Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. (tried this question a lot and I am just missing something)(plz help)
x = −3y2 + 9y − 6, x = 0
Answers
The expression on the left side describes a parabola. Factorize it to determine where it crosses the y-axis (i.e. the line x = 0) :
-3y² + 9y - 6 = -3 (y² - 3y + 2)
… = -3 (y - 1) (y - 2) = 0
⇒ y = 1 or y = 2
Also, complete the square to determine the vertex of the parabola:
-3y² + 9y - 6 = -3 (y² - 3y) - 6
… = -3 (y² - 3y + 9/4 - 9/4) - 6
… = -3 (y² - 2•3/2 y + (3/2)²) + 27/4 - 6
… = -3 (y - 3/2)² + 3/4
⇒ vertex at (x, y) = (3/4, 3/2)
I've attached a sketch of the curve along with one of the shells that make up the solid. For some value of x in the interval 0 ≤ x ≤ 3/4, each cylindrical shell has
radius = x
height = y⁺ - y⁻
where y⁺ refers to the half of the parabola above the line y = 3/2, and y⁻ is the lower half. These halves are functions of x that we obtain from its equation by solving for y :
x = -3y² + 9y - 6
x = -3 (y - 3/2)² + 3/4
x - 3/4 = -3 (y - 3/2)²
-x/3 + 1/4 = (y - 3/2)²
± √(1/4 - x/3) = y - 3/2
y = 3/2 ± √(1/4 - x/3)
y⁺ and y⁻ are the solutions with the positive and negative square roots, respectively, so each shell has height
(3/2 + √(1/4 - x/3)) - (3/2 - √(1/4 - x/3)) = 2 √(1/4 - x/3)
Now set up the integral and compute the volume.
\(\displaystyle 2\pi \int_{x=0}^{x=3/4} 2x \sqrt{\frac14 - \frac x3} \, dx\)
Substitute u = 1/4 - x/3, so x = 3/4 - 3u and dx = -3 du.
\(\displaystyle 2\pi \int_{u=1/4-0/3}^{u=1/4-(3/4)/3} 2\left(\frac34 - 3u\right) \sqrt{u} \left(-3 \, du\right)\)
\(\displaystyle -12\pi \int_{u=1/4}^{u=0} \left(\frac34 - 3u\right) \sqrt{u} \, du\)
\(\displaystyle 12\pi \int_{u=0}^{u=1/4} \left(\frac34 u^{1/2} - 3u^{3/2}\right) \, du\)
\(\displaystyle 12\pi \left(\frac34\cdot\frac23 u^{3/2} - 3\cdot\frac25u^{5/2}\right) \bigg|_{u=0}^{u=1/4}\)
\(\displaystyle 12\pi \left(\frac12 u^{3/2} - \frac65u^{5/2}\right) \bigg|_{u=0}^{u=1/4}\)
\(\displaystyle 12\pi \left(\frac12 \left(\frac14\right)^{3/2} - \frac65\left(\frac14\right)^{5/2}\right) - 12\pi (0 - 0)\)
\(\displaystyle 12\pi \left(\frac1{16} - \frac3{80}\right) = \frac{12\pi}{40} = \boxed{\frac{3\pi}{10}}\)
Related Questions
HELP ASAPPP
What is 60% of 40?
Answers
Answer:
The answer is 24
Step-by-step explanation:
(dont mind this bit, i have to have more letters to post x)
whats the same about online school and in-person
for a project pls help thanks :)
Answers
Answer:
You're both learning the same stuff, only in different environments.
Step-by-step explanation:
It doesn't matter about the different environments you are in, you are still learning the same schoolwork that you are required to do.
Answer:
The similarity between online school and in- person is the material learned and how it is taught.
Step-by-step explanation:
=
Simple interest is given by the formula A P+Prt. Where A is the balance of the
account after t years, and P is the starting principal invested at an annual percentage rate
of r, expressed as a decimal.
Bill is investing $26,000 into a savings account that pays 2.6% simple interest. How long will
it take for this investment to double in value?
It will take
Round your answer to the nearest tenth.
years for this investment to double in value.
Answers
The investment will take 38.5 years to double in value.
What is simple interest?
The principal of a loan or the first deposit into a savings account serves as the basis for simple interest. Simple interest doesn't compound, therefore a creditor will only charge interest on the principal sum, and a borrower will never be required to pay further interest on the interest that has already accrued.
