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.
I've attached a photo of the question.
Answers
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
Ms. Usman buys about 14.4 gallons of gas every other week. How many gallons of ir gas does she buy in a year?
Answers
Answer:
374.4 gallons
Step-by-step explanation:
there are 52 weeks in a year
every other week means once every two weeks
52 / 2 =26
14.4 gallons x 26 = 374.4 gallons
please help me on this
Answers
Answer:
10/9
Step-by-step explanation:
Solve the equation for z, where z is restricted to the given interval.
[tex]y=7sinz[/tex], for z in [tex][-\frac{\pi }{2}, \frac{\pi }{2} ][/tex]
Answers
[tex]y = 7sin(z)\implies 0 = 7sin(z)\implies 0=sin(z)\implies sin^{-1}(0)=sin^{-1}[sin(z)] \\\\\\ sin^{-1}(0)=z\implies 0=z[/tex]
cant seem to figure this out
Answers
Answer:
log14=1.146
log18=1.255
Step-by-step explanation:
Use log rules to break up the given logs, to be able to use the information give
Using the product rule, log14 = log(7 x 2) = log7 + log2
Substitute .845 + .301 = 1.146
Using the quotient rule, log18 = log(2 x 3 x 3) = log2 + log3 + log 3
Substite .301 + .477 + .477 = 1.255
inverse function of y = 3x
Answers
Answer:
f-1(x) = 1/3x would be the answer
Step-by-step explanation:
Answer:
y= 1/3x
Step-by-step explanation:
inverse function perform the opposite.
3/1 becomes 1/3
A rectangular pool has a length of
4x + 7 feet and a width of 15x - 2 feet. Which
expression represents the perimeter of the
pool?
Answers
PLEASE HELP WILL MARK BRAINLIEST!
Answers
Answer:
Step-by-step explanation:
[tex]12000(1+\frac{.12}{4})^{4*45}=2454040.31463\\=2454040[/tex]
HELP PLS which box has the largest IQR
Answers
Answer:
Box B i think.
Step-by-step explanation:
Given the linear equation y = -3x
identify the initial value and rate of change
Hurry timed test!!
Answers
Answer:
here you go. hope u past the test
Step-by-step explanation:
I hope it's helpful :)
What is the answer to y=-2(-3/2)-9
Answers
Answer:
-6
Step-by-step explanation:
y = -2 x -3/2 - 9
3 - 9 = -6
Hey there!
y= -2 (-3/2) - 9
-2(-3/2) = [3]
y = 3 - 9
y = -6
Therefore, your answer is: y = -6
Good luck on your assignment and enjoy your day!
~Amphitrite1040:)
Add the matrices. Help please
Answers
Answer:
D) [tex]\left[\begin{array}{cc}-1&2&1&9\end{array}\right][/tex]
Step-by-step explanation:
You basically add the corresponding entries for each row and column:
[tex]\left[\begin{array}{cc}1&-4&3&5\end{array}\right]+\left[\begin{array}{cc}-2&6&-2&4\end{array}\right]=\left[\begin{array}{cc}1+(-2)&-4+6&3+(-2)&5+4\end{array}\right]=\left[\begin{array}{cc}-1&2&1&9\end{array}\right][/tex]
Therefore, option D, [tex]\left[\begin{array}{cc}-1&2&1&9\end{array}\right][/tex], is correct.
ms. valdez is preparing dinner for 6 people she bought an 8 pund tukey to rost for dinner and plans to distribute it evenly to each of the 6 people how many pounds of turkey should each person get
a- 1 1/3
b- 1 1/6
c 3/4
1/2
1/4
Answers
Answer:
A
Step-by-step explanation:
8/6 is also 1 1/3.
(assuming she doesn't include herself in the distribution).
You bought a magazine for $2 and some erasers for $5 each. You spent a total of $12. How many erasers did you buy?.
Answers
Answer:
2
Step-by-step explanation:
I got it right bro man my guy I got you
If you sleep 30% of the day. how many hours did you sleep?
answer fast please
Answers
Answer:
7.2 hours
Step-by-step explanation:
24 · 0.3 = 7.2
Round 4.645 to the nearest whole number! Please explain step by step to get marked
Answers
Step-by-step explanation:
to round 4.645 to the nearest whole number consider the tenths’ value of 4.645, which is 6 and equal or more than 5. Therefore, we have to round up: the whole number part of 4.645, 4, increases by 1 to 5, and the decimal point and all digits (.645) are removed.
4.645 rounded to the nearest whole number = 5
What is the GCF of 9y and 12y?
Answers
9 - 1,3,9
12 -1,3,4,12
There is a y in both terms.
So, 3y is the greatest common factor.
Part 1: Nouns
Directions: Write the plural form of each noun.
1. child
2. friend
3. baby
4. lunch
5. bike
Answers
Children
Friends
Babies
Lunches
Bikes
classify the following number 8.4444
Answers
Answer:
rational number
Step-by-step explanation:
Natural numbers:
This is every number in the sequence 1, 2, 3, 4...
