A spherical solid, centered at the origin, has radius 4 and mass density(x,y,z)=6-(x^2+y^2+z^2). Set up the triple integral and find its mass.
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A Spherical Solid, Centered At The Origin, Has Radius 4 And Mass Density(x,y,z)=6-(x^2+y^2+z^2). Set

Answers

Answer 1

There's something very off about this question.

In spherical coordinates,

x² + y² + z² = ρ²

so that

f(x, y, z) = 6 - (x² + y² + z²)

transforms to

g(ρ, θ, φ) = 6 - ρ²

When transforming to spherical coordinates, we also introduce the Jacobian determinant, so that

dV = dx dy dz = ρ² sin(φ) dρ dθ dφ

Since we integrate over a sphere with radius 4, the domain of integration is the set

E = {(ρ, θ, φ) : 0 ≤ ρ ≤ 4 and 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π}

so that the integral is

[tex]\displaystyle \int_{\phi=0}^{\phi=\pi} \int_{\theta=0}^{\theta=2\pi} \int_{\rho=0}^{\rho=4} (6 - \rho^2) \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi[/tex]

Computing the integral is simple enough.

[tex]\displaystyle = \int_{\phi=0}^{\phi=\pi} \int_{\theta=0}^{\theta=2\pi} \int_{\rho=0}^{\rho=4} (6 \rho^2 - \rho^4) \sin(\phi) \, d\rho \, d\theta \, d\phi[/tex]

[tex]\displaystyle = 2\pi \int_{\phi=0}^{\phi=\pi} \int_{\rho=0}^{\rho=4} (6 \rho^2 - \rho^4) \sin(\phi) \, d\rho \, d\phi[/tex]

[tex]\displaystyle = 2\pi \left(\int_{\phi=0}^{\phi=\pi} \sin(\phi) \, d\phi\right) \left(\int_{\rho=0}^{\rho=4} (6 \rho^2 - \rho^4) \, d\rho\right)[/tex]

[tex]\displaystyle = 2\pi \cdot 2 \cdot \left(-\frac{384}5\right) = \boxed{-\frac{1536\pi}5}[/tex]

but the mass can't be negative...

Chances are good that this question was recycled without carefully changing all the parameters. Going through the same steps as above, the mass of a spherical body with radius R and mass density given by

[tex]\delta(x, y, z) = k - (x^2 + y^2 + z^2)[/tex]

for some positive number k is

[tex]\dfrac{4\pi r^3}{15} \left(5k - 3r^2\right)[/tex]

so in order for the mass to be positive, we must have

5k - 3r² ≥ 0   ⇒   k ≥ 3r²/5

In this case, k = 6 and r = 4, but 3•4²/5 = 9.6.


Related Questions

find the greatest common factor GCF and the least common multiple LCM of these Numbers. 8and 9

Answers

Answer:

LCM is [tex]72[/tex]

GCF is [tex]1[/tex]

Step-by-step explanation:

LCM is least common multiplier

LCM of

[tex]8,9=8\times 9\\\\=72[/tex]

GCF is greatest common factor

Factors of [tex]8=1,2,4,8[/tex]

Factors of [tex]9=1,3,9[/tex]

Greatest common factor is [tex]1[/tex]

GCF is [tex]1[/tex]

Hey mathematicians ! Can anyone help me with these problems please ! Will give brainly !

Answers

27) 115=5(5a-2)

115= 25a-10

125=25a

a=5

28) -72=-x-6(x+5)

-72=-7x-30

-42=-7x

x=6

i did two of them. now i gtg so won't do the rest

Trey wants to earn more than $89 trimming trees. He charges $8 per hour and pays S7 in equipment fees. What are the possible numbers of hours Trey could
trim trees?
Use t for the number of hours.
Write your answer as an inequality solved for t.

Answers

Answer:

11 hours

Step-by-step explanation:

8t + 7 > 89

8t > 81 : subtract 7 from both sides

t > 10.125 : divide by 8 for both sides

t = 11 : estimate

Trey has to work 11 hours because 10 hours would only give him $87.

Triangles ABC and EDC are similar.
ACE and BCD are straight lines.
Angle BAC = angle DEC
Angle CBA= angle CDE
AB = = 6 cm, BC = 2.5 cm, CD = 7.5 cm and CE = 10.5 cm.

Answers

Answer:

18cm

Step-by-step explanation:

The ratio is given by: 7.5/2.5=3

DE = 6 × 3 = 18 cm

or

DC/CB = x/AB

7.5/2.5 = x/6

3=x/6

x=6×3=18 cm

Answer is 18. You can use triangle similarity

Which equation best matches the scenario below?

The total cost t is proportional to the number n of items purchased at a constant price of $4.

