Find a potential function for the vector field

Find A Potential Function For The Vector Field

Answers

Answer 1

(a) We want to find a scalar function [tex]f(x,y,z)[/tex] such that [tex]\mathbf F = \nabla f[/tex]. This means

[tex]\dfrac{\partial f}{\partial x} = 2xy + 24[/tex]

[tex]\dfrac{\partial f}{\partial y} = x^2 + 16[/tex]

Looking at the first equation, integrating both sides with respect to [tex]x[/tex] gives

[tex]f(x,y) = x^2y + 24x + g(y)[/tex]

Differentiating both sides of this with respect to [tex]y[/tex] gives

[tex]\dfrac{\partial f}{\partial y} = x^2 + 16 = x^2 + \dfrac{dg}{dy} \implies \dfrac{dg}{dy} = 16 \implies g(y) = 16y + C[/tex]

Then the potential function is

[tex]f(x,y) = \boxed{x^2y + 24x + 16y + C}[/tex]

(b) By the FTCoLI, we have

[tex]\displaystyle \int_{(1,1)}^{(-1,2)} \mathbf F \cdot d\mathbf r = f(-1,2) - f(1,1) = 10-41 = \boxed{-31}[/tex]

[tex]\displaystyle \int_{(-1,2)}^{(0,4)} \mathbf F \cdot d\mathbf r = f(0,4) - f(-1,2) = 64 - 41 = \boxed{23}[/tex]

[tex]\displaystyle \int_{(0,4)}^{(2,3)} \mathbf F \cdot d\mathbf r = f(2,3) - f(0,4) = 108 - 64 = \boxed{44}[/tex]


Related Questions

emily is standing 150 feet from a circular target with radius 3 inches. will she hit the target if her aim is off by 0.2 degrees?

Answers

Answer:

  no

Step-by-step explanation:

The angle subtended by the radius of the target at Emily's distance can be found using the tangent relation.

Application

  tan(α) = opposite/adjacent = (1/4 ft)/(150 ft) = 1/600

The angle is found using the inverse relation:

  α = arctan(1/600) ≈ 0.095°

If Emily's aim is off by 0.2°, she will miss the target by several inches.

__

Additional comment

Emily's projectile will miss her aiming point by ...

  (150 ft)tan(0.2°) ≈ 0.524 ft ≈ 6.28 in

what is 7/8 - 1/2 as a fraction

Answers

Answer: 3/8

Step-by-step explanation:

1/2 is also 4/8 so you subtract 4/8 from 7/8 to get 3/8

One company estimates same-day delivery as more than three less than half the total number of miles. Which graph represents the overall equation represented by this scenario (all points may not apply to the scenario)? Note that the graphs have miles for the independent variable on the x-axis, and the y-axis is a unit of time dependent on the number of miles.

Answers

By using an online graphing calculator, the overall equation represented by this scenario is plotted in the image attached below.

How to determine the graph?

Based on the information provided about this company with respect to its delivery and total number of miles to be covered, we would assign variables as follows:

Let y represent same-day delivery.Let x be the total number of miles.

Next, we would translate the word problem into an algebraic equation by using an inequality:

y > x/2 - 3

Also, we would determine the intercepts as follows:

When x = 0, we have;

y = 0/2 - 3

y = -3.

y-intercept = (0, -3).

For the x-intercept, we have:

When y = 0, we have;

0 = x/2 - 3

x/2 = 3

x = 6.

x-intercept = (6, 0).

By using an online graphing calculator, the overall equation represented by this scenario is plotted in the image attached below.

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Answer:

c

Step-by-step explanation:

c is the answer

So,some please help me with this question !

Answers

Answer:  63 degrees  (choice A)

=============================================================

Explanation:

Angles EBA and DBC are congruent because of the similar arc marking. Both are x each.

Those angles, along with EBD, combine to form a straight angle of 180 degrees. We consider those angles to be supplementary.

So,

(angleEBA) + (angleEBD) + (angleDBC) = 180

( x ) + (4x+12) + (x) = 180

(x+4x+x) + 12 = 180

6x+12 = 180

6x = 180-12

6x = 168

x = 168/6

x = 28

Angles EBA and DBC are 28 degrees each.

