By the divergence theorem, the flux of [tex]\vec F[/tex] across S is equal to the volume integral of [tex]\mathrm{div}(\vec F)[/tex] over the interior of S.
We have
[tex]\vec F(x,y,z) = (x^3+y^3)\,\vec\imath + (y^3+z^3)\,\vec\jmath + (z^3+x^3)\,\vec k \\\\ \implies \mathrm{div}(\vec F) = \dfrac{\partial(x^3+y^3)}{\partial x} + \dfrac{\partial(y^3+z^3)}{\partial y} + \dfrac{\partial(z^3+x^3)}{\partial z} = 3(x^2+y^2+z^2)[/tex]
so that
[tex]\displaystyle \iint_S \vec F(x,y,z)\cdot\mathrm d\vec s = \iiint_T \mathrm{div}(\vec F)\,\mathrm dV = 3 \iiint\limits_{x^2+y^2+z^2\le3} (x^2+y^2+z^2)\,\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]
To compute the volume integral, convert to spherical coordinates with
x = ρ cos(θ) sin(ϕ)
y = ρ sin(θ) sin(ϕ)
z = ρ cos(ϕ)
so that
ρ ² = x ² + y ² + z ²
dx dy dz = ρ ² sin(ϕ) dρ dϕ dθ
The region T is the interior of the sphere S, given by the set
[tex]T = \left\{(\rho,\theta,\phi) \mid 0\le\rho\le3 \text{ and } 0\le\phi\le\pi \text{ and }0\le \theta\le2\pi\right\}[/tex]
So we have
[tex]\displaystyle 3 \int_0^{2\pi} \int_0^\pi \int_0^3 \rho^4 \sin(\phi) \,\mathrm d\rho \,\mathrm d\phi \,\mathrm d\theta \\\\ = 6\pi \left(\int_0^\pi \sin(\phi)\,\mathrm d\phi\right) \left(\int_0^3 \rho^4 \,\mathrm d\rho\right) = \boxed{\frac{2916\pi}5}[/tex]
The required surface integral S, and flux across F·dS; that is [tex]\dfrac{2916\pi }{5}[/tex]
Given that,
Function;[tex]F(x, y, z) = (x^3 + y^3)\vec{i} + (y^3 + z^3)\vec{j} + (z^3 + x^3)\vec{k},[/tex]
S is the sphere with center the origin and radius 3.
We have to determine,
Use the Divergence Theorem to calculate the surface integral S, F.dS that is, calculate the flux of F across S.
According to the question,
By the divergence theorem, the flux of [tex]\vec{F}[/tex]across S is equal to the volume integral of [tex]div(\vec{f})[/tex] over the interior of S.
S is the sphere with center the origin and radius 3.
Therefore,
[tex]F(x, y, z) = (x^3 + y^3)\vec{i} + (y^3 + z^3)\vec{j} + (z^3 + x^3)\vec{k},\\\\= div\vec({F}) = \dfrac{d(x^3+y^3)}{dx} + \dfrac{d(y^3+z^3)}{dx} + \dfrac{d(z^3+x^3)}{dx} = 3(x^{2} + y^{2} + z^{2} )\\\\Then,\\\\\int \int_S \vec{F}(x, y, z) . \vec{ds} = \int \int \int _T div\vec{F}dV= 3 \int \int\int (x^{2} + y^{2} + z^{2} )dx.dy.dz[/tex]
To compute the volume integral, convert to spherical co-ordinate,
[tex]x = p\ cos\theta\ sin\phi\\\\y = p \ sin\theta \ sin\phi\\\\z = p\ cos\phi\\\\[/tex]
Therefore,
[tex]p^2 = x^{2} + y^{2} +z^{2} \\\\dx.dy.dz = x^{2} \ sin\phi \ dp\ d\phi \ d\theta[/tex]
The region T is the interior of the sphere S is given by the set,
[tex]T = {[ p,\theta,\phi}] | \ (0\leq p\leq 3 \ and \ \ 0\leq \phi \leq \pi \ and \ 0\leq 0\leq 2\pi )[/tex]
Then,
[tex]= 3 \int^2_0 \int^\pi _0 \int^3_0 p^4. sin(\phi). dp.d\phi .d\theta\\\\= 6\pi ( \int^\pi _0 sin(\phi).d\phi) (\int^3_0 p^4dp\\\\= \dfrac{2916\pi }{5}[/tex]
Hence, The required surface integral S, F·dS; that is [tex]\dfrac{2916\pi }{5}[/tex]
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Which describes the relationship between the distance and the difference?
Answer:
Which describes the relationship between the distance and the difference? The distance is always the opposite of the difference. ... The distance is the absolute value of the difference.
What is one root of this equation?
2x^-4x+9=0
9514 1404 393
Answer:
1 +i√3.5
Step-by-step explanation:
In vertex form, the equation is ...
