Of all points (x,y,z) that satisfy x + 4y + 22 = 28, find the one that minimizes (x - 1)^2 + (y - 1)^2 + (z – 1)^2 The given function has a minimum at ( ____ ) which satisfies x + 4y +2z= 28. (Simplify your answer.)
Answers
the point that minimizes the function and satisfies the constraint is (2, 3, 3). To find the point that minimizes the function (x - 1)^2 + (y - 1)^2 + (z – 1)^2 among all points that satisfy x + 4y + 22 = 28, we can use Lagrange multipliers.
Let f(x,y,z) = (x - 1)^2 + (y - 1)^2 + (z – 1)^2 and g(x,y,z) = x + 4y + 22 - 28. Then we want to find the values of x, y, and z that minimize f(x,y,z) subject to the constraint g(x,y,z) = 0.
The Lagrange function is L(x,y,z,λ) = f(x,y,z) - λg(x,y,z) = (x - 1)^2 + (y - 1)^2 + (z – 1)^2 - λ(x + 4y + 22 - 28).
To find the minimum, we need to find the critical points of L(x,y,z,λ). Taking partial derivatives and setting them equal to zero, we get:
∂L/∂x = 2(x - 1) - λ = 0
∂L/∂y = 2(y - 1) - 4λ = 0
∂L/∂z = 2(z – 1) - 2λ = 0
∂L/∂λ = x + 4y + 22 - 28 = 0
Solving this system of equations, we get x = 7/3, y = 5/3, z = 1, and λ = 2/3.
Therefore, the function has a minimum at the point (7/3, 5/3, 1) which satisfies x + 4y +2z= 28.
To find the point (x, y, z) that minimizes the function f(x, y, z) = (x - 1)^2 + (y - 1)^2 + (z - 1)^2, subject to the constraint x + 4y + 2z = 28, we can use the method of Lagrange multipliers.
Let g(x, y, z) = x + 4y + 2z - 28. The gradient of f and g must be proportional, i.e., ∇f = λ∇g, where λ is the Lagrange multiplier. This gives us the following system of equations:
1) 2(x - 1) = λ
2) 2(y - 1) = 4λ
3) 2(z - 1) = 2λ
4) x + 4y + 2z = 28
From equation (1), x = 1 + λ/2.
From equation (2), y = 1 + λ.
From equation (3), z = 1 + λ.
Substituting these values into equation (4), we get:
1 + λ/2 + 4(1 + λ) + 2(1 + λ) = 28
Solving for λ, we find λ = 2.
Now, we can find the point (x, y, z) that satisfies the constraint and minimizes the function:
x = 1 + λ/2 = 1 + 2/2 = 2
y = 1 + λ = 1 + 2 = 3
z = 1 + λ = 1 + 2 = 3
So, the point that minimizes the function and satisfies the constraint is (2, 3, 3).
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Related Questions
Write an expression, using an exponent, that is equivalent to 8\times8\times8\times8
.
Answers
Answer:
8^3
Step-by-step explanation:
8*8*8 is equivalent to 8^3 which equals a total of 512 units
Consider the following rational expression: 4y + 16 y+ 4 Step 2 of 2: Find the restricted values of y, if any, for the given rational expression Answer How to enter your answer (opens in new window) 2
Answers
The given rational expression is 4y + 16 y + 4. To find the restricted values of y, we need to identify any values of y that would make the expression undefined.
In this case, the expression is in the form of a sum, so we don't have any denominators that could lead to division by zero. Therefore, there are no restricted values of y for this rational expression.
The expression 4y + 16 y + 4 is defined for all real numbers. We can evaluate it for any value of y without encountering any restrictions.
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you have a cake that is 16 inches by 18 inches.you want each piece to be exactly the same size.how many people can u serve equal sized piece of cake
Answers
The number of people are 72.
What is HCF?Highest Common Factor is the full name for HCF in mathematics.
