Complete the equation for the line

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, <em>x</em> + <em>y</em> + 2<em>z</em> = 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, 4<em>x</em> + 4<em>y</em> + <em>z</em> = 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 <em>x</em>- and <em>y</em>-intercepts, but the second plane's <em>z</em>-intercept is higher, so Plane 2 acts as the roof of the bounded region.

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

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

Gaston is buying frult at a local grocery store. This table shows the cost per pound of each type of fruit. Match each

yuradex [85]

Answer:

bananas=$12.72

grapes=$9.48

oranges=$5.85

apples=$5.36

Step-by-step explanation:

To find the total cost for each fruit you would multiply the number of fruit by the cost.

bananas=3*4.24=$12.72

grapes=4*2.37=$9.48

oranges=3*1.95=$5.85

apples=2*2.68=$5.36

4 0

3 years ago

Each month, Liz pays $35 to her phone company just to use the phone. Each text she sends costs her an additional $0.05. In March

Firdavs [7]

<h3><u>Answer:</u></h3><h3><u>752 & 617</u></h3><h3 /><h3><u>Explanation:</u></h3><h3><u>So if Liz pays $35 each month just to use the phone, we could subtract 35 from the total bill that month to get the the total cost she spent on text messages.</u></h3><h3 /><h3><u>March:$72.60-$35=$37.60</u></h3><h3><u>April: $65.85-$35=$30.85</u></h3><h3 /><h3><u>We can see that in March Liz spent $37.60 on texts in total and in April she spent $30.85 on texts in total. All we have to do is divide the amount of money she spent on texts( $37.60 & $30.85) by the cost of one text message($0.05) to get the amount of texts she sent that month.</u></h3><h3 /><h3><u>March: $37.60/$0.05=752</u></h3><h3><u>April: $30.85/$0.05=617</u></h3><h3 /><h3><u>Therefore, she sent 752 texts in March and 617 texts in April.</u></h3><h3 /><h3><u>~FrxziteTheLxoser~ I hoped I help you :)</u></h3>

<u>If i helped you please give me brainIIlest!</u>

3 0

3 years ago

What is the interest rate $2,200 at 7.5% for 3 1/4 years

Phoenix [80]

Answer:

$536.25

Step-by-step explanation:

First, you need to find what 7.5% of $2,200 is,

(2200/100)7.5=165

then to find how much interest has accumulated over 3 1/4 year you need to multiply the one-year interest rate by the 3.25

(165)3.25=536.25

5 0

3 years ago

PLZ HELP I WILL GIVE BRAINLIEST...

Delicious77 [7]

Answer:

a. 6

b. 7

c. (b) Kira's

why? b. it has the large sample size

Step-by-step explanation:

7 0

3 years ago