A) 4t = n
B) t = 4n
C) t = n ÷ 4
D) 4t = 4n

Answers

Answer:

Definetly t = 4n

Which symbol correctly compares the two angle measures? pi/4 radians_____30°
A. <
O B. >
C. =
I don't know the answer and thats my guess :,)​

Answers

Answer:

B. [tex]\frac{\pi }{4} Rad>30^{0}[/tex]

Step-by-step explanation:

[tex]\frac{\pi }{4} Rad=(180/4)^{0} =45^{0}[/tex]

Hope this helps

Please help, I will give brainliest Health Insurance Expenses
Business
Type
Definition
Costs
Premium
The annual amount employee pays
the insurance company for the policy.
$6,237
Career
College Rea
Deductible
The amount employee pays for
medical expenses before the
insurance pays for anything.
$1,000
Extra Cred
Business
The percentage employee pays for
Co-Payment medical treatment after the
deductible is paid
30%
Kerr Heavyn_He
hbenefitsss
Use the information in the table. If the employer has agreed to pay 45% of the employees' health insurance premium, what is the total annual
premium?
X
Select one:
O A. The premium is $13,860.
OB. The premium is $11,340,
O C. The premium is $5,103.

Answers

Answer:

It is B

Step-by-step explanation:

A coin weighs 3/4 ounces. Laura picked up 6 of these coins. What is the total weight of the coins she picked up? In simplest form.

Answers

Answer:

1 1/2 pound

Step-by-step explanation:

If each coin weighs 3/4 ounces and she picked up 6 then you do 3/4 x 6/1 which equals 18/4 and since there is 16 ounces in a pound it simplifies to 1 2/4 or 1 1/2 pounds

David plays baseball 18 times each month for 3 months. He also plays soccer 15 times each month for 3 months. How many more times does he play baseball than soccer each month?


Plzz help asp

Answers

Answer:

3

Step-by-step explanation:

3 months baseball = 18 x 3 = 54

3 months soccer = 15 x 3 = 45

54-45 = 9

9 / 3 = 3

David plays baseball 3 more times than soccer each month.

David plays 3 more times baseball than soccer each month.

What is ratio?

Ratio basically compares quantities, that means it shows the value of one quantity with respect to the other quantity.

If a and b are two values, their ratio will be a:b,

Given that,

David plays baseball for 3 months, and each month he plays 18 times.

Also, David plays soccer for 3 months, and each month he plays 15 times.

In 3 months, David plays baseball = 18 x 3 = 54

Also, in 3 months, David plays soccer = 15 x 3 = 45

The more number of times David plays baseball than soccer

= 54 - 45

= 9

In 3 months, David plays 9 more times baseball than soccer.

In one month, David plays 9/3 = 3 more times baseball than soccer.

3 more times David plays baseball than soccer each month.

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Write -37.75 as a mixed number.​

Answers

Answer:

[tex]-37\frac{3}{4}[/tex]

Step-by-step explanation:

0.75 is equivalent to three quarters, or (3/4).

3/4 would be the fraction. -37 would be the whole number.

[tex]-37.75 = \boxed{-37\frac{3}{4}}[/tex]

Hope this helps.

Answer:

- 37 3/4

Step-by-step explanation:

What is the function rule for the line?

Answers

Answer: It is f(x)= -3/2x-2

Step-by-step explanation: f(x) is the same as y. The formula for a line is slope-intercept form (y=mx+b). The -2 is the y-intercept, and the line intercepts the graph at -2. The slope is -3/2 because it rises 3 and goes to the left -2.

Determine the period of the function g(x) = (cos ;)

Answers

Answer:

6[tex]\pi[/tex]

Step-by-step explanation:

The period of a sinusoidal function in the form a * cos (b (x + h)) + m is [tex]\frac{2\pi }{|b|}[/tex]. Therefore, we can see that "b" in g(x) is 1/3 and 1/3 substituted into [tex]\frac{2\pi }{|b|}[/tex] is 2[tex]\pi[/tex]/1/3 which is 6[tex]\pi[/tex].

Though it just opened last month, the Mash Mobile food truck has been doing well; they're now serving an average of 200 mashed potato creations every day. If there are 8 ounces of potatoes in each dish, how many 50-pound bags of potatoes does the food truck use every day?

Answers

Answer:

2 50-pound bags

Step-by-step explanation:

[Conversions] 50-pound bags -> 800-ounce bags

[Total potatoes used] 200 mashed potato creations * 8 ounces of potatoes in each dish = 1,600 ounces of potatoes

[Bags used] 1,600 / 800 = 2 bags

Have a nice day!

     I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly. (ノ^∇^)

- Heather

if the poins A(-6,-5),B(-3,-2) and C(3,k) is on the same straight line, find the value of k.