According to the laws of mathematics, the highest positive integer that divides two or more positive integers without leaving a residual is known as the greatest common divisor, or gcd.
What is LCM?Least Common Multiple is the full name for LCM in mathematics.
LCM (a,b) in mathematics stands for the least common multiple, or LCM, of two numbers, such as a and b. The smallest or least positive integer that is divisible by both a and b is known as the LCM.
We need to know the size of the pieces in order to calculate how many people can be fed equal-sized pieces of cake from a 16 by 18-inch cake.
Assume that each piece is a square with sides measuring "x" in length. Each piece's area would be x², and the cake's overall area would be 16 x 18 inches, or 288 square inches.
We must divide the entire area of the cake by the area of each piece to determine the maximum number of pieces that can be cut from the cake:
288 / x² = (16*18) / x² = 288 / x²
Choosing a "x" value that evenly splits into 16 and 18 is necessary if we want every piece to be the exact same size. Since 2 is the greatest possible value, we can divide the cake into 8 rows of 9 squares, yielding a total of 72 pieces.
As a result, a cake measuring 16 inches by 18 inches can be divided into 72 equal-sized pieces, each measuring 2 inches by 2 inches.
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We can serve 18 people equal-sized pieces of cake from a cake that is 16 inches by 18 inches, assuming each piece is a square and we want the pieces to be exactly the same size.
What is area?A two-dimensional figure, form, or planar lamina's area is a measurement of how much space it takes up in the plane.
To find out how many people can be served equal-sized pieces of cake from a cake that is 16 inches by 18 inches, we need to first determine the size of each piece of cake.
The total area of the cake is:
16 inches x 18 inches = 288 square inches
To divide the cake into equal-sized pieces, we need to determine the size of each piece. Let's assume we want each piece to be a square. To find the size of each square, we need to find the square root of the total area of the cake:
√(288 square inches) ≈ 16.97 inches
Since we want the pieces to be exactly the same size, we'll round down to the nearest inch, which gives us:
Each piece of cake will be approximately 16 inches by 16 inches.
To find out how many people can be served equal-sized pieces of cake, we need to divide the total area of the cake by the area of each piece:
288 square inches ÷ (16 inches x 16 inches) = 18
Therefore, we can serve 18 people equal-sized pieces of cake from a cake that is 16 inches by 18 inches, assuming each piece is a square and we want the pieces to be exactly the same size.
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quires 2 1/2 feet of wood. She has a total of 13 feet of wood. What is the greatest number of birdhouses could
Answers
The required number of birdhouses can be made through the 13 feet of wood are 5 birdhouses.
What is the Ratio?The ratio can be defined as the proportion of the fraction of one quantity towards others. e.g.- water in milk.
Here,
A birdhouse consists of 2 1/2 feet of wood,
he has a total of 13 feet of wood.
Number of birdhouses
= ratio of wood total length / used to build one birdhouse
= 13 / [2 /1/2]
= 5.2
≈ 5
Thus, the required number of birdhouses can be made through the 13 feet of wood are 5 birdhouses.
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Help please thanks looks at pic attached
Answers
Answer:
m || n only when the corresponding angles will be equal.
Step-by-step explanation:
Here its only given that vertical angles are equal. But it doesn't prove m || n . If the corresponding angles wouls have been equal , then only we can prove that m || n.
write a function of g whose graph represents the indicated transformation
f(x)=-|8x|-4 vertical shrink by 1/4
Answers
The function of g whose graph represents the indicated transformation is g(x) = -|2x| - 1
How to write a function of g whose graph represents the indicated transformation?The function is given as
f(x) = -|8x| - 4
The transformation is given as
A vertical shrink by 1/4
This is represented as
g(x) = 1/4f(x)
So, we have
g(x) = 1/4 * (-|8x| - 4)
Evaluate
g(x) = -|2x| - 1
Hence, the function of g whose graph represents the indicated transformation is g(x) = -|2x| - 1
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Write an equation in slope-intercept form y = mx + b of a line with a slope of 3/4
and a y-intercept of -2. *
Answers
Answer:
y=3/4x-2
Step-by-step explanation:
The base of a rectangular prism has an area of 252525 square meters. The height of the rectangular prism is 444 meters. What is the volume, in cubic meters, of this rectangular prism?
