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
(ONLY MY FRIENDS RESPOND) What is the Area of a square if it has L= 32cm and W= 20cm?
(Heres a re-made thing for my friends, sorry for the rushed handwriting, I had to do it before my mom saw, hope u guys can read it)
Answer:
640
Step-by-step explanation: and wow
In the last 40 days, Mr. Neptune's newspaper has been delivered late 6 times. What is the experimental probability that it will be delivered late tomorrow?
Answer:
15%
Step-by-step explanation:
The reason for this is simply because if you take 40 and 6, turn it into a fraction, then turn it into a percentage, it turns out to be 15%
consider the two functions which statement is true?
Answer:
A
Step-by-step explanation:
Cinemark is having a summer special.If you get a membership one-time cost is $12.50 plus $2.00 per movie that you watch.Write a linear equation that will show the cost of membership and each movie seen
Answer:
Y=12.50x+2.00
Step-by-step explanation:
The linear equation that will show the cost of membership and each movie seen will be C = 2n + 12.5.
What is a linear equation?A relationship between two or more parameters that, when shown on a graph, produces a linear model. The degree of the variable will be one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
Cinemark is having a summer special. If you get a membership one-time cost is $12.50 plus $2.00 per movie that you watch.
The slope of the line is $2.00 and the y-intercept of the line is $12.50.
Let C be the total cost and n be the number of movies. Then the linear equation will be
C = 2n + 12.5
More about the linear equation link is given below.
https://brainly.com/question/11897796
#SPJ2
The diagram shows two congruent equilateral triangles whose overlap is a hexagon.
The areas of the smaller triangles, which are also equilateral, are 1, 1, 9, 9, 16 and 16, as shown.
16
16
X1
9
What is the area of the inner hexagon?
Answer:
38
Step-by-step explanation:
Need Help with this please!!
Answer:
A is 85 degrees
I need to write an equation for the triangle and solve for x
Graph y-4=2x and identify
the slope and y-intercept.
Answer:
slope is 2 and y-intercept is (0, 4)
Step-by-step explanation:
Answer:
s- 2 y int- 4
Step-by-step explanation:
A pair of jeans that originally cost $40 was marked up 50%. During a sale the jeans were discounted 50%. What
is the sale price?
Can you explain how to show my work step by step
Answer:
20$
Step-by-step explanation:
50% of 40= 20
Number seven plz helppp
Answer:
1,296
Step-by-step explanation:
I multiplied 6 four times.
Last question on homework I’m having trouble with ^_^
Answer:
25 + h
f^2
Step-by-step explanation:
It says the sum of 25 and h which is 25 + h.
Now, it basically says "25 + h" is divided by f cubed. Cubed means to the power of 2 (the exponent), so f cubed would be f^2 (f to the power of 2).
25 + h is divided by f^2
25 + h
f^2
(the reason I put "^2" is because I don't know how to put exponents on here, sorry)
If GEOM is a square, find m2GQE.
Answer:
90 is correct answer of it question
John spent $75 on a shopping trip for new clothes last week. He had expected to spend $100 on clothes,
Approximate Value Exact Value Error Absolute Error Ratio Percent Error
$100
$75
$25
How much was the absolute error in his estimate?
$
Intro
Answer:
Absolute error = $25
Step-by-step explanation:
Given that:
Amount spent by John on shopping trip = $75
Estimated amount to spent on shopping trip = $100
Absolute error = Approximate value - Exact value
Here,
Approximate value = $100
Exact value = $75
Absolute error = 100 - 75
Absolute error = $25
Percentage = [tex]\frac{Absolute\ error}{Exact\ value}*100[/tex]
Percentage = [tex]\frac{25}{75}*100[/tex]
Percentage = 33.33%
Hence,
Absolute error = $25
A laptop has a price $300. The store is giving a 15% discount for the laptop. How much will the laptop cost?
$35 bill before tax
tax:5.8%
Tip:21%
Answer:
If tip goes only on amount but not on tax: $44.38
If tip goes on total including tax: $44.81
Step-by-step explanation:
If the tip is applied to the price before tax, then we need to add 5.8% and 21% of $35 to $35.
tax + tip = 5.8% + 21% = 26.8%
100% + 26.8% = 126.8% = 1.268
1.268 * $35 = $44.38
If the tip is applied to the price plus tax, then we calculate the amount including tax first, then we add the 21% tip based on that price.
Bill before tax: $35
Add 5.8% tax: 1.058 * $35 = $37.03
Add tip to the amount including tax: 1.21 * $37.03 = $44.81
What hazard is present in this image?
Select the best option.
A mat that has upturned edges
Hair clippings on the floor
Tool cord left out
Spills on floor
Answer:
spills on the floor
Step-by-step explanation:
there is liquid of some sort someone could slip and hurt themselves. hope this helps
Plsssss helpppppppppp meeeeeeeeee
Answer:
x = 14, y = 37
Step-by-step explanation:
The angle vertical to (9x + 12 )° and 3x are same- side interior angles and are supplementary, sum to 180°, then
9x + 12 + 3x = 180
12x + 12 = 180 ( subtract 12 from both sides )
12x = 168 ( divide both sides by 12 )
x = 14, so
3x = 3 × 14 = 42
3x and 4y - 10 are adjacent angles and supplementary, then
3x + 4y - 10 = 180
42 + 4y - 10 = 180
4y + 32 = 180 ( subtract 32 from both sides )
4y = 148 ( divide both sides by 4 )
y = 37
Solve the linear equation x + 4 + 3x = 72. State the property that justifies your first step and why you chose it
Please Help!!!!!!!!!