Answers

Answer:

4

Step-by-step explanation:

y = x + 1

yeah-ya.... right?

Answer:

4 is the answer i believe

Step-by-step explanation:

C skips 2 dots, Hope this helps:)

find the quotient 3^1^5/3^3 3^5 3^18 3^12 3^45

Answers

Answer:

It's about drive its about power put in the work put in the hours and take what's ours

Step-by-step explanation:

It's about drive its about power put in the work put in the hours and take what's ours It's about drive its about power put in the work put in the hours and take what's ours It's about drive its about power put in the work put in the hours and take what's ours It's about drive its about power put in the work put in the hours and take what's ours It's about drive its about power put in the work put in the hours and take what's ours

Help help help math math

Answers

Answer:

yes

Step-by-step explanation:

x increases by 2, and y increases by 3 consistently

Answer:

No

Step-by-step explanation:

please mark me brainliest

The triangle and rectangle have the same perimeter.find the value of x.

Answers

Answer:

[tex]x + 5 \: + \: x \: + 6 \: + x + 7 = 2x - 1 + x + 7 + x + 7 + 2x - 1[/tex]

[tex]3x + 18 = 6x + 12[/tex]

[tex]18 = 3x + 12[/tex]

[tex]6 = 3x[/tex]

[tex]x = 6 \div 3 = 2[/tex]

[tex]x = 2[/tex]

I hope this helps you !!

Please help derivitive
Please dont comment if you dont know …

Answers

[tex]\lim \limits_{x \to 0^{+}} \left( \csc x + \dfrac 1x \right)\\=\lim \limits_{x \to 0^{+}} \csc x + \lim \limits_{x \to 0^{+}} \dfrac 1x\\\\=\infty[/tex]

The midpoint of AB is M(1, -4). If the coordinates of A are (-3, -6), what are the coordinates of B

Answers

Answer:

Step-by-step explanation:

x value of M is 1 - (-3) = 4 units to the right of A

B will be another 4 units to the right of M or x = 1 + 4 = 5

y value of M is -4 - (-6) = 2 units above A

B will be another 2 units above M or y = -4 + 2 = -2

B = (5, -2)

An angle of 90° measures (___) Pi radians.

An angle of -pi/5 radians measures ____?

°.

Answers

[tex]\huge \bf༆ Answer ༄[/tex]

Conversion from degree to radian ~

[tex] \boxed{ \sf{ \frac{\pi}{180} \times \theta }}[/tex]

[tex] \sf \dfrac{\pi}{180 } \times 90[/tex]

[tex] \sf \dfrac{\pi}{2} \: \: rad[/tex]

Conversion from radians to degree ~

[tex] \boxed{ \sf{ \frac{180}{\pi} \times \theta}}[/tex]

[tex] \sf \dfrac{180}{\pi} \times - \dfrac{ \pi}{5} [/tex]

[tex] \sf - 36 \degree[/tex]

The required measures of the angles in radians and degree are given as  π/2 radians and -36° respectively.

What are the angle?

Orientation of the one line with respect to the horizontal or other respective line is known as a measure of orientation and this measure is known as the angle.

here,
An angle of 90° measures
in radians,
angle = π/180 × Ф
angle = π/180 × 90 = π/2 radians

Similarly,
An angle of -pi/5 radians
in degrees,
angle = 180 / π × -π/5
angle = -36°

Thus, the required measures of the angles in radians and degree are given as  π/2 radians and -36° respectively.

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List the first 5 multiples and find ALL the factors of 16.

Multiples:
Factors:

Answers

Answer:

Multiples: 16, 32, 48, 64, 80 Factors: 1, 2, 4, 8 and 16

Step-by-step explanation:

hope this helped :3

Getting trained and licensed as a truck driver involves a one-time cost of $450. Gas and insurance end up costing about $2 per mile. For the first delivery, you will get paid $350 plus $6 per mile that you drive. Write 2 equations and solve this system.​

Answers

The question is an illustration of a system of linear equations.

The system of linear equations is: [tex]y=450 + 2x[/tex] and [tex]y = 350 + 6x[/tex]The solution to the equation is (25,500)

Let the number of miles be x.

Getting trained and licensed

One time cost: $450Gas and Insurance: $2 per mile

So, the equation is:

[tex]y=450 + 2x[/tex]

Delivery

First delivery: $350Rate: $6 per mile

So, the equation is:

[tex]y = 350 + 6x[/tex]

Hence, the system of linear equations is:

[tex]y=450 + 2x[/tex] and [tex]y = 350 + 6x[/tex]

To solve the equation, we equate both, as follows:

[tex]450 + 2x = 350 + 6x[/tex]

Collect like terms

[tex]6x - 2x = 450 - 350[/tex]

[tex]4x = 100[/tex]

Divide both sides by 4

[tex]x = 25[/tex]

Substitute 25 for x in [tex]y=450 + 2x[/tex]

[tex]y = 450 + 2 \times 25[/tex]

[tex]y = 450 + 50[/tex]

[tex]y = 500[/tex]

Hence, the solution to the equation is (25,500)

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The radius of the semicircle is 3 in. How long is its perimeter?
Please be accurate and explain.