Answers
Answer:Height:
15.8 meters
Explanation: For a rectangular prism: XXX Volume = height × width × depth and XXX Area (of base) = width × depth ⇒ height = Volume Area = 306.52
19.4 = 15.
Answer: 100 CUBIC METERS
Step-by-step explanation:
Orders of operation Practice
(5^2+)x2-12
Answers
julie needs to cut 4 pieces of yarn, each the same length, and a piece of yarn 7.75 inches long. let x represent the length of each of the equal pieces of yarn that julie decides to cut. what is the equation that can be used to determine the total length of all of the yarn she ends up cutting, y? is the graph of the equation continuous or discrete?
Answers
Answer: She has to cut another piece = 7.75 inches
Total length = 4x + 7.75
This is what y represent, so
y = 4x + 7.75
The graph of the equation is continuous
Step-by-step explanation:
The equation that can be used to determine the total length of all of the yarn she ends up cutting (y) will be y = 7.75x. Then the graph of the equation is continuous.
What is the linear system?A linear system is one in which the parameter in the equation has a degree of one. It might have one, two, or even more variables.
Julie needs to cut 4 pieces of yarn, each the same length, and a piece of yarn 7.75 inches long.
Let x represent the length of each of the equal pieces of yarn that Julie decides to cut.
Then the equation that can be used to determine the total length of all of the yarn she ends up cutting (y) will be
y = 7.75x
Then the graph of the equation is continuous.
The graph is given below.
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Can someone help me with this??
Answers
Answer:
negative 10 less than x
-10 < x
Please help! I am so lost. What is the probability of landing in any yellow region of the square below?
Radius of each circle is 1 unit. Find the exact probability.
Answers
The probability of landing in any yellow region is the area of the yellow region divided by the total area of the square, which is (π + 2) / 4. This is the exact probability.
To find the probability of landing in any yellow region of the square, we need to calculate the area of the yellow regions and divide it by the total area of the square.
Let's consider one of the quarters of the square, which is symmetrical. The yellow region consists of a quarter of a circle with radius 1 unit, and a right-angled triangle with sides of length 1 unit.
The area of the quarter circle is (1/4)π\(r^2\) = (1/4)π(\(1^2\)) = π/4.
The area of the right-angled triangle is (1/2) \(\times\)1 \(\times\)1 = 1/2.
Therefore, the total area of the yellow region in one-quarter of the square is (π/4) + (1/2) = (π + 2)/4.
Since the square has four identical quarters, the total area of the yellow region in the entire square is 4 \(\times\)[(π + 2)/4] = π + 2.
The probability of landing in any yellow region is the area of the yellow region divided by the total area of the square, which is (π + 2) / 4. This is the exact probability.
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Meghan sells handmade sweaters and hats at her store. Let A represent the event that a randomly selected customer buys hat, and let B
represent the event that a randomly selected customer buys a sweater.
What does the notation P(AJB) represent in terms of the context?
A the probability that a customer buys a hat or a sweater
B the probability that a customer buys both a hat and a sweater
C the probability that a customer buys a hat given that the customer has bought a sweater
D the probability that a customer buys a sweater given that the customer has bought hat
Answers
Answer:
Step-by-step explanation:
C the probability that a customer buys a hat given that the customer has bought a sweater.
Find the derivative of the following function. f(t)=t 7
( t
+1) The derivative is f ′
(t)=
Answers
The derivative of the function f(t) = t^7(t + 1) is f'(t) = 8t^6 + 7t^7. To find the derivative of the given function, we will apply the product rule and the power rule of differentiation.
The product rule states that if we have two functions u(t) and v(t), the derivative of their product is given by (u(t)v'(t) + v(t)u'(t)).
