1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
RSB [31]
2 years ago
8

Find the mass of the solid paraboloid Dequals=​{(r,thetaθ​,z): 0less than or equals≤zless than or equals≤8181minus−r2​, 0less th

an or equals≤rless than or equals≤99​} with density rhorho​(r,thetaθ​,z)equals=1plus+StartFraction z Over 81 EndFraction z 81. Set up the triple integral using cylindrical coordinates that should be used to find the mass of the solid paraboloid as efficiently as possible. Use increasing limits of integration.
Mathematics
1 answer:
Lubov Fominskaja [6]2 years ago
8 0

Answer:

M = 5742π  

Step-by-step explanation:

Given:-

- Find the mass of a solid with the density ( ρ ):

                             ρ ( r, θ , z ) = 1 + z / 81

- The solid is bounded by the planes:

                             0 ≤ z ≤ 81 - r^2

                             0 ≤ r ≤ 9

Find:-

Find the mass of the solid paraboloid

Solution:-

- The mass (M) of any solid body is given by the following triple integral formulation:

                           M = \int \int \int {p ( r ,theta, z)} \, dV\\\\

- We can write the above expression in cylindrical coordinates:

                           M = \int\limits\int\limits_r\int\limits_z {r*p(r,theta,z)} \, dz.dr.dtheta \\\\M = \int\limits\int\limits_r\int\limits_z {r*[ 1 + \frac{z}{81}] } \, dz.dr.dtheta\\\\

- Perform integration:

                           M = \int\limits\int\limits_r{r*[ z + \frac{z^2}{162}] } \,|_0^8^1^-^r^2 dr.dtheta\\\\M = \int\limits\int\limits_r{r*[ 81-r^2 + \frac{(81-r^2)^2}{162}] } \, dr.dtheta\\\\M = \int\limits\int\limits_r{r*[ 81-r^2 + \frac{6561 -162r + r^2}{162}] } \, dr.dtheta\\\\M = \int\limits\int\limits_r{r*[ 81-r^2 + 40.5 -r +\frac{r^2}{162} ] } \, dr.dtheta\\\\M = \int\limits\int\limits_r{[ 121.5r-r^2 -\frac{161r^3}{162} ] } \, dr.dtheta\\\\

                           M = 2*\int\limits_0^\pi {[ 121.5r^2-r^3 -\frac{161r^4}{162} ] } |_0^6 \, dtheta\\\\M = 2*\int\limits_0^\pi {[ 121.5(6)^2-(6)^3 -\frac{161(6)^4}{162} ] }  \, dtheta\\\\M = 2*\int\limits_0^\pi {[ 4375-216 -1288] }  \, dtheta\\\\M = 2*\int\limits_0^\pi {[ 2871] }  \, dtheta\\\\M = 5742\pi  kg              

- The mass evaluated is M = 5742π                      

You might be interested in
Jh is the altitude to the hypotenuse gi what is the measure of
iren2701 [21]

Answer:

We are told of a 90 degree angle. plus we are told that GH and HI are similar.

Hence: Angle I is = 45 degree or Option C

5 0
1 year ago
-4= m- 17<br> plz help me out plz
taurus [48]
m=13. All you have to do is add 17 on both sides of the equation
6 0
3 years ago
Read 2 more answers
How do I solve for this
Anon25 [30]

Answer: for 90 degrees it will be 1

for 180 degrees it will be \sqrt{2}/2

Step-by-step explanation:

7 0
3 years ago
A.) it has no reflectional symmetry
coldgirl [10]

Answer:

D.) It has rotational symmetry with an angle of rotation of 90 degrees.

Step-by-step explanation:

i) the shape shown in the figure has reflectional symmetry with four lines of symmetry. There are four lines of symmetry such that each line when drawn divides the shape into two exact halves such that if each half were too be folded across the line of symmetry it would produce the other half exactly.

Therefore Option A.) and Option B.) are not true

ii) It has  point symmetry which means if the shape is rotated by 180 degrees the shape does not change. Therefore Option C.) is not true.

iii) the shape has rotational symmetry because if the shape is rotated though any between 0 degrees and 360 degrees the shape appears to be unchanged.

Therefore the correct options are

D.) It has rotational symmetry with an angle of rotation of 90 degrees.

5 0
3 years ago
Jake has five cats and then he finds ten more cats how many does jake have after he loses 11 cats
AleksandrR [38]

jake has 5 cats

then he finds 10 cats

jake now has 5+10 = 15 cats

now he loses 11 cats

15 -11 =4

jake now has 4 cats

8 0
3 years ago
Read 2 more answers
Other questions:
  • GETT IT RIGHT AND GET BRAINLIEST
    10·2 answers
  • What is the intersection of the sets A = {3, 4, 7, 8, 13} and B= {3, 7, 10, 13, 17}
    5·2 answers
  • What's the improper fraction of 3 and 1/3
    5·2 answers
  • Rachel enjoys exercising outdoors. Today she walked 5 2/3 miles in 2 2/3 hours. What is Rachel's unit walking rate in miles per
    9·2 answers
  • What is the distance between the points (4,3) and (1,1) on the cordinate plane
    11·1 answer
  • Bella has 4 packs of spikes and 8 Beetlebots. The spikes come in packs
    6·1 answer
  • The intersections of 12th St., Constitutional Ave., and Pennsylvania Ave. in Washington, D.C. form a triangle. The Old Post Offi
    15·1 answer
  • ANSWER THIS QUESTION FOR 10 POINTS AND BRAINLEST
    5·2 answers
  • Help plzzzzzzzzzzzzzz ​
    8·1 answer
  • 7. Use the table to answer the following question.
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!