(x) = 1/2(5^(x -3))
Rewrite the function as:
g(x) = 1/2(5^(x -3))
Hence, the function that represents the transformation is: g(x) = 1/2(5^(x -3))
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The correct function representing the transformation is B) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x-2} - 2 \).[/tex]
Step 11. **Vertical Compression by a Factor of One-Half:**
To vertically compress the function by a factor of one-half, we multiply the function by [tex]\( \frac{1}{2} \)[/tex]. This affects the [tex]\( y \)[/tex]-coordinates of all points on the graph. So, the function becomes: [tex]\( \frac{1}{2} \cdot 5^x = \left(\frac{1}{2}\right)5^x \).[/tex]
2. **Shifted to the Left Three Units:**
To shift the function to the left three units, we add 3 to the [tex]\( x \)[/tex] -coordinate. This affects the input values. So, the function becomes: [tex]\( \left(\frac{1}{2}\right)5^{x+3} \).[/tex]
3. **Shifted Down Two Units:**
To shift the function down two units, we subtract 2 from the [tex]\( y \)-[/tex]coordinate. This affects the output values. So, the final transformed function becomes: [tex]\( \left(\frac{1}{2}\right)5^{x+3} - 2 \).[/tex]
Step 2 :Now, let's compare this transformation with the provided options:
A) [tex]\( g(x) = 5^{\left(\frac{n}{2}\right)+3} - 2 \)[/tex]
This option doesn't match the transformation. It introduces a variable [tex]\( n \)[/tex] and doesn't involve the original function [tex]\( 5^x \)[/tex].
B) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x-2} - 2 \)[/tex]
This option matches the transformation. It has the vertical compression by [tex]\( \frac{1}{2} \)[/tex], shifting to the left by three units, and shifting down by two units.
C) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x+5} - 2 \)[/tex]
This option doesn't match the transformation. It shifts the function to the right by five units instead of to the left.
D) [tex]\( g(x) = 5^{\left(\frac{1}{2}\right)^{x-3}} - 2 \)[/tex]
This option introduces a fractional exponent, which doesn't correspond to the given transformations. It also has an incorrect shift to the right by three units.
Therefore, the correct option representing the transformation is B) [tex]\( g(x) = \left(\frac{1}{2}\right)5^{x-2} - 2 \).[/tex]
Complete Question :
The parent function, f(x)=5^x has been vertically compressed by a factor of one-half shifted to the right three units and down two units.
Choose the correct function to represent the transformation
A) g(X)=5^(((n)/(2))+3)-2
B) g(x)=((1)/(2))5^(x-2)-2
C) g(x)=((1)/(2))5^(x+5)-2
D) g(x)=5^(((1)/(2))^(x-3))-2
what is 3w(p +18) equal?
7/6 - 4/3n = -3/2n + 2(n + 3/2)
How to solve -5 3/14 to the thousandths place?? Please help
which function is an even function?
Kayla is knitting red sweaters to sell in her online store. The pattern requires 3 ounces of white yarn to trim each sweater. Kayla has 57 ounces of white yarn. How many sweaters can she trim with the yarn she has on hand?
Answer:
She can trim 19 sweaters.
Step-by-step explanation:
Kayla has a particular pattern in which she needs 3 ounces of white yarn.
That is in order to make a sweater of that particular pattern she requires white wool = 3 ounces
Kayla has total white wool = 57 ounces
one sweater requires = 3 ounces
number of sweater =[tex]\frac{total wool}{wool required for one sweater}[/tex]
=[tex]\frac{57}{3} = 19[/tex]
She can trim 19 sweaters with the Yarn she has on hand
Answer:
She can trim 19 sweaters.
Step-by-step explanation:
The perimeter of a rectangle is 400 yards. What are the dimensions of the rectangle if the length is 80 yards more than the width?
The width is 60 yards and the length, being 80 yards more, is 140 yards.
To find the dimensions of the rectangle when the perimeter is 400 yards and the length is 80 yards more than the width, we can set up two equations based on the formula for the perimeter of a rectangle, which is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width.
Let the width be w. Then the length is w + 80 yards.
Using the perimeter formula:
400 = 2(w + 80) + 2w
400 = 2w + 160 + 2w
400 = 4w + 160
Subtract 160 from both sides:
240 = 4w
Divide by 4:
w = 60 yards
Now, calculate the length:
l = w + 80 = 60 + 80
l = 140 yards
The width is 60 yards, and the length is 140 yards.
Wilbur's widgets, a widget company, produces 100 widgets. its average fixed cost is $5 and its total variable cost is $300. what is the total cost of producing 100 widgets?
a. $300
b. $305
c. $500
d. $800
The total cost of producing 100 widgets will be $800. Then the correct option is D.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
Wilbur's widgets, a widget company, produce 100 widgets. its average fixed cost is $5 and its total variable cost is $300.