Let's apply the product rule to the function f(t) = t^7(t + 1):
f'(t) = (t^7)'(t + 1) + t^7(t + 1)'
Now, let's find the derivative of each term using the power rule:
The derivative of t^7 is given by (t^7)' = 7t^(7-1) = 7t^6.
The derivative of (t + 1) is simply 1.
Substituting these derivatives back into the product rule formula, we have:
f'(t) = 7t^6(t + 1) + t^7(1)
Simplifying further:
f'(t) = 7t^7 + 7t^6 + t^7
Combining like terms:
f'(t) = 8t^6 + 7t^7
Therefore, the derivative of the function f(t) = t^7(t + 1) is f'(t) = 8t^6 + 7t^7.
In summary, the derivative of the function f(t) = t^7(t + 1) is given by f'(t) = 8t^6 + 7t^7.
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help me please someone actually help . if rendom youre getting reported sorry
Answers
Answer: 1.2 x 10^12.
Step-by-step explanation:
200 billion is 2 x 10^11
1.4*10^12 - 2*10^11= 1.2*10^12
The ratio of boys to girls in a class is 5 to 4. If there are 18 students in the class, how many are girls?
Answers
Answer:
8 girls:)
Step-by-step explanation:
there are 10 boys and 8 girls. added the component digits of the ratio (5+4 = 9) divide the total by that number (18 / 9 = 2). Each “unit” of boys or girls consists of 2 people, so 5 units of boys is 10 (5 x 2 = 10) and 4 units of girls is 8 (2 x 4 = 8)
In which quadrant does the terminal side of your angle lie if sec theta < o and cot theta < o ?
Answers
Answer:
The endpoint of an angle
x
lies in the 4th quadrant. Here is why.
First of all, let's recall a few definitions.
Recall the definition of a trigonometric function
sec
(
x
)
:
sec
(
x
)
=
(by definition)
=
1
cos
(
x
)
Recall the definition of a trigonometric function
cot
(
x
)
:
cot
(
x
)
=
(by definition)
=
cos
(
x
)
sin
(
x
)
Step-by-step explanation:
jhyjyujtyuyju
if f and g are twice differentiable and if h(x)=f(g(x)) then h''(x)=
Answers
If h(x) = f(g(x)), where f and g are twice differentiable functions, then h''(x) = f''(g(x)) ×\((g'(x))^{2}\) + f'(g(x)) × g''(x).
To find the second derivative of h(x), we can apply the chain rule. The chain rule states that if we have a composite function h(x) = f(g(x)), then its derivative is given by h'(x) = f'(g(x)) × g'(x).
Differentiating both sides of the equation h'(x) = f'(g(x)) × g'(x) with respect to x, we obtain the second derivative:
h''(x) = (f'(g(x)) × g'(x))' = f''(g(x)) ×g'(x) × g'(x) + f'(g(x)) ×g''(x).
Here, f''(g(x)) represents the second derivative of f(x) evaluated at g(x), and g''(x) represents the second derivative of g(x). The terms g'(x) * g'(x) and g'(x) * g''(x) account for the chain rule.
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You are baking cupcakes for your class. There are 30 total students in your class and you have baked 15 cupcakes. Write and
solve an equation to find the additional number of cupcakes you need to bake in order to have 3 cupcakes for each student.
Write your equation so that the units on each side of the equation are cupcakes per student.
Equation:
= 3
Number of cupcakes: x=
Answers
Answer:
Equation: 15x+30=3
Number of Cupcakes: x = 75
Step-by-step explanation:
30 x 3 - 15 = 75
I hope this helps you :)
The equation is 3=(x+15)/30 and the number of additonal cupcakes to be made is 75.
What is equation?An equation is a relationship between the variables and constants representing how they are related to each other. It includes equal to "=" sign.
It is known that each student will have 3 cupcakes. So, right side of the equation will be 3.
After 15 cupcakes, x more cupcakes are needed. So, the total number of cupcakes is x+15.
