A total of 200 video game players take a survey on their favorite game. Unknown Kingdom

Find the volume of the region between the planes x plus y plus 2 z equals 2 and 4 x plus 4 y plus z equals 8 in the first octant

Alex787 [66]

Find the intercepts for both planes.

Plane 1, x + y + 2z = 2:

$y=z=0\implies x=2\implies (2,0,0)$

$x=z=0\implies y=2\implies(0,2,0)$

$x=y=0\implies 2z=2\implies z=1\implies(0,0,1)$

Plane 2, 4x + 4y + z = 8:

$y=z=0\implies4x=8\implies x=2\implies(2,0,0)$

$x=z=0\implies4y=8\impliesy=2\implies(0,2,0)$

$x=y=0\implies z=8\implies(0,0,8)$

Both planes share the same x- and y-intercepts, but the second plane's z-intercept is higher, so Plane 2 acts as the roof of the bounded region.

Meanwhile, in the (x, y)-plane where z = 0, we see the bounded region projects down to the triangle in the first quadrant with legs x = 0, y = 0, and x + y = 2, or y = 2 - x.

So the volume of the region is

$\displaystyle\int_0^2\int_0^{2-x}\int_{\frac{2-x-y}2}^{8-4x-4y}\mathrm dz\,\mathrm dy\,\mathrm dx=\displaystyle\int_0^2\int_0^{2-x}\left(8-4x-4y-\frac{2-x-y}2\right)\,\mathrm dy\,\mathrm dx$

$=\displaystyle\int_0^2\int_0^{2-x}\left(7-\frac72(x+y)\right)\,\mathrm dy\,\mathrm dx=\int_0^2\left(7(2-x)-\frac72x(2-x)-\frac74(2-x)^2\right)\,\mathrm dx$

$=\displaystyle\int_0^2\left(7-7x+\frac74 x^2\right)\,\mathrm dx=\boxed{\frac{14}3}$

3 0

2 years ago

Nathan has 80 us stamps,64 Canadian stamps,and 32 Mexican stamps. He wants to put the same number of stamps on each page. Each p

Feliz [49]

Answer:

The greatest number of stamps that Nathan can put on each page = 16.

Step-by-step explanation:

Given:

Nathan has:

80 US stamps

64 Canadian stamps

32 Mexican stamps

The stamps need to put on a page such that each page has same number of same country stamps on each page.

To find the greatest number of stamps he can put on each page.

Solution:

In order to find the greatest number of stamps Nathan can put on each page, we will find the G.C.F. of the three numbers.

The numbers are:

$80,64,32$

We will list down the prime factors of each number.

$80=2\times 2\times 2\times 2\times 5$

$64=2\times 2\times 2\times 2\times 2\times 2$

$32=2\times 2\times2\times 2\times 2$

The G.C.F can be given as = $2\times 2\times 2\times 2$ = 16

Thus, the greatest number of stamps that Nathan can put on each page = 16.

6 0

2 years ago

Mr. Wilson bought a bag of birdseed and put half of it in his bird feeder. He split the other half equally among his 4 pet birds

pickupchik [31]

Answer:

each pet bird got 1/8 of the bag of birdseed

Step-by-step explanation:

the easiest way to do this is to divide the bag into eighths. if you divide the bag into eighths, 4/8 of the bag, being half, goes into the bird feeder. the other 4/8 of the bag is divided among Mr. Wilson's four pet birds, leaving each bird to get 1/4 of the bag when split evenly.

hope this helps

5 0

2 years ago

Pa answer po thank you

Arisa [49]

Answer:

hope it helps you

Step-by-step explanation:

To make such a frequency distribution table, first, write the class intervals in one column. Next, tally the numbers in each category based on the number of times it appears. Finally, write the frequency in the final column. A frequency distribution table drawn above is called a grouped frequency distribution table.

4 0

1 year ago

Two pairs of statements about Jeff's housing expenditures are given below. How can both pairs of statements be true?

Irina18 [472]

Explanation:

Pair 1 is true if Jeff's monthly income is $600/20% = $3,000.

Pair 2 is true if Jeff's monthly income is $1200/10% = $12,000.

Both pairs can be true if Jeff's monthly income increased by a factor of 4 in the 20 years from 1990 to 2010.

Obviously, Jeff spent more on housing in 2010. (Fortunately for Jeff, that larger expenditure was a smaller fraction of his income.)

5 0

2 years ago