Answer:
Volume of the Tetrahedron T =[tex]\frac{1}{3}[/tex]
Step-by-step explanation:
As given, The tetrahedron T is bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0
We have,
z = 0 and x + 2y + z = 2
⇒ z = 2 - x - 2y
∴ The limits of z are :
0 ≤ z ≤ 2 - x - 2y
Now, in the xy- plane , the equations becomes
x + 2y = 2 , x = 2y , x = 0 ( As in xy- plane , z = 0)
Firstly , we find the intersection between the lines x = 2y and x + 2y = 2
∴ we get
2y + 2y = 2
⇒4y = 2
⇒y = [tex]\frac{2}{4} = \frac{1}{2}[/tex] = 0.5
⇒x = 2([tex]\frac{1}{2}[/tex]) = 1
So, the intersection point is ( 1, 0.5)
As we have x = 0 and x = 1
∴ The limits of x are :
0 ≤ x ≤ 1
Also,
x = 2y
⇒y = [tex]\frac{x}{2}[/tex]
and x + 2y = 2
⇒2y = 2 - x
⇒y = 1 - [tex]\frac{x}{2}[/tex]
∴ The limits of y are :
[tex]\frac{x}{2}[/tex] ≤ y ≤ 1 - [tex]\frac{x}{2}[/tex]
So, we get
Volume = [tex]\int\limits^1_0 {\int\limits^{1-\frac{x}{2}}_{y = \frac{x}{2}}\int\limits^{2-x-2y}_{z=0} {dz} \, dy \, dx[/tex]
= [tex]\int\limits^1_0 {\int\limits^{1-\frac{x}{2}}_{y = \frac{x}{2}}{[z]}\limits^{2-x-2y}_0 {} \, \, dy \, dx[/tex]
= [tex]\int\limits^1_0 {\int\limits^{1-\frac{x}{2}}_{y = \frac{x}{2}}{(2-x-2y)} \, \, dy \, dx[/tex]
= [tex]\int\limits^1_0 {[2y-xy-y^{2} ]}\limits^{1-\frac{x}{2}} _{\frac{x}{2} } {} \, \, dx[/tex]
= [tex]\int\limits^1_0 {[2(1-\frac{x}{2} - \frac{x}{2}) -x(1-\frac{x}{2} - \frac{x}{2}) -(1-\frac{x}{2}) ^{2} + (\frac{x}{2} )^{2} ] {} \, \, dx[/tex]
= [tex]\int\limits^1_0 {(1 - 2x + x^{2} )} \, \, dx[/tex]
= [tex]{(x - x^{2} + \frac{x^{3}}{3} )}\limits^1_0[/tex]
= 1 - 1² + [tex]\frac{1^{3} }{3}[/tex] - 0 + 0 - 0
= 1 - 1 + [tex]\frac{1 }{3}[/tex] = [tex]\frac{1}{3}[/tex]
So, we get
Volume =[tex]\frac{1}{3}[/tex]
The volume of Tetrahedron will be [tex]\frac{1}{3}[/tex] units.
Given,
The tetrahedron T is bounded by the planes,
[tex]x+2y+z=2......(1)\\ x=2y....(2)\\ x=0......(3) \\ z=0.....(4)[/tex]
From equation (1),
[tex]z=2-x-2y[/tex]
So the limits of z will be from 0 to [tex]2-x-2y[/tex].
Now, from equation (2),
[tex]y=\frac{x}{2}[/tex]
and from equation (1), putting z=0 we get,
[tex]x+2y=2\\y=1-\frac{x}{2}[/tex]
So the limits of y will be from [tex]\frac{x}{2}[/tex] to [tex]1-\frac{x}{2}[/tex].
On solving equation (1), for x we get
[tex]x+x+0=2[/tex] [tex](x=2y \ and \ z=0 )[/tex]
[tex]x=1[/tex].
So the limits of x will be from 0 to 1.