Since there are 30 students, the number of cupcakes per student will be (x+15)/30
Determine the equation and solve it.
(x+15)/30=3
x+15=90
x=75
So, 75 more cupcakes are needed to be baked.
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Find F'(x): F(x) = Sx 3 t^1/3 dt
Answers
The derivative of F(x) is \(F'(x) = x^{(1/3)\).
What is function?A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.
To find the derivative of the given function F(x), we will apply the fundamental theorem of calculus and differentiate the integral with respect to x.
Let's compute F'(x):
F(x) = ∫[0 to x] \(t^{(1/3)} dt\)
To differentiate the integral with respect to x, we'll use the Leibniz integral rule:
F'(x) = d/dx ∫[0 to x] \(t^{(1/3)} dt\)
According to the Leibniz integral rule, we have to apply the chain rule to the upper limit of the integral.
\(F'(x) = x^{(1/3)} d(x)/dx - 0^{(1/3)} d(0)/dx\) [applying the chain rule to the upper limit]
Since the upper limit of the integral is x, the derivative of x with respect to x is 1, and the derivative of 0 with respect to x is 0.
\(F'(x) = x^{(1/3)} (1) - 0^{(1/3)} (0)\)
\(F'(x) = x^{(1/3)\)
Therefore, the derivative of F(x) is \(F'(x) = x^{(1/3)\).
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A rectangular prism-shaped display case is 30 1/2 inches wide, 10 1/2 inches long, and 32 1/4 inches tall.
What is the volume of the display case in cubic inches?
Responses
73 1/4 in³
73 and 1 fourth, in³
320 1/4 in³
320 and 1 fourth, in³
10,328 1/16 in³
10328 and 1 sixteenth, in³
28,372 5/8 in
Answers
The volume of the display case is 2161 1/8 cubic inches.To find the volume of a rectangular prism-shaped display case, we multiply its length, width, and height.
Given that the display case is 30 1/2 inches wide, 10 1/2 inches long, and 32 1/4 inches tall, we can calculate the volume as follows:
Volume = Length * Width * Height
Using the given measurements:
Volume = 10 1/2 inches * 30 1/2 inches * 32 1/4 inches
To simplify the calculation, we can convert the mixed numbers to improper fractions:
Volume = (21/2) inches * (61/2) inches * (129/4) inches
Next, we multiply the fractions:
Volume = (21/2) * (61/2) * (129/4) = 17289/8
The resulting fraction, 17289/8, is an improper fraction. To convert it back to mixed number form:
Volume = 2161 1/8 in³.
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Russ and Vickie are trying to solve this problem: There are 117 students taking buses to the museum. If each bus holds 28 students, how many buses will they need?
Russ says the students need 4 buses. Vickie says they need 5 buses. Who is correct?
Just to tell you, I am a 5th grader.
Answers
Answer:
vicky is correct.
Step-by-step explanation:
117 divided by 28 is 4.178. u can't divide buses so they'll need 5 buses. Hope this helps.
can someone explain?
Answers
Answer: K
Step-by-step explanation:
I. Correct. When t=0, h=c, so if you alter the value of c, the h-intercept changes.
II. Correct, the maximum value of \(-at^2 +bt\) will stay the same, but you are changing c, which is the amount you are always adding to it, meaning the maximum will change.
III. Correct. If you plug into the quadratic formula setting h=0, it is easy to see changing the value of c will change the solutions.
The correct statement regarding the quadratic function is:
K. I, II and III.
What is a quadratic function?A quadratic function is given according to the following rule:
\(y = ax^2 + bx + c\)
If a > 0, it has a maximum value, and if a < 0, it has a minimum value.