Answer:
x=17
Step-by-step explanation:
x+4+3x=72
4x+4-4=72-4
4x/4=68/4
x=17
Associative property of addition
Answer: x = 17
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
What is the simplified form of the expression(x + 6)(7x − 1)?
8x^2 + 5
7x^2 + 41x − 6
7x^2 + 43x − 6
7x^2 − 6
Answer:c
Step-by-step explanation:
Answer:
B: 7x2 + 41x − 6
Step-by-step explanation:
Use the FOIL method to multiply the two expressions.
(First Outer Inner Last)
Then, combine like terms.
(x + 6)(7x − 1) = (x)(7x) + (x)(-1) + (6)(7x) + (6)(-1)
= 7x2 − x + 42x − 6
= 7x2 + 41x − 6
PLEASE HELP!!! I NEED TO TURN THI IN SOON !!!
Question and instructions in the image below!!!
MAKE SURE TO JUSTIFY YOUR ANSWER!!!!
Will mark brainliest if correct!!!
Answer:
3. f(x + 3)Step-by-step explanation:
Point A is 3 units to the left from the graph
The graph, if translated 3 units to the left will include the point A.
Horizontal translation is:
f(x) ⇒ f(x + 3)Correct option is 3.
Answer:
3. f(x + 3)
Step-by-step explanation:
Twice the difference of 1 and a number.
Mario start collecting stamps in January and buys 12 stamps each month.Karen starts collecting stamps in July and buys 18 starts each month. how many months after Karen starts collecting stamps will she have the same number of stamps as Mario?
Answer:
12 months
Step-by-step explanation:
Given that:
Mario:
12 stamps per month
Since January
Karen :
18 stamps per month
Starts on july
Number of stamps collected by Mario till the end of June :
12 * 6 = 72 stamps
For Mario :
72 + 12x
For karen:
18x
Where x is the number of months after karen starts :
Equate both equations
72 + 12x = 18x
72 = 18x - 12x
72 = 6x
x = 72/6
x = 12
After 12 months
Number of months after karen starts, will they have same amount of stamp
The volume of a cylinder is 1767 cm and its height is 11 cm.
What is the length of the cylinder's radius?
Enter your answer in the box.
Answer:
1778 cm
Step-by-step explanation:
7. Find the product. 8. Find the product. 9. Find the product.
Answer: 7) 6/17
8) 28/45
9)24/15
Step-by-step explanation:
Answer:
B is the answer
Step-by-step explanation:
no need to
what is the product of 2 4/7 × 13/12
Answer:
2 11/14
Step-by-step explanation:
First make both of them improper fractions.
therefore,
2 4/7 = 18/7
now simplify.
18/7 × 13/12
18 and 12 get cancelled off as they're multiples of 3 and 6
therefore,
3/7 × 13/2
3× 13 = 39
7×2 = 14
39/14
now make it a proper fraction
2 11/14//
Help pls question is the picture above :))))
What is the measure of angle P?
Answer:
60
Step-by-step explanation:
step 1: 180°-130°= 50
step 2: 180-110= 70
step 3: 70+50= 120
step 4: 180°-120°= 60
Juan received a bank account statement reporting the recent changes
to his account. The statement showed an overall increase in the
account balance of $1,909.15. During the time shown on the
statement, Juan was paid twice and withdrew a total of $1,437.99 to
cover all his bills and expenses. How much (x) is each of Juan's
paychecks?
Answer:
2628.15
Step-by-step explanation:
1909.15+ 1437.99/2=2628.15
In a certain year, the average e-mail user sent or received 111 e-mails each day. The number of e-mails received was 3 more than twice the number
sent. How many were sent and how many were received each day?
Answer:
Number of sent mail = 36
Number of received mail = 75
Step-by-step explanation:
Let,
x be the number of mails sent
y be the number of mails received
According to given statement;
x + y = 111 Eqn 1
y = 2x + 3 Eqn 2
Putting value of y from Eqn 2 in Eqn 1
x + 2x + 3 = 111
3x = 111 - 3
3x = 108
Dividing both sides by 3
[tex]\frac{3x}{3}=\frac{108}{3}\\x = 36[/tex]
Putting x = 36 in Eqn 2
y = 2(36) + 3
y = 72 + 3 = 75
Hence,
Number of sent mail = 36
Number of received mail = 75
Which of the following expressions will result in a product less than zero?
А
6.4 x (-3.5)
B
-7.1 x (2.5)
С
-2.3 (-1.9)
D
9.2 x (4.0)
Answer:
All of the except for D and C
Step-by-step explanation:
A negative times a positive is always negative
Answer:
A,B
Step-by-step explanation: