Mathematics High School

Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y=x^3 and y=9x about the x-axis.
...?

Answers

Answer 1
First solve x³=9x

[tex]x^3-9x=0 \\x(x^2-9)=0 \\x(x-3)(x+3)=0 \\x=0, x=3,x=-3[/tex]

We can ignore x = -3 because it is not in the first quadrant.

So our integral is going to go from x=0 to x=3.

Now we can use the formula
[tex]V=\int_{a}^{b}\pi f(x)^2dx[/tex]
[tex]\int\limits_{0}^{3}(\pi (9x)^2-\pi(x^3)^2)dx \\\int\limits_{0}^{3}(\pi (9x)^2-\pi(x^3)^2)dx \\\pi\int\limits_{0}^{3}(81x^2-x^6)dx \\2\times \int\limits_{0}^{3}[\pi(9x)^2-\pi(x^3)^2]dx \\\pi\left[81\frac{x^3}{3}-\frac{x^7}{7}\right]_{0}^{3} \\ \\\frac{2916\pi}{7}[/tex]
Answer 2
Final answer:

The volume of the solid formed by rotating the region enclosed by y = x^3 and y = 9x about the x-axis in the first quadrant can be found by integrating from 0 to 3, the square of the outer and inner radius, multiplied by π. Solve the integral to get the volume.

Explanation:

To find the volume of the solid formed by rotating the region enclosed by y = x^3 and y = 9x about the x-axis, we need to use the method of discs/washers. The volume V is given by the following integral:

 ∫[a,b] π(r(x)^2 - R(x)^2) dx

For our given curves, r(x) = 9x and R(x) = x^3 since 9x ≥ x^3 for 0 ≤ x ≤ 3. Therefore, we get:

 V = ∫[0,3] π[(9x)^2 - (x^3)^2] dx

Solving this integral will yield the volume of the solid:

 V = π ∫[0,3] (81x^2 - x^6) dx

Calculating this integral will give the volume of the solid.

Learn more about Volume of solid here:

https://brainly.com/question/23705404

#SPJ11

Related Questions

Explain why 17:3 and 68:12 are equivalent ratios

Answers

17 times 4 is 68 and 3 times 4 is 12. 17:3 and 68:12 are equivalent because you can times and devide them into each other

Jason considered two similar televisions at a local electronics store. The generic version was based on the brand name and was three fourths the size of the brand name. If the generic television set is 12 inches by 24 inches, what are the dimensions of the brand name television?
18 inches by 36 inches
6 inches by 12 inches
16 inches by 32 inches
9 inches by 18 inches

Answers

Answer: The answer is 16 by 32.

Step-by-step explanation:

I took the test and this was correct. Explanation is easy because 12 is 3/4 of 16 and 24 is 3/4 of 32.

If the generic television set is 12 inches by 24 inches, the dimensions of the brand name television is, 16 inches by 32 inches

What is a fraction?

Fraction is a mathematical term, which represents the portion or sub-parts of the whole thing. Basically, It has two parts: numerator and denominator.

Numerator is a number which lies on the top side, it indicates the number of equal parts taken and denominator is on the bottom side, it indicates the equal parts of the whole number.

Given that,

Jason considered two similar televisions at a local electronics store

The generic version was three fourths the size of the brand name

The generic television set is 12 inches by 24 inches

The dimensions of the brand name television = ?

Suppose, the dimensions of the brand name are a  inches by b inches

So,

3/4 a   =  12

a   =  16

Similarly,

3/4 b = 24

b  =  32

Hence, the dimensions are 16 inches by 32 inches

To know more about Fraction check:

https://brainly.com/question/8482939

#SPJ2

(3x^2a - 4y^a z^3a) ^2 ...?

Answers

The answer is 9x⁴ᵃ - 24x²ᵃyᵃz³ᵃ + 16 y²ᵃz⁶ᵃ


(a - b)² = a² - 2ab + b²
(3x²ᵃ - 4yᵃz³ᵃ)² = (3x²ᵃ)² - 2 * 3x²ᵃ * 4yᵃz³ᵃ + (4yᵃz³ᵃ)² =
                         = 3²x²ᵃ*² - 2 * 3 * 4 x²ᵃ * yᵃz³ᵃ + 4²yᵃ*²z³ᵃ*² =
                         = 9x⁴ᵃ - 24x²ᵃyᵃz³ᵃ + 16 y²ᵃz⁶ᵃ

Catie is making bracelets for her friends using small rubber bands. She has 364 rubber bands and she wants to make 6 bracelets. How many rubber bands will she use for each bracelet? How many rubber bands will be left over?

