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) = (3x2 − 2y2)i + (4xy + 4)j

Looks like either [tex]f''(x)=4x^2[/tex] or [tex]f''(x)=\frac4{x^2}[/tex]...

In the first case, integrate both sides twice to get

[tex]f''(x)=4x^2\implies f'(x)=\dfrac43x^3+C_1\implies f(x)=\dfrac13x^4+C_1x+C_2[/tex]

Then the initial conditions give

[tex]f'(1)=2\implies 2=\dfrac43\cdot1^3+C_1\implies C_1=\dfrac23[/tex]

[tex]f(1)=5\implies 5=\dfrac13\cdot1^4+C_1\cdot1+C_2\implies C_2=4[/tex]

so that the particular solution is

[tex]f(x)=\dfrac{x^4}3+\dfrac{2x}3+4[/tex]

If instead [tex]f''(x)=\frac4{x^2}[/tex], we have

[tex]f''(x)=\dfrac4{x^2}\implies f'(x)=-\dfrac4x+C_1\implies f(x)=-4\ln|x|+C_1x+C_2[/tex]

[tex]f'(1)=2\implies 2=-\dfrac41+C_1\implies C_1=6[/tex]

[tex]f(1)=5\implies 5=-4\ln|1|+C_1\cdot1+C_2\implies C_2=-1[/tex]

[tex]\implies f(x)=-4\ln|x|+6x-1[/tex]

Answer:

[tex]\boxed{\sf B. \ All \ real \ numbers}[/tex]

Step-by-step explanation:

The domain is all possible values for x.

[tex]f(x)=(\frac{1}{4} )^x[/tex]

There are no restrictions on x.

The domain is all real numbers.

Answer:

B.All real number

hope you have unterstand

Hey there! I'm happy to help!

To find the volume of a cube, you simply cube the side length (multiply it by itself three times). This is because all of the sides of a cube are the same and if you multiply the length by the width by the height it is the same number multiplied by itself three times.

We already know that the volume is 91.125 cubic inches. To find the side length, we simply do the cube root on our calculator, which tells us what number we cube to get 91.125.

∛91.125=4.5

Therefore, the side length of the sandbox is 4.5 inches.

I hope that this helps! Have a wonderful day! :D

Answer:

2^11

Step-by-step explanation:

since the bases are the same, we can add the exponents

a^b * a^c = a^(b+c)

2^6 * 2^5

2^(6+5)

2^11

Answer:

30 31 64 59 58 33 54 77 56 41 (arrange it)

30 31 33 41 54 56 58 59 64 77 (done!)

Mean: Find the number in the middle (54+56)/2= 110/2 = 55

Mode: None

Mean: (30+31+33+41+54+56+58+59+64+77)/10=503/10= 50,3

Answer:

2x^2 + x - 6 = rectangular garden: length = 2x – 3 feet; width = x + 2 feet

Step-by-step explanation:

(2x - 3)(x + 2) = 2x^2 + x - 6 =

2x^2 + 4x - 3x - 6 = 2x^2 + x - 6 =

2x^2 + x - 6

You get the original equation from the two sides multiplied. :)

Hope this helps, have a good day.

The length and width of the rectangle will be (2x – 3) and (x + 2). Then the correct option is D.

Let W be the rectangle's width and L its length.

The area of the rectangle is the multiplication of the two different sides of the rectangle. Then the rectangle's area will be

Area of the rectangle = L × W square units

The area is 2x² + x – 6 square feet. Then the factor of the equation is given as,

A = 2x² + x – 6

A = 2x² + 4x – 3x – 6

A = 2x(x + 2) – 3(x + 2)

L × W = (2x – 3)(x + 2)

The length and width of the rectangle will be (2x – 3) and (x + 2). Then the correct option is D.

More about the area of the rectangle link is given below.

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Answer:

Step-by-step explanation:

Distance walked by Arthur = 5/8 miles

Distance ride  by Jonathan = 8 times that of Arthur

it means that Distance rode on bus by Jonathan is 8 multiplied by Distance walked by Arthur

Distance rode on bus by Jonathan = 8 * Distance walked by Arthur

Distance rode on bus by Jonathan = 8 * 5/8 = 5 Miles Answer

Answer:

[tex]\frac{3}{8}[/tex]

Step-by-step explanation:

To simplify the fraction [tex]\frac{24}{64}[/tex], we need to find its greatest common factor.

Both 24 and 64 are divisible by 2, but that's not the biggest.

Both 24 and 64 are divisible by 4, but that's not the biggest either.

Both 24 and 64 are divisible by 8, and that's the highest we can go.

So:

[tex]\frac{24\div8}{64\div8} = \frac{3}{8}[/tex].

Hope this helped!

Answer:

finding the value of x first

2x + 3x + 10 = 180 (linear pair)

5x = 180 - 10

x = 170 / 5

x = 34

3x = 102

Answer:

To find the volume of this cube, you would have to multiply 1.08 by 5.25 by 3.3 feet. If you did this, you would get: 18.711 feet^3. This is the volume of the rectangular prism.

Hope this helped!

Answer:

D. The z scores are numbers without units of measurement.

Step-by-step explanation:

Z-scores are without units, or are pure numbers.

Answer:

  18√91 +54√3

Step-by-step explanation:

Name the point at the top of the pyramid "A", the point at the left front corner "B", and the one in the center of the hexagonal base "C". Then right triangle ABC is shown. The "base" BC of that triangle is the same measure as the front edge (6), because the diameter of a regular hexagon is equal to twice the side length.

Using the Pythagorean theorem, we can find the face edge length to be ...

  AB^2 = BC^2 +AC^2

  AB^2 = 6^2 +8^2 = 100

  AB = √100 = 10

If we call the midpoint of the front edge "D", then we need to find the length of AD in order to determine the face area. Again, we can use the Pythagorean theorem.

  AB^2 = BD^2 +AD^2

  AD^2 = AB^2 -BD^2 = 10^2 -3^2 = 91

  AD = √91

The area of one of the 6 lateral faces is ...

  A = (1/2)bh = (1/2)(6)√91 = 3√91

The area of one of the 6 equilateral triangles that make up the base is ...

  A = (√3)/4·s^2 = (√3)/4(6^2) = 9√3

Then the total area of the pyramid is ...

  total area = 6 × (face area + partial base area)

  = 6(3√91 +9√3)

  total area =  18√91 +54√3

Answer:

243

Step-by-step explanation:

The general term for this arithmetic sequence is:

a(n) = -2 + 5(n - 1).

Then a(50) = -2 + 5(49) =   243

Answer:

63 people.

Step-by-step explanation:

If you have a contagious disease and met with 9 different people each day for 7 days, that'll be 63 people that have gotten infected. 9 x 7 = 63. Hope this helps you!

Answer:

  6

Step-by-step explanation:

Jana had a total of 3+3+3+3 = 12 muffins. Half that number is 3+3 = 6 muffins.

Jana put 6 muffins on the plate.

Step-by-step explanation:

Let minimum score on the final exam to earn an A be X

[tex]mean \: = \frac{sum \: of \: observation}{number \: of \: observation} [/tex]

[tex]90 = \frac{83 + 97 + 89 + 82 + x}{5} [/tex]

Further solving :

X = 99 marks

You can put [tex]n[/tex] different elements in order in [tex]n![/tex] different ways.

So, you can visit 12 different cities in [tex]12!=479001600[/tex] different ways.

Answer: 479,001,600

Step-by-step explanation:

There are 12 ways to go to the first place, 11 for the second, ten for the third, and so on. So 12! Means 12x11x10x9x8x7x6x5x4x3x2x1.

Answer:

3.75

Step-by-step explanation:

[tex]v = lbh \\ 2.5 \times 1.5 \times 1 \\ = 3.75[/tex]

The volume of the rectangular prism will be 3.75 cubic meters.

Let the prism with a length of L, a width of W, and a height of H. Then the volume of the prism is given as

V = L x W x H

A box is 1 m high, 2.5 m long, and 1.5 m wide.

Then the volume of the rectangular prism will be

V = L x W x H

V = 1 x 2.5 x 1.5

V = 3.75 cubic meters

Thus, the volume of the rectangular prism will be 3.75 cubic meters.

More about the volume of the rectangular prism link is given below.

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Answer:

(a) P(E∪F)= 0.8

(b) P(Ec)= 0.4

(c) P(Fc)= 0.7

(d) P(Ec∩F)= 0.8

Step-by-step explanation:

(a) It is called a union of two events A and B, and A ∪ B (read as "A union B") is designated to the event formed by all the elements of A and all of B. The event A∪B occurs when they do A or B or both.

If the events are not mutually exclusive, the union of A and B is the sum of the probabilities of the events together, from which the probability of the intersection of the events will be subtracted:

P(A∪B) = P(A) + P(B) - P(A∩B)

In this case:

P(E∪F)= P(E) + P(F) - P(E∩F)

Being P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.1

P(E∪F)= 0.6 + 0.3 - 0.1

P(E∪F)= 0.8

(b)  The complement of an event A is defined as the set that contains all the elements of the sample space that do not belong to A.  The Complementary Rule establishes that the sum of the probabilities of an event and its complement must be equal to 1. So, if P (A) is the probability that an event A occurs, then the probability that A does NOT occur is  P (Ac) = 1- P (A)

In this case: P(Ec)= 1 - P(E)

Then: P(Ec)= 1 - 0.6

P(Ec)= 0.4

(c) In this case: P(Fc)= 1 - P(F)

Then: P(Fc)= 1 - 0.3

P(Fc)= 0.7

(d)  The intersection of two events A and B, designated as A ∩ B (read as "A intersection B") is the event formed by the elements that belong simultaneously to A and B. The event A ∩ B occurs when A and B do at once.

As mentioned, the complementary rule states that the sum of the probabilities of an event and its complement must equal 1. Then:

P(Ec intersection F) + P(E intersection F) = P(F)

P(Ec intersection F) + 0.1 = 0.3

P(Ec intersection F)= 0.2

Being:

P(Ec∪F)= P(Ec) + P(F) - P(Ec∩F)

you get:

P(Ec∩F)= P(Ec) + P(F) - P(Ec∪F)

So:

P(Ec∩F)= 0.4 + 0.3 - 0.2

P(Ec∩F)= 0.8

Answer: x>3

Step-by-step explanation:

x-5>2

x>+5-2

x>3

Answer:

Question 18: B. 104

Question 19: [tex] x = \frac{3}{2} [/tex]

Step-by-step Explanation:

Question 18:

Step 1: express the inverse relationship with an equation

[tex] y = \frac{k}{x^2} [/tex] ,

where k is constant

y = 26 when x = 4,

Constant, k, = [tex] y*x^2 = k [/tex]

[tex] k = 26*4^2 = 416 [/tex]

The equation would be [tex] y*x^2 = 416 [/tex]

Step 2: use the equation to find y when X = 2.

[tex] y*x^2 = 416 [/tex]

[tex] y*2^2 = 416 [/tex]

[tex] y*4 = 416 [/tex]

Divide both sides by 4

[tex] \frac{y*4}{4} = \frac{416}{4} [/tex]

[tex] y = 104 [/tex]

Question 19:

[tex] \frac{x}{3} = \frac{x + 2}{7} [/tex]

Cross multiply

[tex] x(7) = 3(x + 2) [/tex]

[tex] 7x = 3x + 6 [/tex]

Subtract 3x from both sides

[tex] 7x - 3x = 3x + 6 - 3x [/tex]

[tex] 4x = 6 [/tex]

Divide both sides by 4

[tex] \frac{4x}{4} = \frac{6}{4} [/tex]

[tex] x = \frac{3}{2} [/tex]

Answer: D.) 52

Explanation: I guessed and got it right lol

Subtracting a negative is the same as adding a positive. So 2-(-8) is really 2+8 = 10.

With something like 2-8, we start at 2 and move to the left 8 units to arrive at -6 on the number line. When we do 2-(-8), we start at 2 and move 8 units in the opposite direction since -8 is the opposite of 8.

In terms of money, you can think of a negative number as an IOU or it represents the amount of debt. Writing -8 means you are 8 dollars in debt. If we subtract away debt, then we have less of it and effectively its the same as adding dollars to your pocket. Subtracting away 8 dollars of debt is the same as adding 8 dollars to your pocket, which is one interpretation of how 2-(-8) is the same as 2+8.

Answer:

Put 25 in the box.

Step-by-step explanation:

Apply the exponent rule: (ax)^n = a^n × x^n

So we have:

(5x)^2 = 5^2 × x^2

= 25x^2

Best Regards!

Answer:

249.14 mm³

Step-by-step explanation:

r = diameter/2

= 7.8 /2

volume = 4/3 π r³

= 4/3 * 3.15 * (7.8/2)³

= 249.14 mm³

Answer:

C.

On a coordinate plane, a line goes through (negative 2, 2) and (negative 1, negative 2).

The line parallel to the line 8x + 2y = 12 will be a line that goes through (-2, 2) and (-1, -2). The correct option is C.

An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.

Given that the equation of the line is 8x + 2y =12. First, calculate the slope of the line if the slope of the line is the same as the equation of the given line then the two lines will be parallel.

8x + 2y = 12

2y = -8x + 12

y =-4x + 6

Take points (-2, 2) and (-1, -2) and find the slope of the line.

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

Slope = ( -2 - 2 ) / ( -1 + 2 )

Slope = -4

Therefore, the line parallel to the line 8x + 2y = 12 will be a line that goes through (-2, 2) and (-1, -2). The correct option is C.

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Answer: x ≤ 120

Step-by-step explanation: To get x by itself in this inequality, since it's being divided by -6, we must multiply both sides by -6, just like we would if we were solving an equation, but here is the trick you have to watch out for with inequalities.

When you multiply or divide both sides of an inequality by a

negative, you must switch the direction of the inequality sign.

So our second step in this problem reads x ≤ 120.

Please give this idea your full attention.

Even the most advanced algebra students will sometimes forget to switch the direction of the inequality sign when multiplying or dividing both sides of an inequality by a negative.

Answer:

x ≤ 120

I hope this helps!

Answer:

Step-by-step explanation:

You must multiply the first two equation which is 5x+1 and x this will give you

5x^2 + x. The bold goes in the first box.

Then do the same thing for 2x+1 and x+1 which will give you 2x^2+2x+x+1 or 2x^2+3x+1. This goes in the second box.

In the last box you will add 5x^2 +x and 2x^2 +3x +1, which gives you 7x^2 +4x +1.

Goes in the last box.

Hope this helps you.

Answer:

[tex]\frac{784}{15} \pi[/tex]

Step-by-step explanation:

According to the given situation, the calculation of volume of the solid is shown below:-

Here we will consider the curves that is

[tex]x = 7y^2, x = 7[/tex]

Now, rotating the line for the line x which is equals to 7

[tex]7y^2 = 7\\\\y^2 = 1\\\\ y = \pm1[/tex]

So, the inner radio is

7 - 7 = 0

and the outer radius is

[tex]7y^2 - 7\\\\ = 7(y^2 - 1)[/tex]

Now, the area of cross section is

[tex]A(y) = \pi(7(y^2 - 1))^2\\\\ = 49\pi(y^4 - 2y^2 + 1)[/tex]

The volume is

[tex]V = \int\limits^1_{-1} A(y)dy[/tex]

now we will put the values into the above formula

[tex]= \int\limits^1_{-1} 49\pi(y^4 - 2y^2 + 1)dy\\\\ = 49\pi(\frac{y^5}{5} - \frac{2y^3}{3} + y)^{-1}\\\\ = 49\pi(\frac{1}{5} - \frac{2}{3} + 1 + \frac{1}{5} - \frac{2}{3} + 1)\\\\ = 49\pi(2 + \frac{2}{5} - \frac{4}{3} )\\\\ = 49\pi(\frac{30+6-20}{15} )\\\\ = \frac{49\pi}{15} (16)[/tex]

After solving the above equation we will get

[tex]= \frac{784}{15} \pi[/tex]

Answer:

(2.13982638787^3) x 2