The volume of tetrahedron will be,
[tex]V=\int\limits^1_0 \, dx \int\limits^{1-\frac{x}{2} }_{\frac{x}{2} } \, dy \int\limits^{2-x-2y}_0 \, dz[/tex]
[tex]V=\int\limits^1_0 \, dx \int\limits^{1-\frac{x}{2} }_{\frac{x}{2} } \, dy [2-x-2y-0][/tex]
[tex]V=\int\limits^1_0 \, dx (1-2x+x^2)\\[/tex]
[tex]V=1-1+\frac{1}{3}[/tex]
[tex]V=\dfrac{1}{3}[/tex]
Hence the volume of tetrahedron is [tex]\frac{1}{3}[/tex] units.
For more details on Triple integration follow the link:
https://brainly.com/question/12632797
I need help I cant find my answer
Factor: -12 - 20
Answer 240
Step-by-step explanation: You multiply -12 x -20 which makes a positive bcuz negative x negative is positive and get 240
For every two old homes in a neigborhood there are 3 new ones. Write the ratio of old homes to new homes.
Answer:
2:3
Step-by-step explanation:
Solve: m = 5 – (-10).
GIVING BRAINLIST to people who is correct!
Answer:
15
Step-by-step explanation:
Alex took 5 exams in class. His scores on the first four exams were 75%, 68%,
92%, and 100%. What does Alex need to score on the fifth exam to have an
average of 85%?
90
086
94
82
What is the square root of X with an exponent of 7?
Answer:
[tex] {x}^{3} \sqrt{x} [/tex]
Step-by-step explanation:
hope it helps ☺️
suggestions : Photomath application
it will be explained you about how to find solutions.
3/4g=−12
Help me plz
Answer:
g=-16
Step-by-step explanation:
Answer:
-16=g
Step-by-step explanation:
so -12 is basically -12/1
SO if you out them togther
3/4g=-12/1
Proportion
So you cross multiply
–48=3g
Now divide
g= -16
A book 28 cm tall leans against the sidewall of a bookcase. The bottom of the book starts to slide away from the wall at a rate of 2 cm/sec. How fast is the top of the book moving down the wall when the bottom of the book is 12 cm from the wall?
Answer:
This is hard
Step-by-step explanation:
Answer:
[tex]\frac{3sqrt10}{10}[/tex]
Step-by-step explanation:
Which equation represents the same
linear function as the table below.
*
у
-4
5
0
N
N
N-
3
-
A
A y=2x-2
B 3x + 4y = 2
C -3x – 4y = -8
D y+4=-(4-5)
Answer:
I don't know child I wish I did child
In the following equation, what is the y-intercept? y = -3.1 +7 03 o7
Answer:
the y-intercept, b, is 7
Step-by-step explanation:
Compare the given y = -3.1 +7 to y = mx + b. I must assume that you meant
y = -3.1x +7. It becomes evident that the slope, m, is -3.1 and the y-intercept, b, is 7.
Someone help worth 5 points please help!
Answer:
10
Step-by-step explanation:
Solve the simultaneous equations
2x + 4y = 22
2x + 2y = 15
Answer:
1. x= 11+-27
2.x = 7.5 + -1y
Step-by-step explanation:
1. Move all terms containing x to the left, all other terms to the right.
Add '-4y' to each side of the equation.
2x + 4y + -4y = 22 + -4y
Combine like terms: 4y + -4y = 0
2x + 0 = 22 + -4y
2x = 22 + -4y
Divide each side by '2'.
x = 11 + -2y
Simplifying
x = 11 + -2y
2.Move all terms containing x to the left, all other terms to the right.
Add '-2y' to each side of the equation.
2x + 2y + -2y = 15 + -2y
Combine like terms: 2y + -2y = 0
2x + 0 = 15 + -2y
2x = 15 + -2y
Divide each side by '2'.
x = 7.5 + -1y
Simplifying
x = 7.5 + -1y
WILL GIVE BRAINLIEST TO FIRST TO RESPOND Shape 1 is similar to shape 2
Answer:
both are quadrilaterals
Step-by-step explanation:
Jamal will cut a piece of wood that is 2 1/2 feet long into 1/4-foot sections. How many sections will result?