Answers

do 364 divided by 6 and then whatever is left over is your second answer so the answers would be
60 rubber bands per bracelet
and
4 left over
Well, since 6 goes into 364 sixty times, Catie will use 60 rubber bands for each bracelet. She will be left with 4 rubber bands left over. 

Order these numbers from least to greatest. 8 7/11 , 8.17 , 8.174 , 173/20

Answers

8.17,8.174,8 7/11, 173/20

The numbers are ordered from least to greatest by first converting them all to decimals, resulting in the order: 8.17, 8.174, 8 7/11 (8.6363), and 173/20 (8.65).

First, we need to convert all the numbers to similar formats, either as decimals or fractions, to compare them accurately. Here's what we have:

8 7/11 = 8 + 7/11 = 8.6363 (rounded to four decimal places)8.17 (already a decimal)8.174 (already a decimal)173/20 = 8.65 (as a decimal)

Now, we can easily order these from least to greatest:

8.178.1748 7/11 (8.6363)173/20 (8.65)

Sinea spent $8 on games and $12 on souveniers. if she spends $26 on games, how much will she spend on souveniers?

Answers

there should always be an interval of $4 so just take $26 - $4 = $22 that will be spent on souvenirs.
Hope this will help.

a. Show that the system of rational expressions is closed under addition, or give an example showing the contrary.

Answers

A "rational expression" is a fraction in which the numerator and the denominator are both polynomials. The sum of two such expressions will always produce another such expression, viz., 
p(x)/q(x) + r(x)/s(x) = [p(x)s(x) + q(x)r(x)] / [q(x)*s(x)]. 
The numerator and denominator of the sum will both be polynomials when fully expanded

Line PQ is parallel to BC and splits AB into lengths of x + 8 and x. The other side, AC, is split into lengths of x + 13 and x + 1. What is the length of AC? Image Not to Scale.

Answers

Thank you for posting your question here at brainly. I hope the answer will help you. Feel free to ask more questions.
If a straight line connecting two sides of a triangle is parallel to its third side then the straight line divides these sides proportionally.

If the lengths of the segments are listed in an corresponding order, then it implies that you have this proportion:

x + 8/x = x + 1/x +13

Contradiction. This equation has no solutions in real positive numbers. It means that the lengths are listed in the inverse order. Then the proportion is
x + 8/x = x +13/ x+1


Answer:

AC = 18.

Step-by-step explanation:

The problem is about proportions, because we have two parallels line being intercepted by two transversal. So, the proportion would be:

[tex]\frac{x+8}{x}= \frac{x+13}{x+1}[/tex]

Solving for [tex]x[/tex]:

[tex]\frac{x+8}{x}= \frac{x+13}{x+1}\\(x+1)(x+8)=x(x+13)\\x^{2} +8x+x+8=x^{2} +13x\\9x+8=13x\\8=13x-9x\\4x=8\\x=\frac{8}{4}=2[/tex]

Now, to find the length of AC, we just have to use the value found and replace it in the proper expression:

[tex]AC=x+13+x+1=2x+14=2(2)+14=4+14=18[/tex]

Therefore, the length of AC is 18.

What is 24.051 rounded to the nearest hundredth? How did you solve it?

Answers

This is like late elementary/early middle school stuff where you learn this.
First off, what number is in the position of the hundereth place? 0? No. 1? No. 5 yes! Now how i learned how to round is you take  the number to the right of whatever you are being asked about and round it up or down. Now if the number (in this case it is 1) is lower than 5, you round down (or keep it the same) but if it is 5 or higher you round up. since 1 is lower then 5, that 5 in the hundereths place stays the same. 

The number of pizzas eaten at the class party is equal to one-fourth the number of students, plus two more pizzas for the adults. If nine pizzas are eaten, how many students are there?

Answers

P = 1/4S + 2 

9 = 1/4S + 2
-2             -2
--------------------

7 = 1/4S
4(7=1/4S)

28=S

There are 28 students int the class

The diameter of a circular garden is 16 feet. What is the approximate area of the garden?