The extreme value is \((x_v,y_v)\), in which:
\(x_v = -\frac{b}{2a}\)\(y_v = -\frac{\Delta}{4a}\)\(\Delta = b^2 - 4ac\)The solutions are:
\(x_1 = \frac{-b + \sqrt{\Delta}}{2a}\)\(x_2 = \frac{-b - \sqrt{\Delta}}{2a}\)In this problem, we have a function h(t). Changing the coefficient c, the h-intercept h(0) changes. Looking at the formulas in the bullet point, the value of \(\Delta\) changes, meaning that both the maximum value \(y_v\) and the t-intercepts \(t_1\) and \(t_2\) will change, so option K is correct.
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Wich measurement is the closest to the area of the triangle in square centimeters???? :(
Answers
Dndndjjfkf
Jfkkfjf
Jfkkf
How do I solve 12a+3c=2a+6, for a
Answers
Answer:
14a+9c
Step-by-step explanation:
Answer:
Step-by-step explanatio
All other units of measure are defined and compared to the four basic measurements of: _____?
Answers
The four basic measurements that serve as the foundation for other units of measure are length, mass, time, and temperature.
Length is a measure of distance between two points, typically measured in meters (m) within the International System of Units (SI). Mass is the amount of matter in an object, measured in kilograms (kg) in the SI system. Time refers to the duration or interval between events and is measured in seconds (s) within the SI system.
Lastly, temperature is a measure of the average kinetic energy of particles in a substance, represented in degrees Celsius (°C) or Kelvin (K) in the SI system.
All other units of measure are defined and compared to these four basic measurements. For instance, speed is derived from the combination of length and time (e.g., meters per second). Similarly, force is derived from mass and acceleration, which itself is derived from the change in velocity over time (e.g., newtons).
The combination and comparison of these fundamental measurements enable scientists, engineers, and other professionals to quantify and analyze various phenomena in the physical world.
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A system of equations has infinitely many solutions. If 2y-4x6 is one of the equations, which could be the other equation? Oy=2x+6 Oy=4x+6 Oy=-2x-3 O-y=-4x+6
Answers
The other equation is -y = -2x - 3.
The correct option is C.
To have a system of equations with infinitely many solutions, the two equations must be dependent and represent the same line or have the same slope and y-intercept.
The given equation is 2y - 4x = 6.
By comparing the given equation to the options:
a) y = 2x + 6
The slope of this equation is 2, which is different from the slope of the given equation (-2).
Therefore, this equation does not represent the same line or have the same slope and y-intercept.
b) y = 4x + 6
The slope of this equation is 4, which is different from the slope of the given equation (-2).
Therefore, this equation does not represent the same line or have the same slope and y-intercept.
c) y = -2x - 3
The slope of this equation is -2, which is the same as the slope of the given equation (-2). However, the y-intercept of this equation is -3, which is different from the y-intercept of the given equation (0).
d) y = -4x + 6
This equation does not represent the same line or have the same slope and y-intercept.
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Wasn’t there for class so I don’t really know how to do this someone help :(
Answers
Answer:
Step-by-step explanation:
Substitute (x, y) value in 5x - 4y and if you get 17 then (x , y) is the solution.
If you are getting 17, then (x, y) is not a solution.
1) (1 , -3)
5x - 4y = 5*1 - 4*(-3)
= 5 + 12
= 17
(1, - 3) is a solution.
2) (-6, 2)
5x - 4y = 5*(-6) - 4*2
= -30 - 8
= -38 ≠ 17
So, (-6 , 2) is not a solution.
3) (-3 , -8)
5x - 4y = 5*(-3) - 4* (-8)
= -15 + 32
= 17
So, (-3 , -8) is a solution.
4) (2 , 9)
5x - 4y = 5*2 - 4 *9
= 10 - 36
= -16 ≠ 17
So, (2 , 9) is not a solution
Can someone just help me find the volume of this shape!! Please I need it asap
Answers
Answer: 648cm^3
Step-by-step explanation:
Volume=area of base * height
Area of base: 0.5*9*24=108
108*6=648cm^3
Work out the size of angle x
Answers
Answer:
104 degrees
Step-by-step explanation:
90 + 54 + 132 = 276
360 - 276 = 104