Use the Theorem of Pappus to find the exact volume of a general cone created by rotating a triangle with vertices ( 0 , 0 ) , ( 1 , 0 ) , and ( 0 , 8 ) around the y -axis.
Answers
The exact volume of a general Cone is [tex]\frac{8\pi}{3}[/tex]
To solve this problem, we will make use of the second theorem of Pappus. This theorem is concerned with finding the volume produced when a plane shape is rotated around an axis ( the volume of the solid of revolution ).
The theorem states:
The volume of a solid of revolution obtained by rotating a lamina F (the triangle) about a non-intersecting axis (the y-axis) lying in the same plane is equal to the product of the area A of the lamina F and the distance d traveled by the centroid of F.
That is
[tex]V=A\times d[/tex]
When the centroid travels, it forms a circumference d given by,
[tex]d=2\pi r[/tex]
Therefore
[tex]V=A\times 2\pi r[/tex]
From the diagram below,
The vertices (or, corners) of the triangle are A(0, 0), B(1, 0), and C(0,8).
G is the centroid of the triangle. The centroid of the triangle is the geometric center of the triangle. The centroid would be the balance point if a pin were to be used to balance the triangle.
r is the distance from the centroid to the axis of rotation ( the y-axis in this case). It is like saying the radius of rotation.
The things we need to find are A (the Area) and r.
Calculate A:
[tex]A=\frac{1}{2} \left| \Delta \right|[/tex]
[tex]\Delta = \left|\begin{array}{cc} x_{B}-x_{A} & y_{B}-y_{A} \\ x_{C}-x_{A} &y_{C}-y_{A}\end{array}\right|= \left|\begin{array}{cc} 1-0 & 0-0 \\ 0-0 & 8-0\end{array}\right|= \left|\begin{array}{cc} 1 & 0 \\ 0 & 8\end{array}\right|[/tex]
[tex]\Delta =8\times 1 -0\times 0=8[/tex]
[tex]A=\frac{1}{2} \left|8\right|=\frac{1}{2}\times 8=4[/tex]
Calculate r:
[tex]r=\overline{x}=\frac{x_A+x_B+x_C}{3}=\frac{0+1+0}{3}=\frac{1}{3}[/tex]
Finally, Calculate the volume, V:
[tex]V=A\times 2\pi r=4 \times2\pi\times\frac{1}{3}=\frac{8\pi}{3}[/tex]
Learn more about centroids: https://brainly.com/question/20305516
Related Questions
Jordan counts 53 ducks on a pond. Then she
counts 3 more. What patterns do you notice in
the numbers Jordan counts?
50 51 52 53 54 55 56 57 58 59 60
Answers
Answer:
Step-by-step explanation:
goes up by 1 each time
The pattern of Jordan is that, she goes up by 1 , each time.
It is given that Jordan counts 53 ducks on a pond and then she counts 3 more by following a same pattern each time.
What is the pattern that she follows ?
What will be the value if we add 1 for 4 times.
The value will be 4.
The numbers Jordan counts are
50, 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59, 60
If we check
51 - 50 = 1
52-51= 1
.
.
.
60-59 = 1
and so on.
Every time we get a difference of 1 count.
Thus , the pattern followed by Jordan is that, she goes up by 1 , each time.
To learn more about sequence click here ,
https://brainly.com/question/12246947
#SPJ2
I really need this so please hurryy for a test btw ppppp
Answers
Which of the following is the definition of geometric sequence?
A. an ordered set of numbers
B. an element in a sequence
C. a sequence in which terms are given by multiplying the previous term by a common ratio
D. a sequence in which terms are not given by multiplying the previous term by a common ratio
Answers
Answer:
Geometric sequence is a sequence in which terms are given by multiplying the previous term by a common ratio.
Answer:
a sequence in which terms are not given by multiplying the previous term by a common ratio
Step-by-step explanation:
Please help will mark brainliest PLZZZZZZzzz
Answers
Answer:
see explanation
Step-by-step explanation:
The reciprocal of a number n is [tex]\frac{1}{n}[/tex]
(a)
1 [tex]\frac{6}{7}[/tex] ← change to an improper fraction
= [tex]\frac{13}{7}[/tex]
Then its reciprocal is [tex]\frac{1}{\frac{13}{7} }[/tex] = [tex]\frac{7}{13}[/tex]
(b)
The reciprocal of [tex]\frac{7}{13}[/tex] is [tex]\frac{1}{\frac{7}{13} }[/tex] = [tex]\frac{13}{7}[/tex] = 1 [tex]\frac{6}{7}[/tex]
(c)
Each is the reciprocal of the other
Solve the following expression when
v = 10 and e= 5
2v ÷ e + 3
Answers
Answer: 7
Step-by-step explanation:
[tex]2(10) \div 5 +3\\\\20 \div 5 + 3\\\\4+3\\\\\boxed{7}[/tex]
(14+y)+3 please im stuck on this question
Answers
Answer: Just add the 14+3
= Y17
Step-by-step explanation:
Which is the graph of the following equation?
x2 = 25
Can someone please help me ASAP!
Answers
Answer:
D
Step-by-step explanation:
x^2=25
[tex]\sqrt{x^{2}[/tex] = [tex]\sqrt{25}[/tex]
x= 5
please help?!’ answerr
Answers
Answer:
The choice Draw 2
Step-by-step explanation:
[tex]3(8 - 4x) < 6(x - 5) \\ 24 - 12x < 6x - 30 \\ 24- 12x + 12x < 6x + 12x - 30 \\ 24 < 18x - 30 \\ 24 + 30 < 18x - 30 + 30 \\ 54 < 18x \\ 54 \div 18 < 18x \div 18 \\ 3 < x \\ x > 3[/tex]
I hope I helped you^_^
in science class, Marco recorded numbers. from an experiment as 12.87 12.13 12.5 and 12.08 which numbers are closer to 12 ? which are closer to 13?
Answers
Answer:
12.87, 12.5 is closer to 13
12.08, 12.13 is closer to 12
Step-by-step explanation:
Least to Greatest:
12
12.0812.1312.5 12.8713
If sin x=0.96, find tan x
Answers
Step-by-step explanation:
sin x= P/H
sin x = 0.96 = 96/100 = 24/25.
Let the ratio of the sides be x.
B²=H²-P²
B²=(25x)²-(24x)²
B²=625x²-576x²
B²=49x²
B=√49x²
B= 7x.
cos x = 7x/25x = 7/25.
therefore, tan x = sin x / cos x = (24/25)/(7/25) = 24/25×25/7 = 24/7 .
hope this helps you.
The volume of a cube is found by:
1. l x w x h
2. l x w
3. w x h
Answers
Answer:
1)
Step-by-step explanation:
l x w and w x h are all formula for a rectangle, not a cube.
For instance, let's take l x w. If we change the height it will definitely change the volume, but according to 2) it won't. So, 2 is wrong.
We can do the same process for 3) to show it's wrong.
So, the answer is not 2) or 3), so it must be [tex]\boxed{1)}[/tex]
In which number is the value of the 5 ten times greater than the value of the 5 in 654,321? 658,321 536,401 65,892 58
Answers
Answer:
536,401
Step-by-step explanation:
the 5 is in the ten thousands column in 654,321. you have to times the number by ten to see what column it would be in, 6,543,210. The 5 is now in the hundred thousands column. The only number there with the 5 in in the hundred thousands column is 536,401.
-11/16*(-4/33) van you please help me answer this question?
Answers
Answer:
1/12
Step-by-step explanation:
[tex] - \frac{11}{16} \times - \frac{4}{33} [/tex]
if you know how to cancel, you can cancel the 4 and 16. The 4 becomes a 1 and 16 becomes a 4. Next cancel the 11 and 33. the 11 becomes a 1 and the 33 becomes a 3.
[tex] - \frac{1}{4} \times - \frac{1}{3} [/tex]
multiply, two negatives make a positive
[tex] \frac{1}{12} [/tex]
Andrew and Kate have $30 to spend on dinner, tax, and gratuity at Mo's Restaurant, Tax is 0%, and they will give a 15%
tip on the total bill after taxes. Their dinner costs $21.00. Which statement correctly explains whether Andrew and Kate
have enough money to pay their bill?