A.
50 square feet

B.
100 square feet

C.
201 square feet

D.
803 square feet

Answers

[tex]A = \pi r^2[/tex]

radius = half the diameter  ------> 16/2 = 8

[tex]A = \pi (8^2) = 64 \pi [/tex]

pi = 3.14

64*3.14 = 201
Well I believe that the equation for the area of a circle is [tex] \pi r^{2} [/tex]. Pi*Radius^2 
The radius of the garden is 8ft because it's half of the diameter. 8^2 is 64. Multiply that by 3.14 and you will get approximately 201 square feet so the answer is:

C) 201 square feet

#1: The theoretical yield of a reaction is 5.00 moles of potassium permanganate (KMnO4). If the reaction actually produces 695.2 g KMnO4, what is the percent yield of the reaction?

A. 78%

B. 83%

C. 88%

D. 120%

Answers

Final answer:

To calculate the percent yield, the actual yield in moles of potassium permanganate was found to be 4.40 moles from 695.2 g, using the molar mass of KMnO4. Then, the percent yield was calculated using the theoretical yield of 5.00 moles, resulting in 88%.

Therefore, the correct answer is C. 88%

Explanation:

To calculate the percent yield of the reaction, we first need to know the moles of potassium permanganate (KMnO4) that correspond to 695.2 g. We use the molar mass of KMnO4 which is 158.04 g/mol to convert grams to moles.

Actual yield in moles = mass (g) / molar mass (g/mol) = 695.2 g / 158.04 g/mol = 4.40 moles of KMnO4

Now, with the theoretical yield of 5.00 moles, we can find the percent yield using the formula:

Percent yield = (actual yield / theoretical yield) × 100%

Percent yield = (4.40 moles / 5.00 moles) × 100% = 88%

Therefore, the correct answer is C. 88%

a piggy bank contains only nickels and quarters. The total value in the back is $3.80. Write an equation in standered form that models the possible combinations of nickels and quarters in the piggy bank

Answers

Ax + By = C....standard form

0.05n + 0.25q = 3.80....I just used the variables (n) for nickels and (q) for quarters....but u could use x and y if u want to. But if u do this, be sure and label ur variables

77/8 as a mixed number

Answers

77/8 as a mixed number is 9 5/8
9  5  
    8
--------------------------
is 77/8 as a mixed number

True or false.
1) Adjacent angles are supplementary?
2) Same-side interior angles are supplementary?
3) Vertical angles are congruent?

Answers

Supplementary angles are those that add up to 180 degrees, which means 1 is correct while 2 is false. 3 is true. Two vertical angles are same and therefore congruent.

one pipe fills a storage pool in 8 hours. A second pipe fills the same pool in 4 hours when a third pipe is added and all three are used to fill the pool,it takes only 2 hours. find how long it takes the third pipe to do the alone.

Answers

The answer is 8 hours.

Identify the maximum and minimum values of the function y = 10 cos x in the interval [-2π, 2π].

Answers

We know that the maximum and minimum values of the function y = cos x in the interval [-2π, 2π] are 1 and -1 respectively.
 
The graph of y = 8 cos x is vertical stretch by a factor of 8 units. Therefore the graph y=cos x, = the maximum and minimum values of the function y = cos x in the interval [-2π, 2π]. So they are 8 and -8 respectively.  

max = 8,  ∈[−2π,2π]
                                                    = Solution 
min = −8.  ∈[−2π,2π]

If the average cost of hotel stay per night is about $85 plus 18% of hotel tax, estimate the total stay. Round answer to the hundredths place

Answers

$100 what i dos was 85 + 18% =100.3

Haley is at the store buying some supplies for an art project. She decides to buy some colored pencils for $3.95 and a drawing tablet. The total cost of the supplies before sales tax is $6.94.

Which equation could not be used to find the cost of the drawing tablet?

$3.95 + x = $6.94
$6.94 + x = $3.95
x = $6.94 - $3.95
$6.94 - x = $3.95

Answers

$6.94+x=$3.95
If you want an explanation let me know
B: $6.94 + x = $3.95
because
colored pencils for =$3.95
a drawing tablet = x/$2.99
The total cost of the supplies = $6.94
so it's not $6.94 + $2.99 = $3.95 because $6.94 + $2.99 = $9.93
sorry for the long text up there but the it needed to be more then 20 characters long and this is not B: $6.94 + x = $3.95

A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (Do not use mixed numbers. Round your answers to three decimal places.)

(a) How fast is the top of the ladder moving down the wall when the base is given below?
15 feet away from the wall
_____ ft/sec

20 feet away from the wall
_____ ft/sec

24 feet away from the wall
_____ ft/sec

...?