Whole numbers:
This is all of the above, but it also includes 0.
Integers:
All of the above, but this also includes negative numbers.
Rational numbers:
A rational number is any number than can be represented as a fraction of 2 integers. All of the above are also rational numbers, and below are some other examples:
[tex]1.125, \frac{1}{3}, -5\frac{2}{7}[/tex]
Irrational numbers:
Basically anything that wasn't covered above. These cannot be represented as a fraction of 2 integers.
[tex]\pi, \sqrt{3}[/tex]
=========================
You don't make it clear if this is a repeating decimal or not, but either way, this is a rational number as you can represent it as a fraction:
[tex]8.4444=\frac{21111}{2500}[/tex]
In the case that it was a repeating decimal, those are also rational numbers. Take this for example:
[tex]0.3333333...[/tex]
This is rational:
[tex]\frac{1}{3}=0.333333...[/tex]
Draw two lines with a slope of 1 / 2. What do you notice about the lines?
Answers
Answer:
The lines would be parallel.
Step-by-step explanation:
lines with the same slope are parallel.
25% of Santa's 9 reindeer were playing reindeer games. How many of Santa's reindeer were NOT playing reindeer
games?
Answers
Answer:
75%, or 7
Step-by-step explanation:
100-25=75
75% of 9 is 6.75 (because 9x0.75)
so we can round to 7, unless there's a decapitated reindeer involved
Answer: 6.75
Step-by-step explanation:
for this math problem we need to find the percentage and subtract that from the original number ( 9 ) . . .
what is 25% if 9 ? 2.25
what is 9 - 2.25 ? 6.75
The scale of a blueprint is 3 inches : 10 feet. On the blueprint, a house is 9 inches
a
long. What is the length in feet of the actual house?
Answers
Answer:
30/5=68x6=48 feet
Step-by-step explanation:
divide the demesions
ChallengE
See attachment and answer :)
Answers
[tex]\displaystyle{\sf\:4\:\sqrt{48}\:-\:\dfrac{5}{2}\:\sqrt{\dfrac{1}{3}}\:+\:6\:\sqrt{3}}[/tex]
Answer :[tex]\displaystyle{\boxed{\red{\sf\:4\:\sqrt{48}\:-\:\dfrac{5}{2}\:\sqrt{\dfrac{1}{3}}\:+\:6\:\sqrt{3}\:=\:\dfrac{127\:\sqrt{3}}{6}}}}[/tex]
Step-by-step-explanation:
We have given an expression.
We have to simplify the expression.
The given expression is
[tex]\displaystyle{\sf\:4\:\sqrt{48}\:-\:\dfrac{5}{2}\:\sqrt{\dfrac{1}{3}}\:+\:6\:\sqrt{3}}[/tex]
[tex]\displaystyle{\implies\sf\:4\:\sqrt{16\:\times\:3}\:-\:\dfrac{5}{2}\:\sqrt{\dfrac{1}{3}}\:+\:6\:\sqrt{3}}[/tex]
We know that,[tex]\displaystyle{\boxed{\pink{\sf\:\sqrt{a\:\times\:b}\:=\:\sqrt{a}\:\times\:\sqrt{b}\:}}\:\cdots\sf\:a\:,\:b\:\geq\:0}[/tex]
[tex]\displaystyle{\implies\sf\:4\:\sqrt{16}\:\times\:\sqrt{3}\:-\:\dfrac{5}{2}\:\sqrt{\dfrac{1}{3}}\:+\:6\:\sqrt{3}}[/tex]
[tex]\displaystyle{\implies\sf\:4\:\times\:4\:\sqrt{3}\:-\:\dfrac{5}{2}\:\sqrt{\dfrac{1}{3}}\:+\:6\:\sqrt{3}}[/tex]
We know that,[tex]\displaystyle{\boxed{\blue{\sf\:\sqrt{\dfrac{a}{b}}\:=\:\dfrac{\sqrt{a}}{\sqrt{b}}\:}}\:\cdots\sf\:b\: > \:0}[/tex]
[tex]\displaystyle{\implies\sf\:16\:\sqrt{3}\:-\:\dfrac{5}{2}\:\times\:\dfrac{\sqrt{1}}{\sqrt{3}}\:+\:6\:\sqrt{3}}[/tex]
[tex]\displaystyle{\implies\sf\:16\:\sqrt{3}\:-\:\dfrac{5}{2}\:\times\:\dfrac{1}{\sqrt{3}}\:+\:6\:\sqrt{3}}[/tex]
[tex]\displaystyle{\implies\sf\:\dfrac{16\:\sqrt{3}\:\times\:2\:\sqrt{3}\:-\:5}{2\:\sqrt{3}}\:+\:6\:\sqrt{3}}[/tex]
[tex]\displaystyle{\implies\sf\:\dfrac{16\:\times\:2\:\sqrt{3}\:\times\:\sqrt{3}\:-\:5}{2\:\sqrt{3}}\:+\:6\:\sqrt{3}} \\ \\\displaystyle{\implies\sf\:\dfrac{32\:\times\:3\:-\:5}{2\:\sqrt{3}}\:+\:6\:\sqrt{3}}[/tex]
[tex]\displaystyle{\implies\sf\:\dfrac{8\:\times\:4\:\times\:3\:-\:5\:+\:(\:6\:\sqrt{3}\:\times\:2\:\sqrt{3}\:)}{2\:\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{8\:\times\:12\:-\:5\:+\:(\:6\:\times\:2\:\times\:\sqrt{3}\:\times\:\sqrt{3}\:)}{2\:\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{8\:\times\:12\:-\:5\:+\:(\:12\:\times\:3\:)}{2\:\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{8\:\times\:12\:-\:5\:+\:12\:\times\:3}{2\:\sqrt{3}}} \\ \\ \\ \displaystyle{\implies\sf\:\dfrac{8\:\times\:12\:+\:12\:\times\:3\:-\:5}{2\:\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{12\:(\:8\:+\:3\:)\:-\:5}{2\:\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{12\:\times\:11\:-\:5}{2\:\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{132\:-\:5}{2\:\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{127}{2\:\sqrt{3}}} \\ \\ \\ \displaystyle{\implies\sf\:\dfrac{127}{2\:\sqrt{3}}\:\times\:\dfrac{\sqrt{3}}{\sqrt{3}}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{127\:\sqrt{3}}{2\:\times\:3}} \\ \\ \\\displaystyle{\implies\sf\:\dfrac{127\:\sqrt{3}}{6}} \\ \\ \\ \displaystyle{\therefore\:\underline{\boxed{\red{\sf\:4\:\sqrt{48}\:-\:\dfrac{5}{2}\:\sqrt{\dfrac{1}{3}}\:+\:6\:\sqrt{3}\:=\:\dfrac{127\:\sqrt{3}}{6}}}}}[/tex]
Solve the equations below exactly. Give your answers in radians, and find all
possible values for t in the interval 0 (less than or equal to ) t (less than or equal to) 2 pi
If there is more than one
answer, enter your solutions in a comma separated list.
(a) sin (t) = - square root of 3 divide by 2
when t=
(b) cos (t) = 1/2
when t=
Answers
Answer:
See below (along with unit circle attached)
Step-by-step explanation:
[tex]sin(t)=-\frac{\sqrt{3}}{2}[/tex]
[tex]t=\frac{4\pi}{3},\frac{5\pi}{3}[/tex]
[tex]cos(t)=\frac{1}{2}[/tex]
[tex]t=\frac{\pi}{3},\frac{5\pi}{3}[/tex]
It really helps to use the unit circle! I've attached it below for your convenience. Keep in mind that a point on the unit circle is defined by [tex](cos(\theta),sin(\theta))[/tex] where [tex]\theta[/tex] is measured in radians.
Help bell bell help help help
Answers
Answer:
answer is ‐10
Step-by-step explanation:
(x+14) =(5x-74)pick like terms together (x+5x)=(+14‐74)(6x)=(‐60)divide by 6answer is ‐10I NEED HELP!!
If f(x) = x 3/2 , then f ' (4) =
A. -6
b. -3
c. 3
d. 6
e. 8
Answers
Answer:
f(x)=x(3/2)
f(4)=4*3/2
6
solve for x??????????
Answers
Answer:
x=4
Step-by-step explanation:
you take both equations, and make them equal to each other. so....
5x-6=3x+2 subtract 3x from both sides
2x-6=2 then add 6 to the other side.
2x=8 then divide 8 by 2.
X=8
Find the least common denominator for these fractions. Enter your answer in
the space provided.
Answers
Answer: 27
Step-by-step explanation: LCM of 3 and 27 is 27.
write a quadratic equation in standard form.
1. -6 and 4 are solutions
2. 2 and 5 are solutions
1 and 2 are two different problems
Answers
Answer:
1. x^2 +2x -24 = 0
2. x^2 -7x +10 = 0
Step-by-step explanation:
If you look at the factored and expanded forms of a quadratic with solutions p and q, you see ...
(x -p)(x -q) = x^2 -(p+q) +pq
The x-term coefficient is the opposite of the sum of solutions.
The constant is the product of solutions.
This knowledge lets you write down the standard form equation with no particular effort.
1. x^2 +2x -24 = 0
2. x^2 -7x +10 = 0
Help please and thanks!
Answers
okay. first you must get the properties of a triangle. all three angles are 180 degrees in total. the angle of a straight line is also 180 degrees. both the triangle and the straight line shares the same angle ACB. so you can conclude that 5x-35 = 2x+58
-2x from both sides and 3x-35= 58
so 3x = 58+ 35= 93
x= 31
ABC = 31x2 = 62
BCA = 180-58-62= 60 degrees
please give me brailiest . thanks!
Answer:
x =31°
ABC is 62°
BCA IS 60°
I HOPE THIS IS HELPING
Simplify the ratio:
42 : 28 : 21
Please explain step by step to get marked as brainliest!
Answers
Answer:
6:4:3
Step-by-step explanation:
Use 7 to simplify