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n200080 [17]
3 years ago
9

a) find center of mass of a solid of constant density bounded below by the paraboloid z=x^2+y^2 and above by the plane z=4.(b) F

ind the plane z=c that divides the solid into two parts of equal volume. This plane does not pass through the center of mass.The answers in the back of the book are:(a) x-bar=y-bar=z-bar=(8/3)(b) c=2(sqrt(2))

Mathematics
1 answer:
slava [35]3 years ago
5 0

Answer: x-bar = y-bar = 0 whereas z-bar = 8/3

               M= (c^2)/8 which is intern equal to 2\sqrt{2}

Step-by-step explanation:

           Find the area, by setting the limits as

               = 4\cdot \int _0^{\frac{\pi }{2}}\int _0^2\int _{r^2}^4\:rdzdrd\theta

                =4\cdot \int _0^{\frac{\pi }{2}}\int _0^2r\cdot \:4-r^3drd\theta

                =4\cdot \int _0^{\frac{\pi }{2}}4d\theta

                 =8\pi

Therefore;

Mxy=  \int _0^{2\pi }\int _0^2\int _{r^2}^4\:zrdzdrd\theta

       z-bar = 8/3

M= 8\pi dividing it into two volume gives us = 4\pi

means  4\pi =\int _0^{2\pi }\int _0^{\sqrt{c}}\int _r^c\:rdzdrd\theta

             4\pi =\left(\pi c^2-2\pi \frac{c^{\frac{3}{2}}}{3}\right)

              c=2\sqrt{2}

       

                 

                 

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Step-by-step explanation:

The second equation in system B is only in terms of y, so we need to use elimination to eliminate the x term from the second equation in system A.

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5 (-x − 2y = 7)

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d. Based on the cumulative frequency polygon, about 75% of the employees accumulated how many miles or less?

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Step-by-step explanation:

The given frequency distribution representing the number of frequent flier miles accumulated by the employees in a company.                        

Frequent Flier Miles (000) Frequency

0                     up to 3 5

3                      up to 6 12

6                      up to 9 23

9                     up to 12 8

12                   up to 15 2

Total                       50

a. How many employees accumulated less than 3,000 miles?

The number of employees accumulated less than 3,000 miles= 5

So the number of employees accumulated less than 3,000 miles is 5.

b. Convert the frequency distribution to a cumulative frequency distribution.

We make the cumulative frequency distribution from the frequency distribution by adding up a frequency and all other frequencies below it.                

Frequent Flier Miles Frequency Cumulative frequency

0 up to 3                                5 5

3 up to 6                                      12 17

6 up to 9                                      23 40

9 up to 12                               8 48

12 up to 15                               2 50

Total 50

d. Based on the cumulative frequency polygon, about 75% of the employees accumulated how many miles or less?

8.5 thousands of miles, found by starting at 75 percent on the vertical axis on the right, over to the polygon and then down to the horizontal axis.

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