P=$26,000 r=2.6%=0.026
A=2P=2(26,000)=$52,000
A=P+Prt
Substitute the known values
52,000=26,000+(26,000×0.026t)
52,000-26,000=676t
On simplifying, we get t= 38.46 years
Rounded number to the nearest tenth=38.5 years
To learn more about the simple interest from the given link
https://brainly.com/question/20690803
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Convert to vertex form: f(x) = x2 + 16x + 64
Answers
Answer:
Step-by-step explanation:
hello :
note : the vertex form is :
ax²+bx+c = a(x+(b)/(2a))²- delta/4a² and delta = b²- 4ac
in this exercice you have : a =1 b = 16 and c=64
continu ..........
What is the slope of the line that cuts through the points (1, -5) and (3, 5)?
a. 5
C. 1
b. -5
d.
1
5
Please select the best answer from the choices provided
OA
C
AR
15
Answers
Answer:
a. 5
Step-by-step explanation:
To find slope, subtract the y's.
5 - -5 is 10.
Put that on the top of a fraction.
Subtract the x's.
3 - 1 is 2, put that on the bottom of the fraction.
So the slope is 10/2. This simplifies to 5 (because 10÷2 is 5).
The slope is 5.
Answer the question above.
Answers
9. Use the graph of the function f(x) = x³ – 7x² + 10x to
x`
identify its relative maximum and minimum.
da
-4
8
8
2
K
T
2
da
y
8
+w
8
maximum = 4.1, minimum = -8.2
maximum = 0.9, minimum = 3.8
Answers
The extremas of the function f(x) = x³ - 7x² + 10x are given as follows:
Relative maximum: (0.88, 4.061).Relative minimum: (3.786, -8.209).What are the relative minimums and the relative maximums of a function?The relative minimums of a function are given by the points in which the function's behavior changes from decreasing to increasing, that is, where the function curves down.The relative maximums of a function, meanwhile, are given by the points in which the function's behavior changes from increasing to decreasing, that is, where the function curves up.More can be learned about extremas of a function at https://brainly.com/question/9839310
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here’s the repostttt
Answers
1. we have to find what number lie at A and B , and that too between and 1 and 2 , so these two numbers must be decimal numbers!! So let's find the number of scales between 1 and 2 , we get 8 scales !! So the question is how much place 1 scale will occupy ?? we have to find certain numbers between 1 and 2 , suppose between 1 and 2 we have 1 unit and those 1 unit occupies 8 scales !! so much place 1 scale occupies ?? to get it divide 1 by 8 and we get 0.125 !! so between 1 and 2 ,each blank space or scale occupies 0.125 unit!! now what numbers lie at A ? count the number of scale between 1 and A , we get 3 scales ! now multiply 3 by 0.125 , we get 0.375 , add 1 to 0.375 , we get 1.375 !! hence 1.375 lies at A ! now we have to find which number lies at B ? count the number of blanks or scale , we get 4 ! multiply 4 by 0.125 and we get 0.5 ! add this 0.5 to 1 , we get 1.5 ! therefore the number 1.5 lies at B !!
2. for 2nd number solution refer to the attachment! !
Which number is a perfect square?
O A. 32
O B. 144
O C. 1,000
<
O D. 36,000
Please help
Answers
Answer:
The answer is 144
Step-by-step explanation:
Because the square root of 12 is 144 making a whole number.
what's the slope of (-3,3) and (-1,2)
Answers
Answer:
Step-by-step explanation:
Slope is defined as
m=(y2-y1)/(x2-x1)
We are given two (x,y) points of (-3,3) and (-1,2) so
m=(3-2)/(-3- -1)
m=1/-2
m=-1/2
helppp pleaseee ty! :(
Answers
Answer:
A
Step-by-step explanation:
11. Which expression is equivalent to -2x + 5 ?10 - 4x - 5 + 2x3(2x + 1) - X-13x + 5- X - 10O -x - 9 + 4x + 4
Answers
Simplify each expression as follows:
\(\begin{gathered} 10-4x-5+2x=10-5-4x+2x \\ =(10-5)+(-4x+2x) \\ =5-2x \end{gathered}\)\(\begin{gathered} 3(2x+1)-x-1=3(2x)+3(1)-x-1 \\ =6x+3-x-1 \\ =6x-x+3-1 \\ =5x+2 \end{gathered}\)\(\begin{gathered} 3x+5-x-10=3x-x+5-10 \\ =2x-5 \end{gathered}\)\(\begin{gathered} -x-9+4x+4=-x+4x-9+4 \\ =3x-5 \end{gathered}\)Therefore, the only expression that is equivalent to -2x+5 is 10-4x-5+2x.
Find the value of x in the shown figure
Answers
Explanation:
I added 30+10 which equalled 40 and then i noticed the circle which would equal 360 and subtracted it from the 40 however that got me more then i needed since it wasnt one of the choices. Then i thought hmm maybe its a supplementary angle which usually equals 180 so i subtracted it from 40 which got me 140! Like i said i dont know if im right or not :(
If 8 Superscript y Baseline = 16 Superscript y + 2, what is the value of y?
Answers
Answer:
10. Where was the medical mission held? A. At Barangay Tatalon Marikina B. At Barangay Almanza Las Piñas City C. At Barangay Pamplona Las Piñas City D. At Barangay CAA BF INT'L Las Piñas City
what is the product of 6(x^2)*6x-1/6(x+1)
Answers
Answer:
(x-1)(6x-1)
Step-by-step explanation:
this is the answer i think
What is the ratio of the length of the longer side to the length of the shorter side?
24 in.
4ft.
Answers
option:-
\( \boxed{ \rm{B ) \: 6:1}}\)\( \: \)
Given:-
\( \rm{Length \: of \: rectangle = \bold{ 24in. \: }}\)\( \: \)
\( \rm{Breadth \: of \: rectangle = \bold{ 4ft. }}\)\( \: \)
To find:-
radio of the length of the longer side to the shorter side.\( \: \)
Solution:-
\( \rm{ratio = 24 : 4 }\)\( \: \)
\( \rm\frac{A}{B} = \frac{24}{4} \)\( \: \)
\( \cancel \frac{24}{4} \)\( \: \)
\( \frac{6}{1} \)or
\( \underline{ \boxed{ \rm \red{ \: \: 6 : 1 \: \: }}}\)\( \: \)
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
~How to find ratio?
Ratios compare two numbers, usually by dividing them. If you are comparing one data point (A) to another data point (B), your formula would be A/B. This means you are dividing information A by information B.━━━━━━━━━━━━━━━━━━━━━━━━━━━━
\( \: \)
hope it helps! :)
Write in standard form 5×100+4×19+6×1+2×1/10+8×1/1000
Answers
Answer:
Step-by-step explanation:
i think this is the answer
576.628
Answer:
\(\frac{72776}{125}\)
Step-by-step explanation:
500+76+6+1/5+1/125
500*125/125 +76*125/125 + 6*125/125 +25/125 +1/125
62500+9500+750+25+1/125
\(\frac{72776}{125}\)
or 582.208
or
\(582 \frac{26}{125}\)
translate & solve: 616% of what number is 4620
Answers
Answer:
250
Step-by-step explanation:
1540*100 divided by 616
WILL MARK AS BRAINLEST
Prove that in a triangle an exterior angle is equal to the sum of the two interior angles not adjacent to it.
Answers
Answer:
Brainlest two angle adjacent will mark
In a newspaper, it was reported that yearly robberies in Springfield were up 40% to 154 in 2012 from 2011. How many robberies were there in Springfield in 2011?
quick plss
Answers
Answer:
The answer is 110
Step-by-step explanation:
If you go on wiki you will find a hold lot,of things you need to know about yearly robberies.And thank your for choosing brainly and having your trust upon other user just like me place what ever answer you want and a brainly user will get back to you in about 10-15 minutes is not your one of luckey ones
C 5) Identify 2/10 as a decimal. * O 10.2 2.10 O .2 o O .02
Answers
Answer:
0.2
Step-by-step explanation:
7(4p+2q+3r) apply the distributive property to create an equivalent expression
Answers
Answer:
28p + 14q + 21r
Step-by-step explanation:
7(4p + 2q + 3r) ← multiply each term in the parenthesis by the 7 outside
= 28p + 14q + 21r
Write the equation in slope-intercept form for the line that passes through (5, 4) and (6,-1).
SHOW YOUR WORK
Answers
Answer:
y = -5x + 29
Step-by-step explanation:
Slope-Intercept Form of the Line
Given a line of slope m and y-intercept b, the equation of the line can be written as:
y = mx + b
The slope can be calculated by using the formula:
\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)
Where (x1,y1) and (x2,y2) are the points through which the line passes.
We are given the points (5,4) and (6,-1), thus:
\(\displaystyle m=\frac{-1-4}{6-5}=\frac{-5}{1}=-5\)
Now we use one of the points to find the value of b, for example (5,4):
4 = -5(5) + b
b = 25+4 = 29
Thus, the equation of the line is:
y = -5x + 29
Identify the theorem or postulate that is related to the measures of the angles in the pair, and find the unknown angle measures.
m/1 = (5x +15)°, m/4= (30x - 60)°
Answers
To identify the answer we will use Angles Theorem which is m∠1 = 150. When the transversal line cuts through the two parallel lines, it creates the alternate exterior angles 1 and 8. Both are consistent with the alternate exterior angles theorem. As a result, m1 = m8 and m1 = (50x)°.
m∠8 = 150
Setting both equal, find x 50x = 150.
50x/50 = 150/50 x = 3 when you divide both sides by 50.
Unknown stance
m∠1 = (50x) = 50*3 = 150°
What exactly is a mathematical postulate?A proposition—also referred to as an axiom—that is accepted as true without evidence. Lemmas and theorems arise from postulates, which is their fundamental structure. Five tenets, referred to as Euclid's postulates, form the foundation of Euclidean geometry as a whole.
An unsupported assertion is referred to as a postulate. A postulate is also referred to as an axiom. For instance, you would believe Pam if she claimed that all of her siblings are at least five foot one tall if you knew that she is five feet tall and that all of her siblings are taller than her.
The fundamental tenet of consumer theory holds that consumers choose the bundle from the available options that provides the greatest level of satisfaction. On the set of bundles, assumptions are made that ensure the existence of a comprehensive, reflexive, and transitive preference order.
To learn more about mathematical postulate refer to:
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What is the volume of a cylinder with a radius of 2 ft and a height of 8 ft.
Use 3.14 for pi, round your answer to the nearest hundredth if necessary, and do not include units.
Answers
Answer:
100.48
Step-by-step explanation:
The volume of a cylinder is given by
V = pi r^2 h where r is the radius and h is the height
V = 3.14 ( 2)^2 * 8
V = 3.14 (4)(8)
V = 100.48
5/6 + (-4/9) - (-2), is this right??
Answers
Answer:
YES SIR OR I MEANT YES MAM
She determined that the solution is (–3, –3). Is she correct? If not, explain why. Yes, she is correct. No. She switched the x and y values of the intersection point when writing the solution. No. She used the wrong y-intercepts when graphing the equations. No. She used the wrong slopes when graphing the equations.
Answers
Answer: No she used the wrong slopes when graphing the equations
Step-by-step explanation:
Answer:
D
Step-by-step explanation: No. She used the wrong slopes when graphing the equations.
i need the
Domain:
and
Range:
Answers
Answer:
I don't know what kind of form your assignment wants it in, but the domain (span over the x axis) is -4<=x<=5, and the range is -3<=x<=5. Perhaps write it is as "-4 to 5" for domain and "-3 to 5" for range.
Step-by-step explanation:
the sum of 9/10 and 5/6
Answers
Answer:
Step-by-step explanation:
9/10+5/6 =
27/30+25/30 =
52/30 = 1 22/30 =1 11/15
Answer: 26/15
Step-by-step explanation:
1. First find the common denominator, which is 30
2. Add them
3. Simplify (common factor of 2)
\(\frac{27}{30} + \frac{25}{30} \\\\\frac{52}{30} \\\\\\\frac{26}{15}\)
I NEED HELP PLS HURRY
1. Write each expression with a single exponent:
a. (10^7)²
b. (10^9)³
c. (10^6)³
d. (10^2)³
e. (10³)²
f. (10^5)^7
Answers
b. 10^27
c. 10^18
d. 10^6
e. 10^6
f. 10^35
multiply the exponents.
Answer:
We use the rule,
\((a^b)^c = a^{bc}\)
a. (10^7)²
\((10^7)^2 = 10^{(7)(2)} = 10^{14}\)
10^14
b. (10^9)³
\((10^9)^3 = 10^{(9)(3)} = 10^{27}\)
10^27
c. (10^6)³
\(10^{(6)(3)}= 10^{18}\)
10^18
d. (10^2)³
\(10^{(2)(3)} = 10^{6}\)
10^6
e. (10³)²
\(10^{(3)(2)}=10^{6}\)
10^6
f. (10^5)^7
\(10^{(5)(7)} = 10^{35}\)
10^35
Step-by-step explanation: