Find the point on the paraboloid z = 4x2 + 4y2 which is closest to the point
(4; 5; 6).

Answers

Answer 1

Using the method of Lagrange multipliers: the Lagrangian is

$L(x,y,z,\lambda) = (x-4)^2+(y-5)^2+(z-6)^2 - \lambda (z-4x^2-4y^2)$

Fix λ > 0.

That is, we are minimizing the squared distance of a point (x, y, z) on the paraboloid to the point (4, 5, 6). We can use the squared distance in place of the proper Euclidean distance because both f(x) and √f(x) have critical points at the same point x ; plus the math works out much more easily.

Find the critical points of the Lagrangian:

$\dfrac{\partial L}{\partial x} = 2(x-4) + 8\lambda x = 0 \\\\ \dfrac{\partial L}{\partial y} = 2(y-5) + 8\lambda y = 0 \\\\ \dfrac{\partial L}{\partial z} = 2(z-6) - \lambda = 0 \\\\ \dfrac{\partial L}{\partial\lambda} = -z+4x^2+4y^2 = 0$

Solving for λ in terms of x, y, or z gives

$\displaystyle \lambda = \frac1x-\frac14 = \frac5{4y} - \frac14 = 2z-12$

so right away, we get

$\dfrac5{4y} - \dfrac14 = \dfrac1x - \dfrac14 \implies y=\dfrac{5x}4 \\\\ 2z-12 = \dfrac1x-\dfrac14 \implies z = \dfrac1{2x} + \dfrac{47}8$

Substitute these into the constraint and solve for x, then for y and z :

$\dfrac1{2x} + \dfrac{47}8 = 4x^2 +4\left(\dfrac{5x}4\right)^2 \implies 82x^3 - 47x - 4 = 0$

This has 3 roots, but we take the lone positive solution for x because otherwise λ would be negative. This root is x ≈ 0.7965.

Then it follows that the closest point on the paraboloid to (4, 5, 6) is the critical point (0.7965, 0.9956, 6.5028).

Related Questions

Jan is making a logo for her company. She begins with a 6-cm-by-12-cm rectangle. Two points along the bottom side divide that side into three equal pieces. A point on the top side divides that side into two equal pieces. The logo itself is the total shaded area, as shown. What is the area of the shaded logo, in square cm?

Answers

The area of a shape is the amount of space the shape occupies

The area of the shaded region in square cm is 48 square cm

How is the area of the logo calculated

The logo is a trapezoid with the following dimensions

Height = 12 cmParallel sides: 6 cm and 2 cm

The area of a trapezoid is:

Area = 0.5 * Height * (Sum of parallel sides)

Substitute known values

Area = 0.5 * 12 cm * (6cm + 2cm)

Evaluate

Area = 48 square cm

Hence, the area of the logo i.e. shaded region is 48 square cm

Answers

The statement "The roots of the blues were in New York City" is NOT true.

among the late 19th and early 20th centuries, the blues first appeared among African American communities in the southern United States, particularly Mississippi.

African American musical traditions such field hollers, work songs, and spirituals were among those that influenced the development of the blues. African Americans utilised it to express their joys and achievements as well as their struggles and tribulations.

Even though the blues eventually spread to other parts of the United States and the world, its roots are firmly planted in the rural South.

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When parallel lines are intersected by a transversal, which of the following angle pairs are supplementary?

Answers

Answer:

if two parallel lines are cut by a transversal,then, exterior angles on the same side of the transversal are supplementary!

Step-by-step explanation:

I don't really know how to explain it but I hope this helped!

HELP ASAP! Which of the following tables represents a linear function?

Table A -
x −2 −1 0 1 2
y 5 2 1 2 5

Table B -
x −2 −1 0 1 2
y 5 3 1 −1 −3

Table C -
x 3 3 0 3 3
y −2 −1 0 1 2

Table D -
x 0 1 2 3 4
y 0 −1 2 −3 4

Answers

Answer:

Table B -

x −2 −1 0 1 2y 5 3 1 −1 −3

Step-by-step explanation:

You want to know which of the four offered tables represents a linear function.