2(x² -2x +1) +7 = 0
2(x -1)² +7 = 0
Then the solutions are ...
(x -1)² = -7/2
x = 1 ±i√3.5
One solution is 1+i√3.5.
How many solutions does the nonlinear system of equations graphed
below have?
Y=-1/64(x-16)^2+4 Y=-1/2x-4
9514 1404 393
Answer:
2
Step-by-step explanation:
The graphs have two (2) points of intersection. Each point of intersection is a solution to the system of equations. The coordinates of that point satisfy both equations simultaneously.
There are 2 solutions to the graphed system of equations.
Point B is between points A and C. AB = 2x - 3, BC = 7x - 10, and AC = 32.
What is the value of x?
helppppp maybee pplsss
Answer:
CD and BE
BC and DE
...........
need help......................................... I DO ANYTHING
Step-by-step explanation:
Additive inverse are numbers with the same number but different signs. For example, 1 and -1 are additive inverse.
An additive inverse and it original number add to zero.
For example,
[tex]1 + ( - 1) = 0[/tex]
Using this info, w is -0.8
0.8 and -0.8 are equidistant from 0.
-0.8 is 2 mark to the right of -1.
So this means that 0.8 is 2 mark to left of 1.
Examples of rounding?
Answer:
Rounding means making a number simpler but keeping its value close to what it was.
Step-by-step explanation:
Example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80. ... But 76 goes up to 80.
show that there do not exist scalar c1 c2 and c3 c1(-2,9,6)+c2(-3,2,1)+c3(1,7,5)= (0,5,4)
This is the same as showing the following system of equations doesn't have a solution:
[tex]\begin{cases}-2c_1-3c_2+c_3 = 0 \\ 9c_1 + 2c_2 + 7c_3 = 5 \\ 6c_1 + c_2 + 5c_3 = 4\end{cases}[/tex]
or in matrix form,
[tex]\begin{bmatrix}-2&9&6\\-3&2&1\\1&7&5\end{bmatrix}\begin{bmatrix}c_1\\c_2\\c_3\end{bmatrix} = \begin{bmatrix}0\\5\\4\end{bmatrix}[/tex]
The quickest way to check if there is a solution is to check whether the coefficient matrix is invertible. If its determinant is 0, then it is not invertible.
And the quickest way to show that the determinant is 0 is by observing that the third row is a linear combination of the first two rows:
(-2, 9, 6) - (-3, 2, 1) = (-2 + 3, 9 - 2, 6 - 1) = (1, 7, 5)
So there are indeed no such scalars c₁, c₂, and c₃.
Helen and Rebecca baked some muffins in the ratio 2:5. After Helen baked another 12 muffins, the ratio become 1:2. How many muffins did Helen bake altogether?
=========================================================
Explanation:
x = some positive whole number
2x = amount Helen baked initially
5x = amount Rebecca baked initially
Notice that the ratio 2x:5x reduces to 2:5 after dividing both parts by x.
So (2x)/(5x) = 2/5
Helen then bakes another 12 muffins, bumping the 2x up to 2x+12. The ratio is now 1:2, which means,
(amount Helen bakes)/(amount Rebecca bakes) = 1/2
(2x+12)/(5x) = 1/2
2(2x+12) = 5x*1 .... cross multiply
4x+24 = 5x
24 = 5x-4x
24 = x
x = 24
From here, we then can say,
Helen = 2x+12 = 2*24+12 = 48+12 = 60Rebecca = 5x = 5*24 = 120Helen baked 60 muffins and Rebecca baked 120 muffins. We see that 60:120 reduces to 1:2 after dividing both parts by the GCF 60.
Furthermore, note how
2x = 2*24 = 485x = 5*24 = 12048:120 reduces to 2:5 after dividing both parts by 24This helps confirm we have the correct answer.
[15-(|-6+2|+ squareroot9)]^2
can anyone explain the steps for how to solve it too? i’m confused :(
Answer:
1936
Step-by-step
[tex][15 - (1 - 6 + 21 + \sqrt{9} )] ^{2}[/tex]
use BIDMAS
(1 - 6 + 21 + [tex]\sqrt{9}[/tex]) = (1 - 6 + 21 + 3) = (1 - 30) = -29
[15 - -29] = [15 + 29] because two minuses together make a plus
[tex]44^{2}[/tex] = 1936
Evaluate the exprssion when a= -5
a2+6a-9
Answer:
-49
Step-by-step explanation:
Substitute the a for -5
2(-5)+6(-5)-9
Multiply:
-10+-30-9
Add:
-40-9
Subtract (Add negatives):
-49
Hi there!