Then the equation is given as,
y = 5x + 300
Where 'x' is the number of widgets and 'y' is the total cost.
Then the total cost of producing 100 widgets will be given as,
y = 5 × 100 + 300
y = 500 + 300
y = $800
The total cost of producing 100 widgets will be $800. Then the correct option is D.
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!!!!!!!I NEED HELP WITH THESE PROBLEMS. THANK YOU!!!!!
1. Mike's salary is $6,500 per month. He saves 1/5 of his monthly salary and spends the rest.
How much money does Mike save every month?
$ ?
How much money does Mike spend every month?
$ ?
2. Fill in the missing values that makes the equation true.
−2/5×30= ?/2 x -4 = ?
3. Alex takes 1 hour and 30 minutes to get to the office.
How many hours does Alex spend getting to the office over 4 days?
? hours
4.Fill in the missing values that make the equation true.
−4/7×49= ?/6 x -14 = ?
5. Drag the equivalent expressions into the appropriate columns.
5(1/5+3c) 1/4(1/2 - 4c)
(5+15c), (1+15c), (1/8 - c), (1/8 - 4/4c), (-1/8 + -c), (5/5 + 15c)
6. Drag the equivalent expressions into the appropriate columns.
a(b + c) -x(m - n)
(ab - ac), (ab + ac), -(xm - xn), (-xm + -xn)
7. Part 1
18 × ? = 1
Part 2
Select all the statements that explain why the missing value you chose for Part A makes the equation true.
Answer Choices
A. The product of a rational number and its multiplicative inverse is always 1.
B. The sum of a rational number and its additive inverse is always 0.
C. The product of two positive rational numbers is always positive.
D. The product of any two rational numbers is always 1.
E. It is the multiplicative inverse of the multiplier.
F. It is the additive inverse of the multiplier.
8. This is an equation.
(-7)(-17)=1
Select all of the statements that explain why this equation is true.
Answer Choices
A. The fraction -17 is the multiplicative inverse of -71 .
B. The fraction -17 is the additive inverse of -71 .
C. The product of any two rational numbers is always 1.
D. The product of two negative rational numbers is always positive.
E. The product of a rational number and its multiplicative inverse is always 1.
F. The sum of a rational number and its additive inverse is always zero.
the product of two consecutive even positive integers is 10 more than seven times the larger . find the integers?
Evaluate the series 50 + 10 + 2 + . . .
This graph shows a portion of an even function. Use the graph to complete the table of values.
1, 1, 3, 3 - I just did it on e2020 and got it correct
**I really need help on this! Please answer!*
The table shows the scoring system for quarterbacks in Jeremy's fantasy football league. In one game, Jeremy's quarterback had 2 touchdowns passes, 16 complete passes, 7 incomplete passes, and 2 interceptions. How many total points did Jeremy's quarterback score?
**QUARTERBACK SCORING**
Touchdown pass - 6
Complete pass - 0.5
Incomplete pass - -0.5
Interception - -1.5
The total number of points = 13.5 points
What is the total amount of points gained?
We are given the scoring method as:
Touchdown pass: 6
Complete pass : 0.5
Incomplete pass : -0.5
Interception : -1.5
Now, we are told that Jeremy's quarterback had the following:
2 touchdowns passes,
16 complete passes,
7 incomplete passes,
2 interceptions.
Thus:
Total points = (2 * 6) + (0.5 * 16) + (-0.5 * 7) + (-1.5 * 2)
Total points = 13.5 points
of the 40 students who auditioned for the school play, 3/4 of them were called back for a second audition. How many students were called back for another audition.
Consider the exponential function f(x) = 3 and its graph. Which statements are true for this function and graph? Check all that apply. The initial value of the function is . The growth value of the function is . The function shows exponential decay. The function is a stretch of the function f(x) = . The function is a shrink of the function f(x) = 3x. One point on the graph is (3, 0).
The function shows exponential growth a stretch of the function f(x) = (1/3)^x .
What are exponential functions?When the expression of a function is such that it involves the input to be present as the exponent (power) of some constant, then such a function is called an exponential function. Their usual form is specified below. They are written in several equivalent forms.
Given that the exponential function f(x) = 3
The initial value of the function is 2.
The base of the function is 3.
When the intial value is where x=0
where x=0, the value of f(0)=3
The growth is 1/3 but it is decay, that might be correct, or not because it is decay.
It is a stretch of the function f(x)=(1/3) power of x
when x=3, then f(3)=1/9
The function shows exponential growth a stretch of the function f(x) = [tex](1/3)^x[/tex].
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A and B are vertical angles. If A = 4x - 10 and B = 6x - 30, find mA.