Differences

A linear function will have differences in y that have the same ratio to differences in x for all pairs of values in the table.

Evaluation

Table A has x-differences that are 1 between adjacent terms. The y-differences are negative for the first pair of table values, but positive for the last pair (2 -5 vs 5 -2).

Table B has x-differences that are 1 between adjacent terms. The y-differences are -2 between any pair of adjacent terms, so their ratio to x-differences is -2/1 = -2, a constant. Table B represents a linear function.

Table C has x-differences that change from 0 to -3 to +3 while y-differences are constant at +1. The ratios of differences are not constant.

Table D has x-differences of +1, and y-differences that alternate in sign and magnitude.

Only Table B represents a linear function.

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An important part of the customer service responsibilities of a company relates to the speed with which troubles in residential service can be repaired. Suppose that past data indicate that the likelihood is 0.70 that service troubles can be repaired on the same day they are reported. For the first five troubles reported on a given day, what is the mean of the distribution used in this problem?

Select one:

A.
350

B.
3.5

C.
0.35

D.
0.7143

E.
3500

Answers

The mean of the distribution used in this problem is 3.5.The answer is B. 3.5.

To find the mean of the distribution used in this problem, we need to calculate the expected number of troubles that can be repaired on the same day out of the first five reported. The likelihood of a trouble being repaired on the same day is given as 0.70.

The expected number of troubles repaired on the same day can be calculated by multiplying the probability of each event by the number of trials. In this case, we have five trials (the first five troubles reported) and a probability of 0.70 for each trial.

Expected number of troubles repaired on the same day = Probability of repair on the same day * Number of trials

= 0.70 * 5

= 3.5

This means that, on average, out of the first five troubles reported, we can expect 3.5 of them to be repaired on the same day based on the past data and the given likelihood. So, the correct answer is B. 3.5.

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How many square feet of outdoor carpet will we need for this hole?

Answers

Answer:39 sq. ft

There are two rectangles, so the combined areas will equal the complete carpet amount needed for the hole. Area of a rectangle: length x width. 11(3)+2(3)= 39

Answer:

39 square feet.

Step-by-step explanation:

You can split the outdoor carpet into 2 rectangles, one being the 11 ft x 3 ft and the 3 ft by the 2ft.

Let's solve the area for the 11 ft x 3 ft by multiplying the length and width:

11 x 3 = 33

Now let's solve the rectangle on the bottom by multiplying the length by the width again:

3 x 2 = 6

Once we have found the areas for our 2 rectangles, we can simply add them to find the total amount of square feet for the whole shape:

33 + 6 = 39

39 square feet

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WILL MARK BRINLIST Which pairs of angles are congruent because they are vertical angles? Check all that apply. 7373 34 56 21 and 24 pls don't joke

Answers

Answer:

B, D represent pairs of vertical angles.

___

Other pairs may be congruent, but they are not vertical angles. The lines that look parallel must be assumed to be NOT parallel, because they are not marked in the diagram as being parallel. Thus ∠8 ≠ ∠12, for example.

- sqdancefan

What is the length of side a?

Line a to b is 5
line b to c is a

Answers

Answer: which side is a? I can’t answer the question without knowing the sides

Step-by-step explanation:

Half a number increased by 10 is 12. Find the number.

Answers

Answer:

Step-by-step explanation:

Equation:

1/2x + 10 = 12

Subtract 10 from both sides:

1/2x = 2

Divide by 1/2 from both sides:

x = 4

-Chetan K

Point: (2,7)
Reflected across y-axis: (-2,7)
Reflected across x-axis: ????
My options for the answer is
(-7,-2)
(-2,7)
(2,-7) which answer is it I have already tried the first one and it said it was wrong.

Answers

Answer:

(2,-7)

Step-by-step explanation:

Reflection rule for across the x-axis: $(x,y)\rightarrow (x,-y)$

Using the rule and point (2,7):

$(2,7)\rightarrow(2,-7)$

(2,-7) should be the correct answer.