This means angle D = 3x+5 = 3*28+5 = 89

-----------

Then we have one last set of steps to finish things off.

Focus entirely on triangle DBC. The three interior angles add to 180. This is true of any triangle.

D+B+C = 180

89 + 28 + C = 180

117+C = 180

C = 180 - 117

C = 63 degrees

[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]

[tex]\qquad❖ \: \sf \: \angle C = 63 \degree[/tex]

[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]

[tex]\qquad❖ \: \sf \:x + 4x + 12 + x=180°[/tex]

( Angle EBA= Angle DBC, and the three angles sum upto 180° due to linear pair property )

[tex]\qquad❖ \: \sf \:6x + 12 = 180[/tex]

[tex]\qquad❖ \: \sf \:6x = 180 - 12[/tex]

[tex]\qquad❖ \: \sf \:6x = 168[/tex]

[tex]\qquad❖ \: \sf \:x = 28 \degree[/tex]

Next,

[tex]\qquad❖ \: \sf \: \angle C + \angle D + x = 180°[/tex]

[tex]\qquad❖ \: \sf \: \angle C +3x + 5 + x = 180°[/tex]

[tex]\qquad❖ \: \sf \: \angle C +4x = 180 - 5[/tex]

[tex]\qquad❖ \: \sf \: \angle C +4(28) = 175[/tex]

( x = 28° )

[tex]\qquad❖ \: \sf \: \angle C +112 = 175[/tex]

[tex]\qquad❖ \: \sf \: \angle C = 175 - 112[/tex]

[tex]\qquad❖ \: \sf \: \angle C = 63 \degree[/tex]

[tex] \qquad \large \sf {Conclusion} : [/tex]

[tex]\qquad❖ \: \sf \: \angle C = 63 \degree[/tex]

From the diagram below, if the tree is 34 ft. tall, and the angle of elevation from point B to the top of the tree is 26 °, find the distance that the tree is from point B. (Round to the nearest whole foot.)

Answers

Given the height of the tree and the angle of elevation from point B, the distance between the tree is from point B is approximately 70ft.

What is the distance between the tree and point B?

Given the data in the question;

Height of tree opposite angle of elevation = 34ftAngle of elevation θ = 26°Distance between tree and point B| Adjacent = ?

Since the scenario form a right angle triangle, we use trig ratio.

tanθ = Opposite / Adjacent

tan( 26° ) = 34ft / x

We solve for x
x = 34ft / tan( 26° )

x = 34ft / 0.4877

x = 70ft

Given the height of the tree and the angle of elevation from point B, the distance between the tree is from point B is approximately 70ft.

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is there anybody to solve this need step by step solution details

Answers

The circumference of the circles from biggest to smallest are; 30π, 15π and 7.5π

How to find the area of a circle?

The formula for area of a circle is;

A = πr²

In geometry, the area enclosed by a circle of radius r is πr². Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

The area of a circle formula is useful for measuring the region occupied by a circular field or a plot. Suppose, if you have a circular table, then the area formula will help us to know how much cloth is needed to cover it completely.

The area formula will also help us to know the boundary length i.e., the circumference of the circle. Does a circle have volume? No, a circle doesn't have a volume

Area of entire circle is 706 cm².

Thus;

πr² = 706

r² = 706/π

r = √(706/π)

r ≈ 15

Circumference of biggest circle = 2πr

Circumference of biggest circle = 2π * 15

Circumference of biggest circle = 30π

Circumference of second biggest circle = 2π * (15/2)

Circumference of second biggest circle = 15π

Circumference of smallest circle = 2π * (15/4)

Circumference of smallest circle = 7.5π

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6. Find the values of x and y. 30° 17

Answers

Answer:

X = 17√3

Y = 34

Step-by-step explanation:

TO calculate the values of X and y we use SOHCAHTOA.

Calculating for X

tan 30 = opposite/adjacent

tan30 = 17/X

Cross multiply

tan30 ×X = 17

Divide bothsides by tan30

Note: tan30= 1/√3

X = 17/1/√3

X=17√3

X= 29.4

X = 29 ( approximately)

Calculating for y

[tex]sin30 = \frac{Opposite}{Hypothenus} \\ \\ sin30 = \frac{17}{y} [/tex]

Cross multiply

[tex]sin30 \times y = 17[/tex]

Divide bothsides by sin30

[tex] \frac{sin30 \times y}{sin30} = \frac{17}{sin30} \\ \\ y = \frac{17}{0.5} = 34 \\ y = 34[/tex]


[tex] \: \: \: \: \: \: \: \: \: \: \: n\\ evaluate \: \: \: \: \: Σ \: (nCi)\\ \: \: \: \: \: \: \: \: \: \: \: \: i = 0[/tex]
Evaluate the summation​

Answers

Assuming you mean

[tex]\displaystyle \sum_{i=0}^n {}_nC_{i}[/tex]

where

[tex]{}_n C_i = \dbinom ni = \dfrac{n!}{i! (n-i)!}[/tex]

we have by the binomial theorem

[tex]\displaystyle (1 + 1)^n = \sum_{i=0}^n {}_nC_{i} \cdot 1^i \cdot 1^{n-i}[/tex]

so that the given sum has a value of [tex]\boxed{2^n}[/tex].

A restaurant manager has the option of a 30-year loan of $417,000 at an annual interest rate of 3.85% or the same interest rate but on a loan for 15 years.
(a)
Calculate the monthly payment for each loan. (Round your answers to the nearest cent.)
30-year $
15-year $
(b)
Calculate the savings in interest by using the 15-year loan. (Round your answer to the nearest cent.)
$
(c)
The term of the 15-year loan is one-half the term of the 30-year loan. Is the monthly payment for the 15-year loan twice that of the 30-year loan?
Yes
No
(d)
Is the interest savings for the 15-year loan more or less than one-half of the interest paid on the 30-year loan?
more
less

Answers

a) The monthly payment for each loan is as follows:

30-year $1,954.93

15-year $3,053.25

b) The savings in interest by using the 15-year loan is $154,189,20 ($286,774.20 - $132,585).

c) No. the monthly payment for the 15-year loan is not twice that of the 30-year loan as the loan term.

d) The interest savings for the 15-year loan are more than one-half of the interest paid on the 30-year loan.

How are the calculations for periodic payments done?

The calculations for the monthly payments, including interests can be carried out using an online finance calculator, as follows:

30-year Loan:

N (# of periods) = 360 months (12 x 30 years)

I/Y (Interest per year) = 3.85%

PV (Present Value) = $417000

FV (Future Value) = $0

Results:

PMT = $1,954.93

Sum of all periodic payments = $703,774.80 ($1,954.93 x 360)

Total Interest = $286,774.20 ($703,774.80 - $417,000)

15-year Loan:

N (# of periods) = 180 months (12 x 15 years)

I/Y (Interest per year) = 3.85%

PV (Present Value) = $417000

FV (Future Value) = $0

Results:

PMT = $3,053.25

Sum of all periodic payments = $549,585 ($3,053.25 x 180)

Total Interest = $132,585 ($549,585 - $417,000)

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a box with a square base has length plus girth of 108 in. What is the length of the box if its volume is 2200 in^3?

Answers

The missing length of the box with square base is 20.370 inches.

What is the missing length of the box?

This box can be represented by a right prism, whose volume (V), in cubic inches, is the product of the area of the base (A), in square inches, and the height of the box (h), in inches. The area of the base is equal to the square of the side length (l):

V = l² · h      (1)

If we know that V = 2 200 in³ and l = 108 in, then the height of the box is:

h = V / l²

h = (2 200 in³) / (108 in)

h = 20.370 in

The missing length of the box with square base is 20.370 inches.

Remark

The statement presents typing mistakes, correct form is shown below:

A box with a square base has a side length of 108 in. What is the length of the box if its volume is 2 200 in³?

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As part of a summer internship, five people -- Cindy, Damaris, Eugenio, Fareed, and Guzal -- are to be assigned to floors 1-5 in a dormitory. Each person will occupy his or her own entire floor and no other people will be in the dormitory. The assignment of people to floors must follow the following rules: Eugenio lives immediately above Damaris. Cindy is not on the first floor. Fareed does not live immediately below Damaris. Guzal lives either on the first floor or the fifth floor. Which one of the five people could be assigned to live on any of the five floors in the dormitory?

Answers

Based on the information, it is expected Fareed can be assigned to live on any of the floors.

What condition limits Fareed's assignment?

The only condition that limits the floor Fareed can be assigned to is that he cannot be immediately below Damaris. This means that as long as Damaris is not immediately above him, he can be assigned to all floors. Here are two possible arrangements:

Eugenio (5th floor)DamarisCindyFareedGuzal (1st floor)

Guzal (5th floor)EugenioDamarisCindyFareed (1st floor)

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A COUPLE PLAN TO HAVE SIX CHILDREN. HOW MANY POSSIBLE OUTCOMES ARE IN
THE SAMPLE SPACE?

Answers

Each child has 2 possibilities: boy or girl

2*2*2*2*2*2 =2^6 =64

please help :,)
(geometry involving arc lengths)

Answers

Answer:

bx/a (3rd choice)

Step-by-step explanation:

The length of an arc, s, in a circle of radius r, and central angle Θ given in radians is

s = rΘ

Circle M has radius a. The central angle in radians of the sector is Θ. The length of the arc is x.

x = aΘ

Solve for Θ:

Θ = x/a      Eq. 1

Circle N has radius b. The central angle in radians of the sector is Θ. The length of the arc is s.

s = bΘ

Solve for Θ:

Θ = s/b      Eq. 2

Since Θ = Θ, then equate the right sides of Equations 1 and 2 above.

x/a = s/b

Multiply both sides by ab.

abx/a = abs/b

bx = as

as = bx

s = bx/a

Answer: bx/a

\left(a-b\right)\sqrt{\frac{2}{a^2-b^2}}

Answers

Answer:

[tex]\left(a-b\right)\sqrt{\frac{2}{a^2-b^2}}[/tex]

=

[tex]\frac{ \sqrt{2} }{a + b} [/tex]

Step-by-step explanation:

[tex]\left(a-b\right)\sqrt{\frac{2}{a^2-b^2}}[/tex]

[tex](a - b)( \frac{ {2}^{ \frac{1}{2} } }{(a + b)(a - b)} )[/tex]

[tex] \frac{(a - b)( {2}^{ \frac{1}{2} }) }{(a - b)(a + b)} = \frac{ \sqrt{2} }{a + b} [/tex]

Choose the equation that
matches the graph.
a. (1/2)x-4
b. y = 4x-1
C.
y = (1/1)
(¹²) ² − 4
-
d.
e.
x-4
y = (1/1)
y = 4x + 1
y = 4x+1

Answers

Answer: d

Step-by-step explanation:

The graph is increasing, so the base must be more than 1.

Eliminate a and c.

Also, the y-intercept is 2.

Eliminate b and e.

This leaves d.

please help with the first one

Answers

Answer:

  y = x +5

Step-by-step explanation:

The equation of a parallel line will have the same x- and y-coefficients, but a different constant. You need to find the constant that makes the line go through the given point.

Application

The given line is y = x +1. The equation you're looking for is

  y = x +c . . . . . . for some new constant c

We want this equation to be true for (x, y) = (-2, 3). Putting these values into the equation, we can solve for c.

  3 = -2 +c

  5 = c . . . . . . add 2

The desired equation is y = x +5.

A worker uses a small table in a notebook to track the cost of materials. On the current job he has used seven sheets of plywood at $43.75 each, two boxes of screws at $4.69 each and a tube of adhesive which cost $9.89, as shown below.

Answers

The total cost of the materials calculated by the worker on the notebook is $325.52

What is an equation?

An equation is an expression that shows the relationship between two or more numbers and variables.

An independent variable is a variable that does not depend on any other variable for its value while a dependent variable is a variable that depends on other variable.

The total cost of materials = 7(43.75) + 2(4.69) + 9.89 = $325.52

The total cost of the materials calculated by the worker on the notebook is $325.52

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Can I just get the answer for this I feel like it’s wrong.

Answers

The initial membership fee for Club A is; $2.5.

The initial membership fee for Club B is; $3.

Hence, Club A has a lower initial membership fee.

What is the initial membership fee for each Club?

It follows from the concepts of linear graphs that the interpretation of the initial membership fee is the y-intercept of the graphs given.

This is so because, it corresponds to the total cost at a point when the number of movies watched is; 0. That is, before any movie is watched.

Consequently, the graph calibrations (including the missing Club A graph) allow the determination of the initial membership fee (y-intercept) as declared above.

Ultimately, Club A has a lower initial membership fee.

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Pls answer these questions ASAP

Answers

From the given descriptions and dimensions, we have;

6. 2000 cm²

7. 81 m²

8. a. 138 m²

b. 538 m²

c. 62 cm²

9. $107.25

Which method can be used to calculate the surface areas and cost?

6. Type of box = Open box with no lid

Side length of the box = 20 cm

Therefore;

Number of faces of the box, painted = 5

Area of a face of the box = 20² = 400

Area of the surface painted, A = 5 × 400 cm² = 2000 cm²

7. Area of the plastic to cover one book, A, is found as follows;

A = 2 × 20 × 15 + 2 × 3 × 15 + 2 × 20 × 3

Which gives;

A = 600 + 90 + 120 = 810

The area of plastic needed to cover 1000 books is therefore;

Total area = 1,000 × 810 cm² = 810,000 cm²

Conversion of cm² to gives;

1 m² = 10,000 cm²

Therefore;

A = 810,000 cm² = 810,000 cm² × 1/10,000 m²/cm² = 81 m²

The area of plastic needed to cover 1000 books is A = 81 m²

8. The surface areas are;

a. A = 2 × 3 × 6 + 5 × 6 + 7 × 6 + 2×3×(3+7)/2 = 138

The surface area is 138 m²

b. A = 2 × 15 × 5 + 2×(10×5 + (6×8)/2) + 2×8×15 = 538

A = 150 + 148 + 240 = 538

The surface area is 538 m²

c. From the congruent marks, we have;

A = 5 × 3 × 3 + 5×1×3 + 2×1×1 = 62

The surface area is 62 cm²

9. From the congruent markings, we have;

A = 2 × (1.5 × 1.5 + (1.5 × 1.3)/2) + 5×2×1.5

A = 6.45 + 15 = 21.45

Surface area of a the canvas, A = 21.45 m²

Cost per unit area of the canvas, c = $5

Cost of the canvas for the tent, C, is therefore;

C = $5/m² × 21.45 m² = $107.25

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PLEASE HELP ALOT OF POINTS

Answers

Answer: B

Step-by-step explanation:

7x+28=7(x+4) how many solutions

Answers

The equation 7x + 28 = 7(x + 4) has infinite many solutions

What are linear equations?

Linear equations are equations that have constant average rates of change, slope or gradient

How to determine the number of solutions?

A system of linear equations is a collection of at least two linear equations.

In this case, the equation is given as

7x + 28 = 7(x + 4)

Open the bracket

7x + 28 = 7x + 28

Subtract 7x from both sides of the equation

28 = 28

Subtract 28 from both sides of the equation

0 = 0

The above equation is true, because both sides of the equation are the same

This means that the equation has infinite many solutions

Hence, the equation 7x + 28 = 7(x + 4) has infinite many solutions

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In 2016, the city of Rio de Janeiro had a population density of 5377 people/km2

What was the population density of Rio de Janeiro in people per square meter

Answers

665 hon hola Que es lo q es hi

2/120=X/80=Y/300

x=? & y=?

Answers

X=1.33

Y=4.9875;approx. 5.0

Answer:

(2/120=X/80)=Y/300

2×80=120×X

X=160/120

X=1.33

X/80=Y/300

But X=1.33

By substituting

X=1.33, We have that

1.33/80=Y/300

80×Y=1.33×300

80Y=399

Y=399/80

Y=399/80

Y=5.0 approx.

The population of a rabbit colony triples every 3 days. The population starts at 10 rabbits. Write an exponential function that will model the population of the colony after, t, days have passed.

Answers

Answer:

DO not working in you

Step-by-step explanation:

PLEASE help me in answering

Answer:

Step-by-step explanation:

The population at various days is as follows since population triples every 3 days

Day     Population

0          10

3          30

6          90

9          270

....         ......

This can be modeled by the general equation

[tex]n_{t} = n_{0}(r)^{t/k}[/tex]

where

[tex]n_{t}[/tex] is the population after t days

[tex]n_{0}[/tex] is the population at start (10)

[tex]r[/tex] is the rate at which population changes ie 3

[tex]t[/tex] is the number days from start

[tex]k[/tex] is the number of days at which the population triples(here k =3 days)

We can check this by plugging in values for each of the variables

At day 0, population = 10(3)⁰ = 10. 1 = 10

Similarly populations for days 3, 6, 9 are:

[tex]\\\\10.3^{3/3} = 10. 3^1 = 10.3 = 30\\10.3^{6/3} = 10. 3^2 = 10.9 = 90\\\\10.3^{9/3} = 10. 3^3 = 10.27 = 270[/tex]

if x varies directly as y, and
X = 24 when y= 21, Find x when
Y=6

Answers

Answer:

x = 48/7

Step-by-step explanation:

There's two good ways to do this problem.

Option 1:

Translate "x varies directly as y" into the equation y=kx

Then you have to find k. After you "reset" your y=kx equation, fill in k and then solve for x. See image.

Option 2:

Translate "varies directly" into a proportion, which is two fractions equal to each other:

x/y = x/y

Fill in the three numbers given and cross multiply and solve to find the fourth number. See image.

will give brainliest

Answers

The answer you are looking for is 6

Find X:
J:58
K:70
L:111
M:121

Answers

Answer:

L

Step-by-step explanation:

interior angle formula so

(number of sides - 2)*180

2*180 = 360

you can also just take an example from a square 4*90 = 360

360-121-58-70 = 111

PLEASE HELP ME OUT WITH THIS QUESTION!

Answers

Answer: Choice 3

Step-by-step explanation:

Oof, this is a tough one. But anyways

Angle 19 + Angle 20 = 180

So Angle 20 = 180 - Angle 19

As l and m are parallel, Angle 20 = Angle 6 from the z rule

So Angle 6 = 180 - Angle 19

Substitute the information given at the start into the equation

2x - 4 = 180 - (5x - 8)

180 - 5x + 8 = 2x - 4

7x = 192

x = 27.42

From what we have found earlier, Angle 20 = Angle 6 so

Angle 20 = 2x - 4 = 2 * 27.42 - 4 = 50.85 rounding it to 50.9

So the answer is 50.9 or Choice 3

Melissa is putting money into a checking account. Let y represent the total amount of money in the account (in dollars). Let x represent the number of weeks Melissa has been adding money. Suppose that x and y are related by the equation y = 550+20x.

Answer the questions below. Note that a change can be an increase or a decrease. For an increase, use a positive number. For a decrease, use a negative number.
What is the change per week in the amount of money in the account?
What was the starting amount of money in the account?​

Answers

Check the picture below.

The graph shows a system of inequalities.

Graph of two inequalities. One is a dashed line increasing from left to right passing through negative 5 comma 0 and 0 comma 5, and it has shading above the line. The second is a dashed upward opening parabola with a vertex at negative 2 comma negative 9 and x-intercepts at negative 5 comma 0 and 1 comma 0. This parabola is shaded on the inside.

Which system is represented in the graph?

y > x2 + 4x – 5
y > x + 5
y < x2 + 4x – 5
y < x + 5
y ≥ x2 + 4x – 5
y ≤ x + 5
y > x2 + 4x – 5
y < x + 5

Answers

The system of inequalities having the given feasible regions and points at (-5, 0), and (0, 5), and (-2, -9), (-5, 0), and (1, 0) is the option;

y > x² + 4•x - 5

y > x + 5

How can the required system of inequalities be found?

The coordinates of the given points on the linear inequality are;

(-5, 0), and (0, 5)

Slope, m, of the linear inequality is therefore;

m = (5 - 0)/(0 - (-5)) = 1

The equation of the inequality is therefore;

y - 0 > 1•(x - (-5)) = x + 5

Which gives;

y > x + 5

Quadratic Inequality;

The coordinates of the points on the quadratic inequality are;

Vertex point = (-2, -9)

x-intercept = (-5, 0), and (1, 0)

Therefore, we have;

y > (x + 5)•(x - 1) = x² + 4•x - 5

y > x² + 4•x - 5

The correct option is therefore;

y > x² + 4•x - 5

y > x + 5

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Answer:

y > x2 + 4x – 5

y > x + 5

Step-by-step explanation:

If you put this system into desmos you will see that it matches the photo shown in the question, and it would need to be both greater than symbols because the dotted lines indicate that it is not included in the solution.

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