»»————- ★ ————-««
I believe your answer is:
-14
»»————- ★ ————-««
Here’s why:
⸻⸻⸻⸻
[tex]\text{Assuming that the expression is: } a^2 + 6a -9[/tex]
⸻⸻⸻⸻
[tex]\boxed{\text{Evaluating...}}\\\\a^2+6a - 9\\-----------\\(-5)^2 + 6(-5) - 9\\\\\rightarrow (-5)^2 = 25\\\\\rightarrow 6 * -5 = -30\\\\25 - 30 - 9\\\\\rightarrow \boxed{-14}[/tex]
⸻⸻⸻⸻
»»————- ★ ————-««
Hope this helps you. I apologize if it’s incorrect.
find the solution to this systems of equations
Answer: I dont know
Step-by-step explanation:
Answer:
x= -y+1
y=2x+z−1
z=−x−2y+ 8 /3
Step-by-step explanation:
I honestly entered it into a algebra calculator which gave me the steps.
Find the equation of the line with the slope 3/2 through the point (-6,2)
Step-by-step explanation:
y-y1=slope (x-x1)
where x1=-6,y1=2 and slope=3÷2
y-2=3÷2{x-(-6)}
y-2=3÷2(x+6)
y-2=(3x+18)÷2 cross multiply
2y-4=3x+18 collect like terms
2y=3x+18+4
2y=3x+22....ans
a system of equations is given below in terms of the scalar parameters a and b.
ax + 2y+ 4z = b
2x +y+3z= a+1
5x-2y+ (a+1)z= 3
(1) Show that if a = -5 or a=2, the system does not have a unique solution.
A system of equations only has a unique solution if all the equations are linearly independent (this means that we have two or more equations representing the same curve).
Here we have the system:
a*x + 2*y + 4*z = b
2*x + y + 3*z = a + 1
5*x - 2*y + (a + 1)*z = 3
Now we want to show that if a = -5 or a = 2, then the system does not have a unique solution.
To see this, we can just replace a by these values and see that the equations are not linearly independent.
for a = 2 we get:
2*x + 2*y + 4*z = b
2*x + y + 3*z = 2 + 1
5*x - 2*y + (2 + 1)*z = 3
Simplifying this:
2*x + 2*y + 4*z = b
2*x + y + 3*z = 3
5*x - 2*y + 3*z = 3
From the second and third equation we can write:
2*x + y + 3*z = 3 = 5*x - 2*y + 3*z
2*x + y + 3*z = 5*x - 2*y + 3*z
2*x + y = 5*x - 2*y
3*y = 3*x
y = x
now that we know this, we can rewrite the equations as:
4*(x + z) = b
3*(x + z) = 3
Now we can take the quotient of the two equations to get:
(4*(x + z))/3*(x + z) = (4/3) = b/3
b = 4
Then the equations become:
4*(x + z) = 4
3*(x + z) = 3
Here we can divide te above one by 4 and the below one by 3 to get:
(x + z) = 1
(x + z) = 1
So both equations represent the same line, thus, the system does not have a unique solution.
Now for a = -5:
-5*x + 2*y + 4*z = b
2*x + y + 3*z = -5 + 1
5*x - 2*y + (-5 + 1)*z = 3
Simplify it:
-5*x + 2*y + 4*z = b
2*x + y + 3*z = -4
5*x - 2*y - 4*z = 3
Now we can rewrite the first equation as:
-5*x +2*y - b = -4*z
and the third one as:
-4*z = 3 - 5*x + 2*y
Then we can write:
-5*x +2*y - b = -4*z = 3 - 5*x + 2*y
-5*x +2*y - b = 3 - 5*x + 2*y
Now we can remove the -5*x in both sides to get:
+2*y - b = 3 + 2*y
Now we can remove the 2*y in both sides to get:
-b = 3
b = -3
Replacing this in our system we get:
-5*x + 2*y + 4*z = -3
2*x + y + 3*z = -4
5*x - 2*y -4*z = 3
Now we can multiply the first equation by -1:
-1*(-5*x + 2*y + 4*z)= -1*(-3)
5*x - 2*y - 4*z = 3
This is the same as the third equation, so here we can see that the first and third equations represent the same line, thus, the system of equations does not have a unique solution.
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If 4/15 of the candies in a bag of Skittles are red and 3/10 are yellow, what fraction of the bag are red or yellow
Add the two fractions together but first rewrite the fractions with a common denominator.
4/15 = 8/30
3/10 = 9/30
8/30 + 9/30 = 17/30
Answer: 17/30
The fraction is 17/30.
What is fraction?fractions are represented as a numerical value, which defines a part of a whole. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.
For example.
The following figure shows a pizza that is divided into 8 equal parts. Now, if we want to express one selected part of the pizza, we can express it as 1/8 which shows that out of 8 equal parts, we are referring to 1 part.