Brainilest Appreciated.

determine the value of X and Y
PLS HELP :)

Answers

Answer:

x=90º

y=64º

Step-by-step explanation:

180-67=113

180-75=105

540-90-113-105=232

232/2=116

180-116=64º

180-90=90º

wanna help a 6th grader out
pt 1

Answers

The answer for the blank question would be 27

Find each value given the following function:

Answers

Answer:

Step-by-step explanation:

1) f(-4) --> if x < or equal to 3

2) 1/(-4)-4

3) The answer is - 1/8

solve for v

pleaseee answer and explain <333

Answers

Hi there! :)

Answer:

$\huge\boxed{v = -27}$

$\frac{v-3}{4} = \frac{9+2v}{6}$

Cross multiply:

$6 ( v - 3) = 4 ( 9 + 2v)$

Distribute:

$6v - 18 = 36 + 8v$

Isolate the "v" variable:

$6v - 18 + 18 = 36 + 18 + 8v$

$6v = 54+ 8v$

$6v - 8v = 54+ 8v - 8v$

$-2v = 54\\\\v = -27$

In a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Paul has scored 82 , 88 , and 87 on the first three. What range of scores on the fourth test will give Paul a C for the semester (an average between 70 and 79 , inclusive)? Assume that all test scores have a non-negative value.

Answers

Scores between 23 & 59 will give Paul a C for the semester (an average between 70 and 79).

The sum of the scores on the first three, hundred point tests

= 82 + 88 + 87 = 257.

The total score required in four 100-point tests to get an average score of 70 = 70X4= 280.

The total score required in four 100-point tests to get an average score of 79 = 79X4= 316.

Therefore, the score on the fourth test should be between (280-257) & (316-57) i.e, 23 & 59 to get an average score of 70 & 79.

Answers

Answer:

A- Brady multiplied the first equation by 5 and the second equation by 3.

B- He can use elimination for his next step. 15y and -15y will cancel each other out, 20x-18x is 2x...50-48 is 2 so 2x=2 divide and x is 1

C- x=1 y=2

Step-by-step explanation:

Solve equation:
-6-3k+8=5

Answers

if you solve the equation you get k=-1

Enter your answer as a simplified fraction by filling in the boxes.

$Enter your answer as a simplified fraction by filling in the boxes.$

Answers

Answer:

51/40

Step-by-step explanation:

-1 3/5 - (-2 7/8)

-8/5 - (-23/8)

-8/5 + 23/8

-64/40 + 115/40

-64+115/40

51/40

Solve for x: −7 < x − 1 < 8

6 < x < 9
−6 > x > 9
6 > x > −9
−6 < x < 9

Answers

[Hello, blueface1728]

Answer:

[D] −6 < x < 9

Step-by-step explanation:

So, if a < u <b then a < u and u <b

-7 < x - 1 and x -1 < 8

Solving

-7 < x - 1 = x > -6

x - 1 < 8 = x < 9

Combine the intervals

-6 < x < 9

Now looking at the answer choice :

6 < x < 9

−6 > x > 9

6 > x > −9

−6 < x < 9

Thus, the answer is [D] −6 < x < 9

Kavinsky

I need help I don’t understand please help .

Answers

The standard form of 2i/1 + 3i is 3/5 + (1/5)i.

Option C is the correct answer.

What is an expression?

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

2i/(1 + 3i)
Rationalize the denominator.

= 2i/(1 + 3i) x (1 - 3i)/(1 - 3i)

= (2i + 6)/(1² - (3i)²) [ i = √-1, i² = -1 ]
= (2i + 6)/(1 + 9)

= (2i + 6)/10

= 2i/10 + 6/10
= i/5 + 3/5

= 3/5 + (1/5)i

Thus,

The standard form is 3/5 + (1/5)i.