Answers

last one. i can do the first part. put x = distance from top of ladder to the floor, y = distance from base of ladder to the wall (draw a picture) then pythagoras sez

x2+y2=25

calculus says

2xx′+2yy′=0

and you are told that

y′=2

so you know

2xx′+4y=0

or

x′=−2yx

Will give brainliest answer. If explained..

A parallelogram and some of its dimensions are shown below.
What is the area of the parallelogram?
A. 100 sq.in.
B. 200 sq.in.
C. 300 sq.in.
D. 600 sq.in.

Answers

It C. and if you need the equation for the area of a parallelogram its A=BH

The area of desert a is eight times the area of desert
b. if the sum of their areas is 900,000 square miles, find the area of each desert

Answers

desert a is 800000 Sq miles and desert b is 100000sq miles

What is the solution set of 2x2 7x 5=0??

Answers

9-7x= 0
x=0
\/////////////

Jason bought 8 new baseball trading cards to add to his collection. the next day his dog ate half of his collection. there are now only 40 cards left. how many cards did jason start with

Answers

This is pretty  simple actually.

If they want how many cards before he added those 8 new ones then most likely 72 but seeming he ate half of his card and 40 isnt half of 72 we take 8+72 which equals 80. Add 40+40 or do 80/2 it comes out to be 40. So 80 cards to begin with.

Jason started with 72 baseball trading cards. To find out how many baseball trading cards Jason started with, let's break down the information given step by step.

1. Jason bought 8 new baseball trading cards: This implies that Jason's collection increased by 8 cards.

Let's denote the number of cards Jason had before buying the new ones as X . After buying the new cards, Jason would have X + 8 cards.

2. The next day his dog ate half of his collection: If his dog ate half of his collection, this means that the number of cards left after the incident is half of what Jason had before. So, after the dog ate half of the collection, Jason would have [tex]\( \frac{1}{2} \times (X + 8) \)[/tex] cards remaining.

3. There are now only 40 cards left: This information tells us that the number of cards left after the incident is 40. So, we have the equation:

[tex]\[ \frac{1}{2} \times (X + 8) = 40 \][/tex]

Now, let's solve for  X:

[tex]\[ \frac{1}{2} \times (X + 8) = 40 \][/tex]

[tex]\[ X + 8 = 40 \times 2 \][/tex]

X + 8 = 80

X = 80 - 8

X = 72

So, Jason started with 72 baseball trading cards.

To verify, let's check the steps:

1. Jason initially had 72 cards.

2. He bought 8 new cards, totaling 72 + 8 = 80 cards.

3. After his dog ate half of his collection, he was left with [tex]\( \frac{1}{2} \times 80 = 40 \)[/tex] cards, which matches the given information.

Therefore, Jason started with 72 baseball trading cards.

The complete question is:

Jason bought 8 new baseball trading cards to add to his collection. the next day his dog ate half of his collection. there are now only 40 cards left. how many cards did jason start with?

What is the volume of a cone with a radius of 18 cm and a height of 10 cm? Use 3.14 for pi. Round your answer to the nearest tenth

Answers

The correct answer is 3391.2cm^3

Why is investing in a mutual fund less risky than investing in a particular company’s stock?

Answers

Because a mutual fund is pretty much an investing agency that is run by an expert economist... which invests in stocks w/ low risk and high return....most of the time.

Simplify 5 log2 k – 8 log2 m + 10 log2 n
...?

Answers

The solution to the problem is as follows:

5log2(k) = log2(k^5) 
8log2(m)= log2(m^8) 
10log2(n)= log2(n^10) 
now to subtract logs, you divide the arguments, keeping the same vase. 
To add logs, multiply the arguments. 

So log2(k^5)- log2(m^8)+ log2(n^10)= log2[k^5* n^10)/m^8] 


I hope my answer has come to your help. Thank you for posting your question here in Brainly. We hope to answer more of your questions and inquiries soon. Have a nice day ahead!

Answer:

your answer is log2 k to the fifth power n to the tenth power over m to the eighth power

Step-by-step explanation:


Find a polynomial function with integer coefficients that has given zeros. (There are many correct answers)
0, 0, 4, 1+i

Answers

[tex]y=(x-0)(x-0)(x-4)(x-(1+i))
\\y=xx(x-4)(x-1-i)
\\y=x^2(x-4)(x-1-i)[/tex]

what is 0.9 as a percentage

Answers

Answer is 90% of 1.
90÷100×1=0.9
Final answer:

0.9 as a percentage is 90 percent.