Yes. They have enough to pay their bill: $30x1.08 - 131 80 and $31 80x1.15 - $38.57,
Yes. They have enough to pay their bill: $21x1.08 - $22 28 and $22.28x1 Why 625,80),
No. They do not have enough to pay their bill: 130 x 1.08 - 531 80 and $31 80x115 -- $38 57,
No. They do not have enough to pay their bill: $21x 108 - $22.28 and $22.28 x 1.15 - $25 60.
Answers
Answer:
If the cost of dinner is less then the money that the Andrew and Kate have then we can say Andrew and Kate have enough money to pay their bill. answer should be "b".
Step-by-step explanation:
Answer: its B.)
Step-by-step explanation:
Whats the answer to this question? ( I'll give Branliest! )
Answers
Answer:
21
Step-by-step explanation:
3^3 + 11(2) - 4(7) = 27 + 22 - 28
f=3,g=2,h=7
Now
3³+11(2)-4(7)27+22-2849-2821A triangle is shown on the graph:
A coordinate plane is shown. A triangle has vertices at negative 6 comma 2, negative 5 comma 5, negative 2 comma 2.
What effect will a 90-degree clockwise rotation have on the triangle? Be sure to address how it could impact the angles, side lengths, and any congruency between the original pre-image and the image.
Answers
9514 1404 393
Answer:
the image is congruent with the original, angles and lengths are unchanged
Step-by-step explanation:
Translation, rotation, and reflection are known as "rigid transformations". Effectively, you can cut the figure from some rigid material, and move it according to any of these transformations without causing it to deform in any way.
All lengths and angles retain their measures when the figure is rotated about any point, through any angle. The rotated figure is congruent with the original.
__
Here, the triangle is defined in the second quadrant. Rotating it 90° clockwise will move it to the first quadrant. The transformation of coordinates is ...
(x, y) ⇒ (y, -x)
P(A)=0.82, P(B)=0.48, P(A not B not)= 0.02, what is P(AuB)?
Answers
Answer:
Step-by-step explanation:
A'∩B' = (A∪B)' (demorgan's law)
.02=(AUB)'
.98=AUB
is this correct? if not please help
Answers
Answer:
The correct answer is [tex]2 *10^3 ; 2,000[/tex]
Picture to text form: 2 x 10^3 ; 2000
Step-by-step explanation:
The answer you chose is incorrect.
Answer: It would be 2* 10^3; 2000
Step-by-step explanation:
Divide the total number of students in the state by the number of students in the school.
806240 / 400 = 2015.6
The closest number would be 2000, written as 2* 10^3.
Carlos ate 1/3 of his caramel pie. Carlos is having a friend come over and wants to give her 1/4 of the pie he has remaining. What fraction of the original pie will he give his friend?
Answers
===================================================
Explanation:
Multiply the denominators to get 3*4 = 12. We'll use this value to set up the number of slices we start with.
If we started with 12 slices, then 1/3 of this is (1/3)*12 = 4 slices. So Carlos ate 4 slices. There are 12-4 = 8 slices left.
1/4 of that value is (1/4)*8 = 2 which is the number of slices given to his friend.
Divide the number of slices given to the friend (2) over the number we started with (12) to get 2/12 = 1/6
-----------------------
Another approach:
If Carlos ate 1/3 of the full pie, then there are 2/3 of it left.
1/4 of this is (1/4)*(2/3) = (1*2)/(4*3) = 2/12 = 1/6 which represents the fractional amount of the original pie given to the friend.
If you worked 50 weeks per year for 50 hours per week for $13/hr, how much money would you earn per year?
Answers
Answer:
32500 per year
Step-by-step explanation:
multiply 50 weeks to 50 hours to get how many hours does 50 week have then multiply the outcome result of the two to the $13 to get the money he/she would earn in a year.
If you vertically compress the absolute value parent function, f(x) = \X\, by a
factor of 4, what is the equation of the new function?
A. g(x) = | 4x1 |
B. g(x) = |x - 41 |
C. g(x) = 4|x|
D. g(x) = 1/4 |x|
Answers
Which of the following does not represent a statistical question?