Given:
4/15 of the candies in a bag of Skittles are red
3/10 are yellow
So, make the denominator of both fraction same by LCM
4/15 = 8/30
3/10 = 9/30
Now,
8/30 + 9/30 = 17/30
Hence, the fraction is 17/30.
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The difference of two numbers is 16, and their sum is 20. What are the two numbers?
Answer:
18 and 2
Step-by-step explanation:
18-2=16
18+2=20
Hi ;-)
[tex]\{x-y=16\\+\{x+y=20\\--------\\2x = 36 \ \ /:2\\\boxed{x=18}\\18+y=20 \ \ /-18\\\boxed{y=2}[/tex]
These numbers are 18 and 2.
Hourly rate: $3.82; Hours: 24; What are the total earnings
Answer:
$91.68
Step-by-step explanation:
$3.82 × 24 = $91.68
Answer: $91.68
Step-by-step explanation:
Given information
Hourly rate = $3.82
Hours = 24
Given expression deducted from the given information
Total earning = Hours × Hourly rate
Substitute values into the expression
Total earning = (24) × (3.82)
Simplify by multiplication
Total earning = $91.68
Hope this helps!! :)
Please let me know if you have any questions
What is the correct way to notate the shaded
region indicated by this Venn Diagram?
The correct notation will be A∪B and it can be read as A union B
In the given Venn diagram we can deduce the following;
U: this is the universal set, it comprises all the elements in set A and set B, as well as the compliments of set A and set B.A: this is include the elements contained in set AB: this include the elements contain in set BA∩B: this includes the elements in common for both set A and set B. This is often read as A intersect B.A∪B: this include all the elements contained in both set A and set B. This often read as A union B.The shaded region in the Venn diagram includes all the elements in set A and all the elements in set B.
Thus, the correct notation will be A∪B and it can be read as A union B.
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A Venn Diagram is a diagram used to define sets.
Here we will find that the notation of the shaded region is: A ∪ B
To know this, we need to define all of the notations in the options:
A ∩ B: This is the intersection of sets A and B, so this is the set of the elements that belong to both sets.A ∪ B: This is the set of all the elements that belong to either A or B (or both)A ⊂ B: This means that set A is included in B¬A (or a set with a line on top): Represents the set of all the elements that do not belong to A.So with these general notations, is easy to conclude that if we have shaded sets A and B completely, then we have the set: A ∪ B
The correct option is C.
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The fifth grade class donation consisted of 1/5 pasta boxes and 6/15 cans of soup. What fraction of their total donation was vegetables?
Answer:
None?
Step-by-step explanation:
Can someone pls help me
I will mark u as Brainliest
Answer:
C
Step-by-step explanation:
because when you reflect it the figure A on the y-axis, it looks the same as A*, and when you move it 5 units down both figures are at the same place, showing that the shapes are congruent.
Class 8 squares and square roots without adding find the sum of a) 1+3+5+7+9+11+13+15+17.
Answer:
given
n=9
a=1
d=3-1=2
Step-by-step explanation:
now , by formula,
sn = n/2(2a+(n-1)d)
= 9/2(2×1+(9-1)×2)
=9/2×18
=81
3) You roll a number cube what's the probability of
rolling a number higher than 4 and then a 2?
Answer:
1/18
Step-by-step explanation:
Probability of getting a number higher than 4=1/3
Probability of getting 2=1/6
Probability of rolling a number higher than 4 and then a 2=(1/3)*(1/6)=1/18
What is the answer for number 74.
Answer:
g
Step-by-step explanation:
Solve the equation. 4x2 + 20 = -7
round 6.7 to the nearest whole number.
Answer:
7[tex]\tt{ \green{P} \orange{s} \red{y} \blue{x} \pink{c} \purple{h} \green{i} e}[/tex]
the square of a positive number is always ________
fill in the blanks
Answer:
the square of a positive number is always positive
Is 66.06 a irrational number
Answer:
No.
Step-by-step explanation:
No because it can be written as a fraction, 6606/10
An equation for the difference between the time is 38.4 - a= 9.59. The others time was
Answer:
Presumably the "others time" is the quantity [tex]a[/tex], which is 28.81
Step-by-step explanation:
Let's solve for [tex]a[/tex] from
[tex]38.4-a=9.59[/tex]
Adding [tex]a[/tex] gives
[tex]a+9.59=38.4[/tex]
Then subtracting 9.59 gives
[tex]a=38.4-9.59=28.81[/tex]
The temperature at the begſnning of the day was - 3°F. The temperature dropped 2°F by the end of the day.
What was the temperature at the end of the day? Use the number line model to find – 3-2.
Answer:
- 5°F
Step-by-step explanation:
- 3°F - 2°F
(-3) - 2
-5
- 5°F
What is 50 divided by 8?
Answer:
6.25 or 6 2/8 or 6 R2
Step-by-step explanation: