Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. (If the vector field is not conservative, enter DNE.)
F(x, y) = (2x − 4y) i + (−4x + 10y − 5) j
f(x, y) =
Answers
The vector field F(x, y) = (2x - 4y) i + (-4x + 10y - 5) j is a conservative vector field. The function f(x, y) that satisfies ∇f = F is f(x, y) = \(x^{2}\) - 4xy + 5y + C, where C is a constant.
To determine whether a vector field is conservative, we check if its curl is zero. If the curl is zero, then the vector field is conservative and can be expressed as the gradient of a scalar function.
Let's calculate the curl of F = (2x - 4y) i + (-4x + 10y - 5) j:
∇ x F = (∂F₂/∂x - ∂F₁/∂y) i + (∂F₁/∂x - ∂F₂/∂y) j
= (-4 - (-4)) i + (2 - (-4)) j
= 0 i + 6 j
Since the curl is zero, F is a conservative vector field. Therefore, there exists a function f such that ∇f = F.
To find f, we integrate each component of F with respect to the corresponding variable:
∫(2x - 4y) dx = \(x^{2}\) - 4xy + g(y)
∫(-4x + 10y - 5) dy = -4xy + 5y + h(x)
Here, g(y) and h(x) are arbitrary functions of y and x, respectively.
Comparing the expressions with f(x, y), we see that f(x, y) = \(x^{2}\) - 4xy + 5y + C, where C is a constant, satisfies ∇f = F.
Therefore, the function f(x, y) = \(x^{2}\) - 4xy + 5y + C is such that F = ∇f, confirming that F is a conservative vector field.
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Related Questions
Binomial Problem:A jury has 12 jurors. A vote of at least 10 out of 12 for "guilty" is necessary for a defendant to be convicted of a crime. Assume that each juror acts independently of the others and that the probability that any one juror makes the correct decision on a defendant is .80. If the defendeant is guitly, what is the probability that the jury makes the correct decision?
Answers
The probability that the jury makes the correct decision is approximately 0.7063.
To solve this problem, we need to find the probability that the jury makes the correct decision if the defendant is guilty. Let's break down the problem into smaller steps.
We know that the probability of a single juror making the correct decision is 0.80. If the defendant is guilty, then the probability of a juror making the correct decision is still 0.80. Therefore, the probability that a single juror makes the correct decision if the defendant is guilty is 0.80.
We can use the binomial distribution formula to determine the probability of at least 10 out of 12 jurors making the correct decision. The formula is:
P(X ≥ k) = 1 - Σ(i=0 to k-1) [n!/(i!(n-i)!) x \(p^i \times (1-p)^{(n-i)}\) ]
where:
P(X ≥ k) is the probability of at least k successes
n is the total number of trials (in this case, 12 jurors)
p is the probability of success in a single trial (in this case, 0.80)
k is the number of successes we want to find the probability of (in this case, 10)
Plugging in the values, we get:
P(X ≥ 10) = 1 - Σ(i=0 to 9) [12!/(i!(12-i)!) x \(0.80^i \times (1-0.80)^{(12-i)}\)]
Using a calculator or software, we can calculate this to be approximately 0.7063.
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Find a non-zero vector x perpendicular to the vectors
[-1] [-1]
v = [-3] and u = [-4]
[ 2] [-2]
[ __ ]
x = [ __ ]
[ __ ]
Answers
The vector x = [10, -8, 1] is perpendicular to vectors v and u.
What is the determinant?
The determinant is a mathematical operation defined for square matrices. It is denoted by the symbol det(A) or |A|, where A represents a square matrix.
To find a vector x that is perpendicular to the given vectors v and u, we can use the cross product.
Let's calculate the cross product of vectors v and u:
v x u = [-3, -4, 2] x [-1, -1, -2]
To compute the cross product, we take the determinant of the following matrix:
| i j k |
| -3 -4 2 |
| -1 -1 -2 |
Expanding the determinant, we have:
v x u = i((-4)(-2) - 2(-1)) - j((-3)(-2) - 2(-1)) + k((-3)(-1) - (-4)(-1))
= i(8 + 2) - j(6 + 2) + k(-3 + 4)
= i(10) - j(8) + k(1)
= [10, -8, 1]
Hence, the vector x = [10, -8, 1] is perpendicular to vectors v and u.