A. How tall are the students in my class?
B. How many hours per week do you practice sports?
C. How much do the students on the football team weigh?
Answers
Answer:
B
Step-by-step explanation:
Because "How Tall are the students in my class?" and "How much do the students oon the team weigh,?" can vary. If you ask a question about one person, it is not a statistical question because the answer cant vary.
An answer will vary with 2 or more things.
Hope I helped.
How many hours per week do you practice sports? does not represent a statistical question because it asks for a fixed quantity, i.e., the number of hours per week someone practices sports, which does not involve variability or differences among data.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
A statistical question is a question that can be answered through data analysis and involves variability or differences among data.
It usually requires collecting data from a sample or population, organizing and summarizing the data, and drawing conclusions from the results.
How tall are the students in my class and How much do the students on the football team weigh represent statistical questions because they involve collecting data on a characteristic or variable of a population, such as height or weight.
These questions can be answered through data analysis and provide information on the variability or distribution of the characteristic in the population.
Hence, Option B does not represent a statistical question because it asks for a fixed quantity, i.e., the number of hours per week someone practices sports, which does not involve variability or differences among data.
To learn more on Statistics click:
https://brainly.com/question/30218856
#SPJ2
Write an equation in slope-intercept form
of the line that passes through the point
(-12,6) and is perpendicular to the line
X= - 3/4y -5 ( What would be my first step to put that equation into slope intercept form?) 
Answers
Hint :-
The product of slopes of two perpendicular lines is -1 . The slope intercept form of the line is [tex] y = mx + c [/tex] .The point slope form of the line is [tex] m(x-x_1) = (y-y_1) [/tex]Solution :-
The given equation to us is ,
[tex]\implies x = \dfrac{-3}{4}y - 5 [/tex]
Convert it to slope intercept form we have ,
[tex]\implies 4x =-3y -20 \\\\\implies 3y = -4x -20 \\\\\implies y =\dfrac{-4}{3}x -\dfrac{20}{3}[/tex]
On comparing to the slope intercept form of the line we have ,
[tex]m = \dfrac{-4}{3}[/tex]As we know that the product of slopes of two perpendicular lines is-1 , henceforth ,
[tex] m_{perp}= \dfrac{3}{4}[/tex]Now using slope point form rewrite the equation of the perpendicular line ,
[tex]\implies y-y_1 = m(x-x_1) \\\\\implies y -6 = \dfrac{3}{4}(x+12)\\\\\implies 4y -24 = 3x + 36 \\\\\implies 3x -4y +36+24=0\\\\\implies \underline{\boxed{ \gray{3x -4y +60=0}}}[/tex]
8% of what number is 1.2
Answers
Mark me brainiest if helped!
Jasmine is solving this equation. What value of x makes Jasmine's equation true?
x/4 = 9
Answers
Answer:
The value of x that makes Jasmine's equation true is 36, because 36 / 4 = 9.
Step-by-step explanation:
brainliest please. . .
What is 23 + 18 than a blank line + 14
Answers
Answer:
27
Step-by-step explanation:
it is 55 23+18=41 +14=55
Find the sum of 12.01 and 3.8
Answers
Answer:
15.81 is the answer.
81 is what percent of 90?
Answers
Answer:
72.9
Step-by-step explanation:
hope it helps you out
please mark brainiest
Answer:
81 Out of 90 is 90%.
Step-by-step explanation:
Follow the below steps to calculate what percent is 81 out of 90 or how to write 81/90 as a percentage.
Step 1: Percentage Formula = Number1/Number2 x 100
Step 2: plugin the 81 and 90 to the percentage formula and we get 81/90 x 100 = 90%
hope this helps!
(can u mark brainliest)
Tell me pls but pls no files
Answers
Answer: C
Step-by-step explanation: When you rotate it 90 degrees it is the same then you put it 3 to the right which makes it opposite. I mean I am pretty Sure.
What is the circumference of a circle when the diameter is 8cm?
Answers
Answer:
C = 25.13cm
C = π d = π · 8 ≈ 25.13274cm
Answer:
the answer is 4cm circumference