I need helpp please help me!!!

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

3 years ago

F(g(1)) if f(x)=5x g(x)=2x-3

RoseWind [281]

Answer:-5

Step-by-step explanation:

g(1)=2*1-3=2-3=-1

f(g(1))=5*-1=-5

7 0

3 years ago

Read 2 more answers

M3L3

eduard

The area of the David's pantry = 10 square feet.

The length of the pantry = 5 feet.

<h3>Define the term area of the rectangle?</h3>

The territory inhabited by a rectangle inside its 4 sides or limits is known as its area.
The area of such a rectangle is determined by its sides. Essentially, the formula calculating area is equivalent to the product of the rectangle's length and breadth.

For the given question.

David's kitchen is 60 square feet in size.
The kitchen is six times the size of the living room.

Thus,

Area of kitchen = 6 x area of the pantry

60 = 6 x area of the pantry

area of the pantry = 60 / 6 = 10 square feet.

Area of the pantry = length x breadth

10 = length x 2

length = 10/2 = 5 feet.

Thus, the length of the pantry is found as the 5 feet.

To know more about the area of the rectangle, here

brainly.com/question/16239445

#SPJ4

6 0

1 year ago

100 POINTS AND MARKED AS BRAINLIEST IF YOU ANSWER THESE.

vazorg [7]

Answer:

Step-by-step explanation:1. Sure. What you are looking for is a way to reduce both 24000 and 50 by the same factors until the 5 becomes a 1. In which case whatever 24000 becomes is your answer.

When 50 is reduced to 1 that's the end point because we are saying 1 unit of whatever the 50 represents contains nnn units of that other number. nnn is the answer.

So first of all we can reduce both numbers by a factor of 10 to get 2400 and 5. Now we reduce both by a factor of 5 to get 480 and 1. And there's your answer 480. ANS. 2. A all i know hope this helps luv

8 0

3 years ago

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In addition to his hourly rate, Mr. Smith is paid $2.20 for each watch he fixes. Write an expression for his income if he fixes

cricket20 [7]

Step-by-step explanation:

y = hourly income

r = number of watches fixed

equation: y = 2.20r + 9.5

5 0

3 years ago