Answers

Answer:

15.42 inches

Step-by-step explanation:

-formula : perimeter of semi circle= πr+2r

r=3 inch

π.3+2.3

=3π+6

≅15.42inch

Answer:

15.425 or 6+3π

Step-by-step explanation:

If the radius of the circle is 3 inches and the diameter is double the radius, we know the diameter is 6 inches. Then we find half of the perimeter of a circle with radius 3. The perimeter formula of a cirle is 2πr. Plugging in the radius we get: 2(3)π or 6π. Then we divide by 2 as it is a semicircle and get 3π or ≈9.425. We add 6 to this and we get the answer of 15.425 or 6+3π.

Factor: 3x4y3 – 48y3

Answers

Answer: 4y3(1x1 - 12)

3x4y3 – 48y3

4y3(1x1 - 12)

Given that f(x) = ax – 5 and f(4) = 15, determine the value of a.

Answers

Given that f(x) = ax – 5 and f(4) = 15, then the value of a is 5

The given function is:

f(x)  =  ax  -  5

If f(4)  =  15, then x = 4

Substitute x = 4 into f(x) = ax - 5

f(4)  =  a(4)  -  5

15   =  4a   -  5

Solve for a by adding 5 to both sides

15  +  5  =  4a  -  5  +  5

20    =  4a

Divide both sides by 4

4a/4   =  20/4

a    =  5

Given that f(x) = ax – 5 and f(4) = 15, then the value of a is 5

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The radius of a circle is 19 miles. What is the circle's circumference?

Answers

Answer:

119.32 miles

Step-by-step explanation:

radius=19 miles

circumference =2×3.14×19 miles

The circle's circumference 119.32 miles

What is circle's circumference ?

Circumference of the circle or perimeter of the circle is the measurement of the boundary of the circle. Whereas the area of circle defines the region occupied by it.  If we open a circle and make a straight line out of it, then its length is the circumference. It is usually measured in units, such as cm or unit m.

GIven that,  radius of a circle is 19 miles

we know , circumference of circle = [tex]2\pi r[/tex]

                                                         = [tex]2\pi \cdot19[/tex]

                                                         = 38 (3.14)

                                                         = 119.32 miles

Hence, circle's circumference 119.32 miles

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x²- 8x-9=0 i need some help over here

Answers

Factorise the equation first of all.

The easiest way to do this is to find two number that add to make the middle (-8) and multiply to make the last number (-9)

Two numbers with this property are -9 and 1

So when you factorise, your equation becomes (x-9)(x+1) = 0

From these you can conclude that for the equation to be equal to 0, either

x-9= 0
So x = 9

Or
x+1 = 0
And
x= -1

Answer:

(x+1)(x-9)

Step-by-step explanation:

Break the equation: x²- 8x-9=0 into groups:

([tex]x^{2}[/tex]+x)+(-9x-9)

Factor X out of [tex]x^{2}[/tex]+x:    x(x+1)

Then Facor -9 out of (-9x-9):  -9(x-1)

This means that you can simplify these 2 equations: (x-9) & (x+1)

So your answer is: (x+1)(x-9)

How are the two triangles congruent? If not congruent, select not
congruent.
1 point
SSS
SAS
ASA
AAS
ОО
Not congruent

Answers

Answer:

SAS

Step-by-step explanation:

there are vertical angles in the middle and the sides are already marked

How do you write 2/5 as a percentage

Answers

Answer:

40%

Step-by-step explanation:

To convert a decimal to a fraction, multiply it to 100

[tex]\frac{2}{5} \times 100[/tex]

[tex]=\frac{2}{5} \times \frac{100}{1}[/tex]

[tex]=\frac{2\times100}{5\times1}[/tex]

[tex]=\frac{200}{5}[/tex]

[tex]=40[/tex]

Therefore, [tex]\frac{2}{5}[/tex] is 40%

Step-by-step explanation:

Explanation:

To get 25 as a percent, we need the denominator to be 100 .

Now that we have a denominator equal to 100 , just look at the numerator and that's your percent.

40%

.40.

.4 or 0.4.

{(23,4),(-3,8),(-25,-21),(20,13),(21,-14)} what is the domain and range ​

Answers

Answer:

Step-by-step explanation:

The domain: { - 25, - 3, 20, 21, 23 }

The range: { - 21, - 14, 4, 8, 13 }

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