Explanation:

To convert a decimal to a percentage, you need to multiply the decimal by 100. In this case, 0.9 times 100 equals 90. Therefore, 0.9 as a percentage is 90 percent.

Learn more about Conversion decimal to percentage here:

https://brainly.com/question/30775491

#SPJ12

two ninths of the eighteen people in the bakery are eating cookies how many people are eating cookies

Answers

Let's start with a proportion. 2/9= x/18. Now cross multiply. 2 x 18= 36. Now divide 36 by 9 which is 4. So 4 people are eating cookies. You could of also done it this way where you know 9 is a factor of 18. You then multiply both the denominator and numerator by 2 to get 4/18. Your answer would still be 4 people

Final answer:

To find out how many people are eating cookies, you multiply the total number of people in the bakery, which is 18, by two-ninths. The result is four, meaning four people are eating cookies.

Explanation:

If two-ninths of the eighteen people in the bakery are eating cookies, we need to calculate 2/9 of 18 to find out how many people are eating cookies. This is a basic fraction multiplication problem.

To solve this, we multiply 18 (the total number of people) by the fraction 2/9:

18 × (2/9) = (18 × 2) / 9

18 × 2 = 36

36 / 9 = 4

Therefore, four people in the bakery are eating cookies.

Other Questions
Which piece of textual evidence from the story "Civil Peace" defends best the analysis that the main protagonist, Jonathan, is optimistic?A. "'I count it as nothing,' he told his sympathizers, his eyes on the rope he was tying. 'What is egg rasher? Did I depend on it last week? Or is it greater than other things that went with the war?'"B. "Jonathan and his family were now completely paralysed by terror. Maria and the children sobbed inaudibly like lost souls."C. "He was normally a heavy sleeper but that night he heard all the neighborhood noises die down one after another."D. "At first he went daily, then every other day and finally once a week, to the offices of the Coal Corporation where he used to be a miner, to find out what was what." As Claudia walks away from a 264-cm lamppost, the tip of her shadow moves twice as fast as she does. What is Claudias height? tim estimates that 605.7 is about 10. Will the actual quotient be greater then or less than 10? explain What is the square root of 69? Which of the following is NOT TRUE about the Supreme Court? A. There is only one chief justice on the Court. B. There are eight associate justices on the Court. C. The chief justice is always the justice who has served the most years on the Court. D. The Supreme Court is the highest court in the land. Y is the midpoint of segment XZ. The coordinates of X are (2, 4) and of Y are (-1, 1). Find the coordinates of Z. ...? What is the author's tone in "The Da-mnation of a Canyon"? A.Embarrassed and shameful B.Hostile and humorous C.Friendly and jovial D.Sad and regretful Thesis statement that includes allegory with Boxer in Animal Farm Mrs. Bennett has graded 20% of her 150 students' papers. How many papers does she still need to finish grading? your bank account balance was $235.24 . After two checks were cashed (each for the same amount) your balance is now $-45.58. what was the amount of each check Anna wants to take fitness classes. She compares two gyms to determine which would be the best deal for her. Fit Fast charges a set fee per class. Stepping Up charges a monthly fee, plus an additional fee per class. What is the system of equations representing these costs? 8 is 40% of what number Triangle XYZ is isosceles. The measure of the vertex angle, Y, is twice the measure of a base angle. What is true about triangle XYZ? Check all that apply? What is the quotation of the division problem -44/4 Read the prompt and then answer the question that follows.Based on your research, identify and analyze a human rights issue in your novel and show how it relates to issues and affects people within the United States and one other country.What two things are you supposed to connect? A historical document in English and a document in another language A constitution from another country and the Constitution of the USA A novel written by your author and another novel written by someone else A human rights issue in your novel and a human rights issue in real life Find a possible formula for a fourth degree polynomial g g that has a double zero at 1 1 g(4)=0 , g(5)=0 , and g(0)=6 Why were water routes a slower means of travel?There were no clear waterways.Travel by water needed better directional instruments.The lack of mechanization or power.They were used for traveling great distances. The Mormons faced public hostility because of which statement best summarizes the lasting importance of the many achievements of ancient Greece?a. they strongly shaped western civilizationb. they helped Greece become a military powerc. they showed the curiosity of the greek mindd. they showed their superiority over the Spartans How does the strength of an electromagnet depend on the current and the number of turns in the coil? Steam Workshop Downloader