Mathematics College

F(x, y, z) = yz i + xz j + (xy + 18z) k c is the line segment from (1, 0, −1) to (5, 6, 1) (a) find a function f such that f = ∇f. f(x, y, z) = (b) use part (a) to evaluate c ∇f · dr along the given curve
c.

Answers

Answer 1
[tex]\nabla f(x,y,z)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j+\dfrac{\partial f}{\partial z}\,\mathbf k=\mathbf f(x,y,z)[/tex]

[tex]\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int yz\,\mathrm dx[/tex]
[tex]\implies f(x,y,z)=xyz+g(y,z)[/tex]
[tex]\dfrac{\partial f}{\partial y}=xz+\dfrac{\partial g}{\partial z}=xz[/tex]
[tex]\implies\dfrac{\partial g}{\partial z}=0[/tex]
[tex]\implies g(y,z)=h(z)[/tex]

[tex]\dfrac{\partial f}{\partial z}=xy+\dfrac{\mathrm dh}{\mathrm dz}=xy+18z[/tex]
[tex]\implies\dfrac{\mathrm dh}{\mathrm dz}=18z[/tex]
[tex]\implies h(z)=9z^2+C[/tex]

[tex]\implies f(x,y,z)=xyz+9z^2+C[/tex]

[tex]\implies\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(5,6,1)-f(1,0,-1)=30[/tex]
Answer 2
Final answer:

Through anti-derivatives, we find a function f(x, y, z) in part (a) of the problem. We then use the fundamental theorem of line integrals in part (b) to evaluate the line integral of ∇f ∙ dr along a curve c, using the function f we found.

Explanation:

This problem lies within the field of vector calculus, specifically dealing with gradient fields and line integrals. Part (a) requires us to find a function f(x, y, z) such that the gradient of f (∇f) is equal to the field F. Since ∇f yields the vector field (Fx, Fy, Fz), we find the anti-derivatives of each component of F to solve for f. For this particular function f, the anti-derivatives are 1/2 * yz^2 + C1 for Fx = yz, 1/2 * xz^2 + C2 for Fy = xz, and 1/2 * xy^2 + 9z^2 + C3 for Fz = xy + 18z. Therefore, the function  that satisfies the requirement is 1/2 * yz^2 + 1/2 * xz^2 + 1/2 * xy^2 + 9z^2 + C, where C is a constant.

Part (b) then requires the evaluation of the line integral of ∇f ∙ dr along a curve c, from point (1,0,-1) to (5,6,1). According to the fundamental theorem of line integrals, if a vector field is conservative (as it is in this case because we found a function f such that F = ∇f), the line integral of a vector field F along a curve c from a to b is simply f(b) - f(a). Therefore, using the function f found in part a, you can substitute the coordinates of points a and b to find the solution to the line integral equation.

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Related Questions

From a group of 3 men and 4 women, a delegation of 2 is selected. let x be the random variable that represents the number of women selected. write the probability distribution and find the expected number of women in the delegation?

Answers

Let random variable x be the delegated number of men.
x values are 0, 1, 2, 3
The corresponding probabilities are:
P(x=0)=  3C2/7C2=3/21
P(X=1)=3C1*4C1/7C2=12/21
P(X=2)=3C2/7C2=3/21
P(X=3)=3C2/7C2=3/21

Hence the expected value will be:
E(x)=[0*3+1*12+2*3+3*3]/21
E(x)=27/21=1.29

Q # 8 please solve the question

Answers

Let the two numbers be x and y

Their sum is 20, so we can write
x + y = 20

Their difference is 14, so we can write:
x- y = 14


Adding the two equations, we get:

2x = 34
x = 17

Using the value of x, in first equation, we get:

17 + y = 20
so,
y = 3

Thus the two numbers are 17 and 3.

So, option a is the correct answer
For this case we have the following system of equations:
 x + y = 20
 x-y = 14
 Where,
 x: real number
 y: real number
 Solving the system we have:
 x = 17
 y = 3
 Therefore, the numbers are:
 17 and 3
 Answer:
 17 and 3
 option 1

Which answer shows a correct order to solve this story problem? Maurine had to pay $2.52 tax on her purchases. She bought 3 books for $12.99 each, 2 pens for $0.69 each, and a magazine for $2.99. How much did Maureen spend? A. Step 1: Add the price of a book and a pen. Step 2: Multiply by 3 and then by 2. Step 3: Add $2.99 to that result. Step 4: Add the tax to the total of the purchases. B. Step 1: Calculate the total price of the 2 pens. Step 2: Add $2.99 to that total. Step 3: Calculate the total price of the 3 books. Step 4: Add the tax to the price of the 3 books C. Step 1: Calculate the total price of the 3 books. Step 2: Calculate the total price of the 2 pens. Step 3: Add $2.99 to the total of the cost of the books and pens. Step 4: Add the tax to the total of the purchases. D. Step 1: Add $2.99 to the amount of the tax. Step 2: Calculate the total price of the 2 pens. Step 3: Calculate the total price of the 3 books. Step 4: Add the prices of the pens to the price of the books.

Answers

The usual choice would be
  C. Step 1: find the price of the books; Step 2: find the price of the pens; Step 3: add the prices of the books, pens, and magazine; Step 4: add the tax.


_____
You can, of course, find and add the constituent prices in any order. Generally, you need to know that subtotal in order to figure the tax, which is then added to that subtotal. If you've ever done any shopping at a store where bills are added manually, or if you pay attention to any cash register receipt, you know this is the process.

What is the distance d across the lake?

Answers

Answer-

The distance d across the lake is 90ft.

Solution-

In the picture,

ΔABC ≈ ΔCDE

Because,

∠ACB = ∠DCE (∵ vertical opposite angles)∠B = ∠D = 90°∠ACB = ∠CED (∵ as other two angles are same and sum of those two angles when subtracted from 180° will result the same)

So, from similarity properties

[tex]\Rightarrow \dfrac{AB}{DE}=\dfrac{BC}{DC}[/tex]

[tex]\Rightarrow \dfrac{120}{20}=\dfrac{d}{15}[/tex]

[tex]\Rightarrow d=\dfrac{120\times 15}{20}=90[/tex]

A lunch in a fancy restaurant is three times as expensive as a lunch in a cafeeria. The lunch in the fancy restaurant costs 36. In five day workweek Mary eats at the fancy restaurant once and in the cafeteria the rest of the days. How much does she spend on lunches in that week

Answers

In this item, we first solve for the cost of eating in a cafeteria. From the given it is stated that the cost of eating in a fancy restaurant is 3 times the cost of eating in cafeteria. Mathematically,

    3x = 36

where x is the cost of eating in cafeteria. 

The value of x from the equation is 12.

In a five-day workweek, she eats at the fancy restaurant once and four times in cafeteria. To determine the total cost,

    T = 4(12) + 36
    
     T = 84

Thus, the answer to this item is 84.
let: 

f = fancy restaurant price
c = cafeteria price

f = 3c

f = 36
c = 36/3 = 12

36 + 12*4 = 84

she spends 84 on lunches for a fiveday workweek

What are the zeros of the function?

F (X) = x^3 - x^2 - 6x

A) -1, 0, and 3

B) -3, 0, and 1

C) -3, 0, and 2

D) -2, 0, and 3

Answers

Try this explanation:
if to re-write the function f(x)=x(x²-x-6), then to make up the equation x(x²-x-6)=0, it is possible to find the zeros requred:
x(x-3)(x+2)=0, x=0;x=-2;x=3.

answer: D -2;0;3
The answer is B. Just finished the test. 

Cards are dealt, one at a time, from a standard 52-card deck. a if the first 2 cards are both spades, what is the probability that the next 3 cards are also spades?

Answers

So we already drew two cards and both of them are spades. So that means there are 50 cards and 11 spades left.

Then third card being spades would be 11 / 50

Now there are 10 spades and 49 cards left.
So the probability for fourth card would be 10 / 49

Fifth card would be 9 / 48.

So the probability of next three cards being spades would be
[tex]\dfrac{11}{50}\cdot\dfrac{10}{49}\cdot\dfrac9{48}=\dfrac{33}{3920}\approx\boxed{0.00842}[/tex]

Hope this helps.

David is making a mosaic using 63 glass tiles 105 ceramic tiles and 84 marble tiles he wants to put the same number of each type of tile and Rose what is the maximum number of times you can have in each row

Answers

The largest number that divides 63, 84, and 105 evenly is 21.

David can put 21 of each kind of tile in a row.

_____
These numbers are 3*21, 4*21, and 5*21, respectively.

Brad has 3 pairs of pants, 7 tee shirts, and 2 pairs of sneakers. How many different outfit combinations can he wear

Answers

HE CAN WEAR 42 DIFFERENT COMBOS

Answer:

42

Step-by-step explanation:


A snow storm left 1.25 feet of snow on the ground overnight. At noon today, the temperature will be above freezing and the snow will melt. If the melting snow is modeled by the equation y = -1 3 x + 1.25, how long will it take for the snow to melt away?

Answers

y = -13x + 1.25
0 = -13x + 1.25
-1.25 = -13x
3.75 = x

then,

3.75 hours  to 3 hours 45 minutes 

What is the angular velocity of a 6–foot pendulum that takes 3 seconds to complete an arc of 14.13 feet? Use 3.14 for π.

Answers

The angular velocity is the linear speed divided by the radius.
  ω = d/(t*r) = (14.13 ft)/((3 s)*(6 ft))
  ω = 0.785 radians/second
There are 180/π degrees per radian, so the angular velocity can also be written as ...
  ω = 45 degrees/second

given that an arithmetic sequence can be expressed as a linear function in the form f(x)=mx+b, write a linear function to describe the sequence {10,15,20,25,30,35...}

Answers

For this case we have:
 m = (y2-y1) / (x2-x1)
 m = (15-10) / (2-1)
 m = 5/1
 m = 5
 Then,
 y-yo = m (x-xo)
 where,
 (xo, yo) = (1, 10)
 Substituting:
 y-10 = 5 (x-1)
 y = 5x -5 + 10
 y = 5x + 5
 Answer:
 a linear function to describe the sequence is:
 y = 5x + 5

Answer:

a linear function to describe the sequence is:

y = 5x + 5

Step-by-step explanation: ik ur on usa te$t prep

Two graphs are shown below:

graph A shows scatter points 1, 1 and 2, 2 and 3, 3 and 4, 4 and 5, 5graph A shows scatter points 1, 4 and 2, 4 and 3, 4 and 4, 4 and 5, 4

Which scatter plot shows a positive correlation between x and y?

Only graph A
Only graph B
Both graph A and graph B
Neither graph A nor graph B

Answers

The answer would be graph A because positive correlation is when the X and Y axis advance together. 

The correct answer is graph A

Find the length of JK and the measure of M

Answers

Comment
You have two different situations, but both of them are solved the same way.

Step One
Solve for the 2 equal sides.
21x - 9 = 3x + 27   Add 9 to both sides.
21x = 3x + 27 + 9
21x = 3x + 36       Subtract 3x from both sides.
21x - 3x = 36
18x = 36               Divide by 18
x = 36/18
x = 2

That's not quite the answer

Step Two
Find JK
JK = 21x - 9
JK = 21*2 - 9
JK = 42 - 9
JK = 33  <<<<<<< Answer

Step Three 
Solve for y
5y + 11 = 12y - 10 Subtract 5y from both sides.
11 = 12y - 5y  - 10
11 = 7y - 10          Add 10 to both sides.
11 + 10 = 7y     
21 = 7y                 Divide by 7
21/7 = y
3 = y

Step Four
Solve for M
M = 12y - 10
M = 12*3 - 10
M = 36 - 10
M = 26  <<<<< Answer
JK = 33
M = 74

Hope this Helped!

;D
Brainliest??

If the numerator of a fraction is increased by 3, the fraction becomes 3/4. If the denominator is decreased by 7, the fraction becomes 1. Determine the original fraction. Which of the following equations represents "If the numerator of a fraction is increased by 3, the fraction becomes 3/4"?

Answers

For this case, the original fraction is:
 x / y
 Where,
 x = numerator
 y = denominator:
 If the numerator of a fraction is increased by 3, the fraction becomes 3/4:
 (x + 3) / y = 3/4
 If the denominator is decreased by 7, the fraction becomes 1:
 x / (y-7) = 1
 Solving the system of equations we have:
 x = 9
 y = 16
 The original fraction is:
 9/16
 Answer:
 the original fraction is:
 9/16
 "If the numerator of a fraction is increased by 3, the fraction becomes 3/4" is:
 (x + 3) / y = 3/4

Question# 12, find the slope

Answers

From the graph we can see that the line is passing through following two points:

(0,1) and (-4, -2)

Slope = m = (-2 - 1)/(-4 -0 ) = -3/-4 = 3/4

So the slope of given line is 3/4

Thus the correct answer is option B
The slope is given by:
 m = (y2-y1) / (x2-x1)
 We choose the ordered pairs:
 (x1, y1) = (0, 1)
 (x2, y2) = (4, 4)
 Substituting values we have:
 m = (4-1) / (4-0)
 m = (3) / (4)
 m = 3/4
 Answer:
 The slope for this case is given by:
 m = 3/4
 option 2

A display that shows how the values in a data set are distributed

Answers

Answer: Line Plot

Step-by-step explanation: A line plot is a display that shows how the values in a data set are distributed.

Answer: histogram

Step-by-step explanation:

Regina's mom bought her some bracelets. There were a total of 85 bracelets in 5 packages.

Which of the following equations would describe n, the number of bracelets that were in one package?

A. n + 5 = 85

B. n ÷ 5 = 85

C. n - 5 = 85

D. n × 5 = 85

Answers

The answer your looking for is B

find 1/9 as many as 3 divided by 5

Answers

Final answer:

To find 1/9 as many as 3/5, you multiply 3/5 by 1/9 to get 3/45, which simplifies to 1/15.

Explanation:

To solve the student's question, we need to find what is 1/9 as many as 3/5. This involves multiplying the fraction 3/5 by the fraction 1/9. Here is the step-by-step calculation:

Multiply the numerator of the first fraction (3) by the numerator of the second fraction (1) to get the numerator of the answer.Multiply the denominator of the first fraction (5) by the denominator of the second fraction (9) to get the denominator of the answer.Combine these to get the answer in fractional form.

The calculation will look like this: (3/5) × (1/9) = 3/45 = 1/15.

Thus, 1/9 as many as 3/5 is 1/15.

Quadrilateral has angles 45 degrees, 150 degrees, 65 degrees, and N degrees

Answers

n=100
other angles=260
all interior angles of a quadrilateral add up to 360

Find a unit vector that is perpendicular to the vector 5, −4

Answers


[tex]( \frac{4}{ \sqrt{41} } , \frac{5}{ \sqrt{41} } )[/tex]

What is the ratio (fraction) of minivans to all vehicles on the forecourt?
A)1/120
B)1/20
C)1/6
D)1/4

Answers

there is a total of 120 and 20 minivans so
20/120
1/6
I hope this helps

do u do keystone

? im just wondering i do

Ezra is training for a track race. He starts by sprinting 100 yards. He gradually increases his distance, adding 5 yards a day for 21 days.

Answers

On the 21st day, he sprints 205 yards.

Is that what you meant for the problem

or do u want to know how many yards he sprinted total for the 21 days
If it is,
he sprinted a total of 3390 yards

On Apex the correct one is 200 yards.

hey can you please help me posted picture of question

Answers

The correct answer is the Option C, which is:

 C. G(x)=(x-2)^3

 The explanation is shown below:

 1. You have the the red graph has the equation F(x)=x^3
 2. If you shifted two units to the right, you must add -2 to the function x^3, as below:

 G(x)=(x-2)^3

 3. If you want to verify this is correct, you need to give values to the variable x and plot each point.
 4. You will see that the correct answer is the option mentioned above.

Given: triangle ABC, AB = BC Perimeter of ΔABC = 50 Perimeter of ΔABD = 40 Find: BD

Answers

The answer is 15. You can use a equation 2x+2y=50 for ABC  and so then x+y=25. x+y+h=40(ABD), so h is 15 units.

The measure of angle ABC and BDC is 80° and 90° respectively.

What is the triangle?

In geometry, a triangle is a closed, two-dimensional shape with three straight sides. A triangle is also a polygon.

Given the ΔABC in which AB ≅ BC, m∠ABD = 40° and BD is the median of ΔABC.

We have to find the measure of angle ∠ABC and ∠BDC.

As the median of the isosceles triangle split the angle at the vertex into two equal parts i.e ∠ABC is twice the angle ∠ABD.

[tex]\rm \angle ABC=2\angle ABD\\\\\angle ABC=2 \times 40\\\\\angle ABC=80[/tex]

Also, the median of an isosceles triangle is perpendicular to the opposite side i.e to the base. Here, BD is perpendicular to AC.

∠BDC=90°

Therefore, the measure of angle ABC and BDC is 80° and 90° respectively.

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Nan and sato are designing a coffee table would you sing 4 inches tile Nan uses 30 tiles and sato uses half as many many total tiles did they use

Answers

the correct question in the attached figure

it is assumed that the tile is square
area of the square=b²
b is the length side of a square
b=4 in
area of the tile=4²----> 16 in²

Part 1)  How many total tiles did they use?
Nao uses 30 tiles
and 
Sato uses (1/2)*30=15
so
Nao and Sato uses----> 30+15----> 45 tiles

the answer Part 1) is
45 tiles

Part 2) If the area of the table is 36 inches by 26 inches, will they have enough tiles for the table?

find the area of the table
area of the table=36*26----> 936 in²

area of one tile=4*4----> 16 in²

if 1 tile------------------> 16 in²
x tiles-------------> 936 in²
x=936/16----> x=58.5 tiles

58.5 > 45
so
there are not enough tiles for the table

the answer part 2) is
there are not enough tiles for the table

Part 3)  If not, how many more will they need?

tiles required=58.5-45------> 13.5-----> 14 tiles

the answer part 3) is
14 tiles

Nan and Sato together used 45 tiles for their coffee table design.

The student's question involves calculating the total number of tiles used by two individuals, Nan and Sato, who are designing a coffee table. Nan uses 30 tiles, and Sato uses half as many tiles as Nan does. To find out how many tiles Sato uses, we divide Nan's number of tiles by 2.

So, Sato uses 15 tiles because half of 30 is 15. To find out the total number of tiles used, we add Nan's tile count to Sato's tile count:

30 tiles (Nan) + 15 tiles (Sato) = 45 tiles in total.

Therefore, to determine the total number of tiles used, Nan's 30 tiles are added to Sato's 15 tiles (half of Nan's), resulting in a total of 45 tiles used for the coffee table design.

PLEASE HELP!

1. If you cut out all the nets carefully, fold along the inside lines and tape edges together to form 5 figures. Describe the figures formed. What shapes are they? Are any the same?

2. What does the fact that the figures fit together to make the prism tell you about their combined volume?

Answers

Well, first of all, A was the only one visibly different as in shape.
The first picture is A. The rest should be in order like this: A, B, E, D,C.
Yes I know I did a crummy job at taping but hey, I tried.

An electrolyte solution has an average current density of 111 ampere per square decimeter \left( \dfrac{\text{A}}{\text{dm}^2}\right)( dm 2 A ​ )left parenthesis, start fraction, A, divided by, d, m, start superscript, 2, end superscript, end fraction, right parenthesis. What is the current density of the solution in \dfrac{\text{A}}{\text{m}^2} m 2 A ​ start fraction, A, divided by, m, start superscript, 2, end superscript, end fraction?

Answers

Average current density of the solution = 1 ampere per square decimeter

We have to find the average current density of the solution in ampere per square meter.

1 decimeter = [tex] \frac{1}{10} [/tex] meter
Squaring both sides we get

1 decimeter square = [tex] \frac{1}{100} [/tex] meter square

Current density = [tex]1 \frac{A}{dm^{2} } =1 \frac{A}{ \frac{1}{100} m^{2} } =100Am^{-2} [/tex]

Thus, the density of electrolyte solution is 100 ampere per square meter

Answer:

Answer is 100

Step-by-step explanation:

A theater received $8,917 in ticket sales for one performance of a play. Tickets for preferred seats cost $45 each and regular seats cost $34 each. All of the theaters 75?preffered seats were sold out that evening. How many tickets were sold?

Answers

Let
P = Number of ticket for preferred seats costing $45 per seat
R = Number of regular seats costing $34 per seat.

Total revenue = P*45 + R*34 =8,917

45P + 34R = 8917

But, number of preferred seats is 75 or P=75 seats
Then,
45*75 +34R = 8917
3375 +34R = 8917
34R = 5542
R = 5542/34 = 163

Total number of seats = P+R = 75+163 = 238 seats. If each seat is sold for one ticket, then 238 tickets wee sold.

what is the value of x sin x + 11 degrees equals cos 2x + 22 degrees enter your answer in the Box

Answers

In geometry, if the angles have the same values of sine and cosine, it implies that the angles are complementary which further means that the sum of their measures should be 90°. With the given expressions for the angles above,

     (x + 11) + (2x + 22) = 90

Simplifying,

    3x + 33 = 90

Further simplification,
    3x = 90 - 33 = 57

The value of x from the equation is 19.

Answer: x = 19

Answer:

it is 19

Step-by-step explanation:

I just took this test and got 100 :)

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