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Which is the graph of f(x)=(2/3)x

Answers

Answer:

Here is the desmos graph

Step-by-step explanation:

What is the surface area of the rectangle pyramid below 13 13 13

Answers

Answer:

Step-by-step explanation:

Assuming that the given dimensions of 13, 13, 13 refer to the base of the rectangular pyramid, we can calculate the surface area of the pyramid as follows:

First, we need to calculate the area of the rectangular base, which is simply length x width:

Area of rectangular base = 13 x 13 = 169 square units

Next, we need to calculate the area of each triangular face of the pyramid. Since the rectangular base has two sets of parallel sides, there are two types of triangular faces: the isosceles triangles on the sides and the right triangles on the front and back.

To calculate the area of the isosceles triangles, we need to first find the length of the slant height, which can be found using the Pythagorean theorem:

a² + b² = c²

where a and b are the base and height of the triangle (both equal to 13 in this case), and c is the slant height.

13² + 13² = c²

338 = c²

c ≈ 18.38

Now that we have the slant height, we can calculate the area of each isosceles triangle using the formula:

Area of isosceles triangle = (1/2) x base x height

Area of isosceles triangle = (1/2) x 13 x 18.38

Area of isosceles triangle ≈ 119.14 square units

To calculate the area of each right triangle, we need to use the same slant height of 18.38, along with the height of the pyramid, which is also 13. Then we can use the formula:

Area of right triangle = (1/2) x base x height

Area of right triangle = (1/2) x 13 x 18.38

Area of right triangle ≈ 119.14 square units

Since there are two of each type of triangular face, the total surface area of the pyramid is:

Surface area = area of rectangular base + 2 x area of isosceles triangle + 2 x area of right triangle

Surface area = 169 + 2 x 119.14 + 2 x 119.14

Surface area = 546.28 square units

Therefore, the surface area of the rectangular pyramid with base dimensions of 13 x 13 and height of 13 is approximately 546.28 square units.

express 2 cos 35 sin 67 as a sum

Answers

2 cos 35 sin 67 can be expressed as the sum of sin(102) and -sin(32).

To express 2 cos 35 sin 67 as a sum, we can use the trigonometric identity for the product of two sine or cosine functions.

Specifically, the identity states that 2 cos A sin B can be written as

sin(A + B) + sin(A - B).

Applying this identity, we have:

2 cos 35 sin 67 = sin(35 + 67) + sin(35 - 67)

Simplifying the expressions inside the sine functions:

sin(102) + sin(-32)

Since the sine function is an odd function,

sin(-x) = -sin(x),

so we can rewrite the equation as:

sin(102) - sin(32)

Therefore, 2 cos 35 sin 67 can be expressed as the sum of sin(102) and -sin(32).

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A researcher is interested in whether or not there is any association between normal method of transportation taken to work and reported mood at the beginning of a typical Monday morning. He classifies participants into three transportation groups (those who: bike/walk, take public transit or drive) and asks them how they typically feel on a normal Monday morning: happy/excited, angry/irritated, or sleepy. Data from his survey on 200 participants are presented below.
Happy Angry sleepy Total
Bike/Walk 20 7 13 40
Public transit 13 5 42 7
Drive 17 18 65 100
Total 50 30 120 200
You would like to test his question of whether or not there is any association between normal method of transportation taken to work and reported mood at the beginning of a typical Monday morning. Calculate a two variable chi-square test using an alpha (α) level of 0.01.
You will use the information above to complete the question parts below for a two variable chi-square test.
Part A: Which of the following represents the appropriate null hypothesis(H0), given your research question of interest?
Part B: Which of the following represents the appropriate alternative hypothesis (H1), given your research question of interest?
Part C: What is your degrees of freedom (df)?
Part D: What is the critical chi-square value (from the Chi-square table) appropriate for your test, testing at an alpha level (α) of 0.01?
Part E: What is your calculated chi-square statistic?
Part F: What is your decision about the null hypothesis based on your test results?

Answers

Answer:

Check the explanation

Step-by-step explanation:

Kindly check the attached image below to see the step by step explanation to the question above.

A researcher is interested in whether or not there is any association between normal method of transportation

If 3x+y+4-(y + 6) = 14 + x, what is the value of x ?

Answers

Answer:

x=8

Step-by-step explanation:

Answer:

3x+y+4-(y + 6) = 14 + x

x=8

PROVIDE STEPS. Brainly and 30 points.