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Can dilations be used to form scale copies of a figure?
Yes
No
Sometimes
This was not in the iReady lesson
Answers
Answer:
sometimes
Step-by-step explanation:
There are several ways that dilation is used in real life. Here are several:
In graphic design. I actually do some graphic design, and I use dilation a lot. It is common to dilate photos to fit the space that you want it to fit.
In police work and crime investigation. Detectives and police dilate photos to see smaller details and evidence.
In architecture. It is normal for architects to make a prototype of their design or building. In order to make the building true to the prototype, they must dilate the scale and measurements.
In the doctor's office. Dilation is used in eye exams so that the eye doctor can view the patient's eye better. After a while it will slowly reduce in size and return back to normal.
Your parents took your family out to dinner . Your parents want to give you the server a 15% tip . If the total amount of the dinner was $42.00 , what will be the amount of the tip ?
Answers
Answer:
The tip will be $6.30
The ordered pairs (1, 1), (2, 4), (3, 9), (4, 16), and (5, 25) represent a function. What is a rule that represents this function?
Answers
Answer:
number multiplied by itself
Step-by-step explanation:
с
B
4 2
E
10
X
'D
A
Solve for x.
Answers
Answer:
answer is x=10 I hope it help
HELP ME PLEASE!!! A ball is thrown in the air from a platform that is 112 feet above ground level with an initial vertical velocity of 32 feet per second. The height of the ball, in feet, can be represented by the function shown where t is the time, in seconds, since the ball was thrown.
h(t)=−16t2+32t+112
Which of the following equations shows the function rewritten in the form that would be best used to identify the maximum height of the ball?
Question 16 options:
h(t)=−16(t−2)2+112
h(t)=−16(t−1)2+80
h(t)=−16(t−1)2+96
h(t)=−16(t−1)2+128
Answers
The ball thrown upwards follows a parabolic path
The equation that shows the function rewritten in the form that would be best used to identify the maximum height of the ball is \(h(t)=-16(t-1)^2+128\)
How to determine the equivalent equationThe equation is given as:
\(h(t)=-16t^2+32t+112\)
Factor out -16
\(h(t)=-16(t^2-2t)+112\)
Express the bracket as a perfect square expression
\(h(t)=-16(t^2-2t + 1 - 1)+112\)
Factor out -4
\(h(t)=-16(t^2-2t + 1)+112 + 16 * 1\)
\(h(t)=-16(t^2-2t + 1)+128\)
Express as a perfect square
\(h(t)=-16(t-1)^2+128\)
Hence, the equivalent expression is \(h(t)=-16(t-1)^2+128\)
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solve for x. round to the nearest tenth,if neccessary.
Answers
Answer:
94.8
Step-by-step explanation:
sin 45 = 67/x
x = 67/sin 45
x = 94.8
What congruency theorem would this be: SSS, SAS or ASA and why?
Answers
EXPLANATION: it has two sides listed and one angle
Write the fact family that
can be made from these
digits.
5 6 11
Answers
Answer:
5+6=11
11-6=5
11-5=6
Step-by-step explanation:
A map of two cities is plotted on a coordinate grid one city is centered at -60,40 while the other is at 40,20 a rest station lies at the midpoint of a straight road joining the centers of each city what is the location of the rest station
Answers
Explanation: I found the median of x and y and put it together
Answer: -10,30
Step-by-step explanation: I drew the graph, and it lined with 30 and interCepted -10
Which integer has the least value (-17,-7,-12,2)
Answers
Here is the setup for a non-traditional casino game: You draw a card from a well shuffled full deck and if the card is a king you win $100. The game costs $2 to play and you decide to play the game until you win the $100. Each time you draw a card you pay $2, and if the card is not a king, the card is put back in the deck, and the deck is reshuffled. How much money should you expect to spend on this game?
The answer is $26 - I just don't know how this amount is derived. Please be specific with which statistical technique is used to determine this.
Answers
You should expect to spend $26 to win $100 playing this casino game.
This is a classic example of a geometric distribution, where each trial has two possible outcomes (win or lose) and the probability of winning remains constant at 1/13 (4 kings in a deck of 52 cards) across all trials.
The expected number of trials to win is 13, so the expected cost to win is 13 times the cost per trial, which is $2, giving an expected cost of $26. Note that this is the expected cost, not the guaranteed cost, and it is possible to spend more or less than $26 before winning.
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Elizabeth records the number of points her favorite basketball team scores in each game. She predicts that the team will score about 120 points in its next game is Elizabeth's prediction reasonable?
pls help it due tomorrow
Answers
Answer:
Based on the data that Elizabeth has gathered about her favorite basketball team's previous game scores, it's difficult to say whether her prediction of 120 points in the next game is reasonable. However, if the team has been consistently scoring around that range in the past few games, then it's possible that Elizabeth's prediction could be accurate. However, it's always important to keep in mind that any prediction or estimate is subject to unexpected changes, so it's always best to take them with a grain of salt.Step-by-step explanation:
yeah
B is the midpoint of AC and E is the midpoint of BD. If A(-9, -4), C(-1, 6), and E(-4, -3), find the coordinates or D
Answers
The coordinates of point D is (- 3, - 7).
How to determine the coordinates of a endpoint of a line segment divided by a midpoint
A midpoint is a point that partitions a line segment in two parts of equal length. First, we must find the coordinates of point B by using the midpoint formula:
B(x, y) = 0.5 · A(x, y) + 0.5 · C(x, y)
B(x, y) = 0.5 · (- 9, - 4) + 0.5 · (- 1, 6)
B(x, y) = (- 4.5, - 2) + (- 0.5, 3)
B(x, y) = (- 5, 1)
And the location of point D is found by using the midpoint formula again:
E(x, y) = 0.5 · B(x, y) + 0.5 · D(x, y)
(- 4, - 3) = 0.5 · (- 5, 1) + 0.5 · D(x, y)
(- 4, - 3) = (- 2.5, 0.5) + 0.5 · D(x, y)
(- 8, - 6) = (- 5, 1) + D(x, y)
D(x, y) = (- 8, - 6) - (- 5, 1)
D(x, y) = (- 3, - 7)
The coordinates of point D is (- 3, - 7).
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A straight has slope equals to 12 and y-intercept equals to -5. What's the equation for this straight line?
Answers
Answer:
\( \large{ \tt{❃ \: EXPLANATION}} : \)
We're provided : Slope of straight line ( m ) = 12 & y - intercept ( c ) = - 5 and We're asked to find out the equation for this straight line.You'll have to know : Equation of straight line in slope intercept form : y = mx + c where m is the slope & c is the y-intercept.\( \large{ \tt{❁ \: LET'S \: START}} : \)
Using Slope - intercept form of equation of straight line :\( \boxed{ \large{ \tt{❈ \: y = mx + c}}}\)
- Plug the values :
\( \large{ \tt{↬y = 12x + ( - 5)}}\)
\( \large{ \tt{↬ \: y = 12x - 5}}\)
\( ↬\large{\tt{12x - 5 = y}}\)
\(↬ \large{ \boxed{ \bold{ \tt{12x - y - 5 = 0}}}}\)
Hence , 12x - y - 5 = 0 is the required equation.\( \tt{✺ \: STUDY \: MORE \: PROCASTINATE \: LESS !\:♪ }\)
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The amount of gasoline sold daily at a service station is uniformly distributed with a minimum of 2,000 gallons and a maximum of 5,000 gallons.
a. Find the probability that daily sales will fall between 2,500 and 3,000 gallons.
b. What is the probability that the service station will sell at least 4,000 gallons.
c. What is the probability that the station will sell exactly 2,500 gallons?
Answers
a) The probability that daily sales will fall between 2,500 and 3,000 gallons is 0.2.
b) The probability that the service station will sell at least 4,000 gallons is 0.3.
c) The probability that the station will sell exactly 2,500 gallons is 0.1.
How to determined the Probability distribution of gasoline sales at a service station?Gasoline sold daily is uniformly distributed between 2,000 and 5,000 gallons.
a) To find the probability that daily sales will fall between 2,500 and 3,000 gallons, we need to find the area under the uniform distribution curve between 2,500 and 3,000.
Since the distribution is uniform, the probability density function (PDF) is a constant:
f(x) = 1 / (5000 - 2000) = 1 / 3000, for 2000 <= x <= 5000.
The probability of the daily sales being between 2,500 and 3,000 gallons is equal to the area under the PDF curve between 2,500 and 3,000:
P(2500 <= X <= 3000) = ∫[2500, 3000] f(x) dx= ∫[2500, 3000] (1/3000) dx= (1/3000) [x]_2500³⁰⁰⁰= (1/3000) (3000-2500)= 1/6= 0.1667Therefore, the probability that daily sales will fall between 2,500 and 3,000 gallons is 0.1667 or 16.67%.
b) To find the probability that the service station will sell at least 4,000 gallons,
we need to find the area under the uniform distribution curve from 4,000 to 5,000 gallons:
P(X >= 4000) = ∫[4000, 5000] f(x) dx= ∫[4000, 5000] (1/3000) dx= (1/3000) [x]_4000⁵⁰⁰⁰= (1/3000) (5000-4000)= 1/3= 0.3333Therefore, the probability that the service station will sell at least 4,000 gallons is 0.3333 or 33.33%.
c) To find the probability that the station will sell exactly 2,500 gallons, we need to find the area under the uniform distribution curve at 2,500 gallons:
P(X = 2500) = 0 (since the probability of a single point is zero for a continuous distribution)
Therefore, the probability that the station will sell exactly 2,500 gallons is zero.
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a rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a​ single-strand electric fence. with m of wire at your​ disposal, what is the largest area you can​ enclose, and what are its​ dimensions?
Answers
To find the largest area you can enclose with a single-strand electric fence and m units of wire at your disposal, you need to optimize the dimensions of the rectangular plot.
Let's denote the length of the rectangular plot as L and the width as W. The perimeter of the plot can be calculated as P = L + 2W.
Since one side of the plot is already bounded by the river, we have L + W = m. Rearranging this equation, we get L = m - W.
To find the largest area, we need to maximize the function A = L * W. Substituting the value of L, we have A = (m - W) * W.
Expanding the equation, we have A = mW - W^2.
To find the maximum value of A, we take the derivative of A with respect to W and set it equal to zero: dA/dW = 0.
Differentiating, we get m - 2W = 0. Solving for W, we have W = m/2.
Substituting this value back into the equation for L, we have L = m/2.
Therefore, the dimensions of the largest area are L = m/2 and W = m/2.
To find the area, we substitute these values into the equation for A: A = (m/2) * (m/2) = m^2/4.
In conclusion, the largest area that can be enclosed is m^2/4, with dimensions L = m/2 and W = m/2.
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Caleb started with 50 bacteria for his science experiment and they doubled in one day. Jenna also started with 50 bacteria for her experiment and after one day, her number of bacteria was equal to 50 to the second power. Choose the statement that is true for the number of bacteria after one day.
A Caleb's experiment had 100 bacteria and Jenna's experiment had 250 bacteria.
B Caleb's experiment had 100 bacteria and Jenna's experiment had 2,500 bacteria.
C Caleb's experiment had 500 bacteria and Jenna's experiment had 500 bacteria.
D Caleb's experiment had 250 bacteria and Jenna's experiment had 500 bacteria.
Answers
Answer: B
Step-by-step explanation:
The second power basically means that you multiply the number by itself that many times depending on the power, and since it's 50^2, the answer is Caleb - 100 | Jenna - 2,500. (Tip: Just multiply 50 x 50)
Which statement is true?
A. All integers are whole numbers.
B. All whole numbers are natural numbers.
C. Some rational numbers are integers.
D. Some irrational numbers are rational numbers.
Answers
C Some rational numbers are
If a student has a monthly income of about $250, how much do you predict he might spend on clothing in a month?
Answers
Answer: depends if there like someone like me 550 but if there responable then probaly 50-80
Step-by-step explanation:
What is the solution of 9|2x – 1| + 4 < 49?
x < –2 or x > 3
x < –3 or x > –2
–2 < x < 3
–3 < x < –2
please help me
Answers
Answer:
-2<x<3
Step-by-step explanation:
I have attached the work required to solve this problem as well as a pretty good explanation to each step.
If the selling prices of 20 pens are equal to the cost price of 25 pens, what is the gain percentage?
Answers
\(\huge\bold\red{Answer:-}\)
No(0)Let the cost price of the 20 pens be ₹100.
Cost of each pen=₹100/20=₹5
Selling price of each 25 pens sold=₹100/25=₹4.
%LOSS:
If cost of each pen. = 5=100%
..selling price of each pen. = 4= ?%.
Cross multiply:5? =400
?=80%
So,%loss= 100%-80%=20%
FOLLOW ME ♥️A locker combination has two nonzero digits, and the digits can be repeated. The first number is 3. What is the probability that the second number is 3?
A. 1/27
B. 1/9
C. 1/8
D. 8/9
WILL GIVE BRAINLIEST
Answers
Answer:
I would say A. 1/27
Use the course calculator to solve this problem:
8 + 17 × 26
Answers
Answer:
450
Step-by-step explanation:
8+17=25
25x26=450
Answer: 500
Step-by-step explanation:
You have to do multiplication first according to PEMDAS so you do 17 times 26 and you get 442 add 8 you get 500.
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Answers
Answer:
It's just 2 1/3 so 7/3
Step-by-step explanation:
1. Solve x and y for the diagram.
Answers
Answer:
B
Step-by-step explanation:
The triangle RUS is similar to RST. 9/sqrt(90)=y/(3+x) and y^2=x^2+81. Solving all this will give x=27 and y=9*sqrt(10)
Value of x is 27 and value of y is \(9\sqrt{10}\).
What is angle addition postulate?The angle addition postulate in geometry states that if we place two or more angles side by side such that they share a common vertex and a common arm between each pair of angles, then the sum of those angles will be equal to the total sum of the resulting angle.
X-value:
<RST = \(90^{0}\)
\(tan( < RSU)= \frac{\overline R \overline U}{\overline S \overline U}\)
\(tan( < RSU) = \frac{3}{9}\)
\(< RSU = 18.435^{0}\)
Angle addition postulate:
<RST = <RSU + <TSU
\(90^{0} =18.435^{0} + < TSU\)
\(< TSU =71.565^{0}\)
\(tan( < TSU) = \frac{\overline T \overline U }{\overline S \overline U}\)
\(tan(71.565^{0})=\frac{x}{9}\)
\(x=27\)
Y-value:
<RST = \(90^{0}\)
\(tan( < RSU)= \frac{\overline R \overline U}{\overline S \overline U}\)
\(tan( < RSU) = \frac{3}{9}\)
\(< RSU = 18.435^{0}\)
Angle addition postulate:
<RST = <RSU + <TSU
\(90^{0} =18.435^{0} + < TSU\)
\(< TSU =71.565^{0}\)
\(cos( < TSU) = \frac{\overline S \overline U }{\overline S \overline T}\)
\(cos(71.565^{0})=\frac{9}{y}\)
\(y=9\sqrt{10}\)
Value of x is 27 and value of y is \(9\sqrt{10}\).
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What value of x is in the solution set of 2(3х – 1) > 4х – 6?
—10
-5
—3
-1
Help me
Answers
Answer:
-1
Step-by-step explanation:
2(3х – 1) > 4х – 6
Distribute
6x -2 > 4x-6
Subtract 4x from each side
6x-2-4x > 4x-6-4x
2x-2 > -6
Add 2 to each side
2x-2+2> -6+2
2x> -4
Divide by 2
2x/2 > -4/2
x > -2
The number is greater than -2
The only number that is greater than -2 is -1
The population of a South America country is about 17,400,000. Which expression shows this number in scientific notation?
Answers
Answer:
1.74x10^7
Step-by-step explanation:
you count how many times you moved the decimal to the right
can i have brainliest? i only need 2 more
Answer:
1.74x10^7
Step-by-step explanation:
The product of the slopes of lines
and
is
.
Answers
Answer:
wrong question correct it..
Answer:
Step-by-step explanation:
you a good
for the t