A)5
B)8
C)10
D)12
Four friends did a job together that paid $85.20. How much should each get if they all get an equal amount? *
Answer:
21.30
Step-by-step explanation:
85.20/4
30% of___ is 15 ?????
Answer:
15 is 30% of 50
Step-by-step explanation:
We have, 30% × x = 15
Multiplying both sides by 100 and dividing both sides by 30,
we have x = 15 × 100/30
x = 50
If you are using a calculator, simply enter 15×100÷30, which will give you the answer.
Please help me out with this one
Step-by-step explanation:
Y = 5
X = 2.3
Use: y = mx + b
(m) is the slope which is 2.3
(b) is the intercept which is 5
Answer: Y = 2.3x + 5
* if I am wrong, I am sorry this is just what I learned:/
pls help, i need the answer by tonight!!
Answer:
Step-by-step explanation:
let distance=d
d²=4²+(2+4)²=16+36=52=4×13
d=2√13
Easy !
Quick !
Easy.
Answer:
2/5 or 40%
Step-by-step explanation:
there are 10 possible ways to choose 2 cards:
19,13
19,10
19,14
19,22
10,13
14,13
22,13
10,14
10,22
14,22
there are only 4 pairs whose sum is greater than 32:
19,14
19,22
13,22
14,22
Answer:
3/5
Step-by-step explanation:
is -2 greater than 4
Answer:
no
Step-by-step explanation:
Answer:
no
Step-by-step explanation:
can someone please help me
ANSWER FOR BRAINLIEST
Answer:
x = 10
Step-by-step explanation:
( 6x - 6 ) + ( 7x + 1 ) + 55 = 180
6x + 7x = 180 + 6 - 1 - 55
13x = 130
x = 10
Check:
6(10) - 6 = 56
7(10) - 1 = 69
55 + 56 + 69 = 180 ✔️
Evaluate the expression 42 + 5/9y
What's 20 – 2 > 16 simplified?
Step-by-step explanation:
To reduce this fraction, simply divide the numerator and denominator by 2 (the GCF). So, 216 = 2÷216÷2 = 18. Thus, 216 is equivalent to 18 in the reduced form. thank you
The distance from any point on a circle to the center of the circle is the ____.
circumference
diameter
area
radius
Answer:
radiusStep-by-step explanation:
The distance from a circle's center to a point on the circle is called the radius of the circle. A radius is a line segment with one endpoint at the center of the circle and the other endpoint on the circle.
Answer:
The distance from any point on a circle to the center of the circle is the radius.
pls help ASAP. 3+4√6÷√3+√2
Step-by-step explanation:
[tex] \frac{3 + 4 \sqrt{6} }{ \sqrt{3 } + \sqrt{2} } = \frac{3 + 9.798}{3.146} = \frac{12.798}{3.146} = 4.068 = 4.07[/tex]
Need correct answer only PLS
Which option identifies the sectors of the economy involved in the following scenario?
Joe owns a family farm in central Illinois. He goes to the bank for a loan for a new combine.
agriculture and services
services and industry
agribusiness and industry
agriculture and industry
Answer:
Step-by-step explanation:
A and c
Answer:
a
Step-by-step explanation:
Hope this helps:)
write an improper fraction 3 4/5
Answer:
19/5
Step-by-step explanation:
hope it helps
Answer:
19/5 = 3 4/5
Step-by-step explanation:
hope this helps
what is the y-intercept of the following equation:2x+y=5?
Answer:
Step-by-step explanation:
92
"What is the scale factor of the following if YZG is dilated to RBP?"
Answer:
1/2
Step-by-step explanation:
Hmmm.
.
.
.
.
.
.
.
.
.
Step-by-step explanation:
11.55 is the greater number
Answer:
The answer is >.
Step-by-step explanation:
11.55 ? 11 1/2=
11.55 ? 11.50
Since 11.55 is greater than 11.50 so the answer will be >.