Answers

Answer:

i got x=24. I might be wrong, please don't be mad.

Step-by-step explanation:

i started with (5x-28)+(x+5)+(2x+11)=180

i combined like terms and got 8x-12=180

then i added 12 to both sides 8x=192

finally i divided by 8 on both side x=24

hopefully this is correct and helped

Answer:

i got x=24. I might be wrong, please don't be mad.

Step-by-step explanation:

i started with (5x-28)+(x+5)+(2x+11)=180

i combined like terms and got 8x-12=180

then i added 12 to both sides 8x=192

finally i divided by 8 on both side x=24

hopefully this is correct and helped

Ciana earns an hourly wage of $30 at her job. In order to purchase her sneakers she will have to take time off work, so each hour away from her job
costs her $30 in lost Income. Assume that ciana travel time is the same each way (to and from the store) and that it will take her 30 minutes once
she reaches a store to complete her shopping. Assume throughout the question that ciara incurs no additional costs other than the sneakers, such as
gas.

complete the following table by computing the opportunity cost of ciana’s time and the total cost of shopping at each location

Answers

Ciana should purchase the skirt at the store across town because the total economic cost will be lowest.

How to determine the opportunity cost?

Ciana makes $30 per hour at her work, and her purchase decision includes the opportunity cost of lost wages:

Total economic cost:

Local store = $114 + [1/4 hours x 2 (round trip) x $30] + (1/2 hours x $30 spent shopping) = $144

Across town = $86 + [1/2 hours x 2 (round trip) x $30] + (1/2 hours x $30 spent shopping) = $131

Neighboring city = $60 + [1 hour x 2 (round trip) x $30] + (1/2 hours x $30 spent shopping) = $135

Ciana should buy a skirt at the store across town. Because it has the lowest total economic cost ($131).

Opportunity cost is the lost benefit or additional cost of choosing one activity or investment over another. Economic costs include both accounting costs and opportunity costs.

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Travis buys 4 bags of M&Ms and a bottle of water. The total cost was $10.75. If the bottle of water cost $1.75, which equation can be used to determine the cost of one bag of M&ms?

Answers

Let's call x each bag of M&M, and y each bottle of water. If they cost $10.75, we can express the following

$4x+y=10.75$

If the bottle of water costs $1.75, then we can replace it to find x

$\begin{gathered} 4x+1.75=10.75 \\ 4x=10.75-1.75 \\ 4x=9 \\ x=\frac{9}{4}=2.25 \end{gathered}$Hence, each bag of M&Ms cost $2.25.

Find P 20 of the following arithmetic sequence: 206, 212, 218, 224....
State:
Po =
d =
P20 =

Answers

Answer:

$p_{20} = 1.25528$

P₀ = 206 and difference d = 6

Step-by-step explanation:

Step(i):-

Given that the arithmetic sequence

206,212,218,224,.....

P₀ = 206 and difference d = 212-206 = 6

The $n^{th}$ term of the sequence

$t_{n} = ar^{n-1}$

Step(ii):-

The $n^{th}$ term of the sequence

$P_{n} = ar^{n-1}$

Put n=20

$P_{20} = 206 ( 6) ^{20-1}$

$p_{20} = 1.25528$

Factor.

Write your answer as a product with a whole number greater than 1.

56r+7

Answers

The factored expression of 56r + 7 is 7(8r + 1)

How to write your answer as a product with a whole number greater than 1?

The expression is given as

56r + 7

Express 56 as the product of 8 and 7

So, we have'

56r + 7 = 7 * 8r + 7

Express 7 as the product of 7 and 1

So, we have'

56r + 7 = 7 * 8r + 7 * 1

Factor out 7 from the expression

56r + 7 = 7(8r + 1)

Hence, the factored expression of 56r + 7 is 7(8r + 1)

Find the point on the paraboloid z = 4x2 + 4y2 which is closest to the point (4; 5; 6).

Answers

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Find the point on the paraboloid z = 4x2 + 4y2 which is closest to the point
(4; 5; 6).