Mathematics College

Find the volume of the solid obtained by rotating the region
bounded by the graphs y=(x-4)^3,the x-axis, x=0, and x=5
about the y-axis? (Express numbers in exact form. Use symbolic
notation and fractions where needed.)

Answers

Answer 1

Answer:

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis, we can use the method of cylindrical shells.

The formula for the volume of a solid obtained by rotating a region bounded by the graph of a function f(x), the x-axis, x = a, and x = b about the y-axis is given by:

V = 2π ∫[a, b] x * f(x) dx

In this case, the function f(x) = (x - 4)^3, and the bounds of integration are a = 0 and b = 5.

Substituting these values into the formula, we have:

V = 2π ∫[0, 5] x * (x - 4)^3 dx

To evaluate this integral, we can expand the cubic term and then integrate:

V = 2π ∫[0, 5] x * (x^3 - 12x^2 + 48x - 64) dx

V = 2π ∫[0, 5] (x^4 - 12x^3 + 48x^2 - 64x) dx

Integrating each term separately:

V = 2π [1/5 x^5 - 3x^4 + 16x^3 - 32x^2] evaluated from 0 to 5

Now we can substitute the bounds of integration:

V = 2π [(1/5 * 5^5 - 3 * 5^4 + 16 * 5^3 - 32 * 5^2) - (1/5 * 0^5 - 3 * 0^4 + 16 * 0^3 - 32 * 0^2)]

Simplifying:

V = 2π [(1/5 * 3125) - 0]

V = 2π * (625/5)

V = 2π * 125

V = 250π

Therefore, the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis is 250π cubic units.

Related Questions

Team A and Team B together won 50% more games than Team C did. Team A won 50% as many games as Team B did. The three teams won 60 games in all. How many games did each team win?

Answers

Let's assign variables to represent the number of games won by each team:

Let x be the number of games won by Team A.
Let y be the number of games won by Team B.
Let z be the number of games won by Team C.

From the given information, we can form the following equations:

Equation 1: x + y + z = 60 (The total number of games won by the three teams is 60.)

Equation 2: x = (1/2)y (Team A won 50% as many games as Team B.)

Equation 3: x + y = 1.5z (Team A and Team B together won 50% more games than Team C.)

Now, let's solve this system of equations:

Substituting Equation 2 into Equation 3, we get:

(1/2)y + y = 1.5z
(3/2)y = 1.5z
y = (1.5z) * (2/3)
y = z

Substituting y = z into Equation 1, we have:

x + y + z = 60
x + y + y = 60
x + 2y = 60

Substituting y = z into Equation 3, we have:

x + y = 1.5z
x + y = 1.5y
x = 0.5y

Now, we can substitute x = 0.5y and y = z into Equation 1:

0.5y + 2y = 60
2.5y = 60
y = 60 / 2.5
y = 24

Substituting y = 24 into x = 0.5y:

x = 0.5 * 24
x = 12

Substituting y = 24 into the equation y = z:

z = 24

Therefore, Team A won 12 games, Team B won 24 games, and Team C won 24 games as well.

A company Charting its profits notices that the relationship between the number of units sold,x, and the profit,P, is a linear. If 170 units sold results in $20 profit and 220 units sold results in $2820 profit, write the profit function for this company.
P=
Find the marginal profit
$

Answers

Step-by-step explanation:

a linear relationship or function is described in general as

y = f(x) = ax + b

Because the variable term has the variable x only with the exponent 1, this makes this a straight line - hence the name "linear".

here f(x) is P(x) :

P(x) = ax + b

now we are using both given points (ordered pairs) to calculate a and b :

20 = a×170 + b

2820 = a×220 + b

to eliminate first one variable we subtract equation 1 from equation 2 :

2800 = a×50

a = 2800/50 = 280/5 = 56

now, we use that in any of the 2 original equations to get b :

20 = 56×170 + b

b = 20 - 56×170 = 20 - 9520 = -9500

so,

P(x) = 56x - 9500

Other Questions
Describe the structure of the Parliament of the United Kingdom. B. Identify FOUR (4) functions of the U.K Parliament. C. Examine THREE (3) differences between the U.S Presidential System and the U.KParliamentary System of Government. what is the biologcal feature to determine a rajidea shark Connect Today to How effective do you think the United Nations would be today if the United States withdrew as a member nation? Which of the following is true about the cerebellum?a. It is part of the immune systemb. It contains the midbrainc. Its near the front of the braind. It has a role in posture You fire a cannon horizontally off a 50 meter tall wall. The cannon ball lands 1000 m away. What was the initial velocity? Determine the Schwartzschild radius of a black hole equal to the mass of the entire Milky Way galaxy (1.1 X 1011 times the mass of the Sun). When one person shouts at a football game, the sound intensity level at the center of the field is 60.8 dB. When all the people shout together, the intensity level increases to 88.1 dB. Assuming that each person generates the same sound intensity at the center of the field, how many people are at the game? A patient of mass X kilograms is spiking a fever of 105 degrees F. It is imperative to reducethe fever immediately back down to 98.6 degrees F, so the patient is immersed in an ice bath. How much ice must melt for this temperature reduction to be achieved? Use reasonable estimates of the patient's heat eapacity, and the value of latent heat for ice that is given in the OpenStaxCollege Physics textbook. Remember, convert temperature from Fahrenheit to Celsius or Kelvin. To repeat a ceremony, often without thought, reduces the ceremony to a ritual.TrueFalse A car is placed on a hydraulic lift. The car has a mass of 1598 kg. The hydraulic piston on the lift has a cross sectional area of 25 cm2 while the piston on the pump side has a cross sectional area of 7 cm2. How much force in Newtons is neededon the pump piston to lift the car? An evacuated tube uses an accelerating voltage of 31.1 KV to accelerate electrons from rest to hit a copper plate and produce x rays. Non-relativistically, what would be the speed of these electrons? Instructions: First, read the correlational study scenarios below. Then identify the variables that correlate with one another, noting which terms need to be operationally defined. Finally, provide operational definitions for those terms. You can be creative with your operational definitionsjust make them clear and specific.Scenario F: People who are involved in an intimate relationship may experience distinct, although related, feelings of liking for each other and love for each other. It is possible to like someone and not love them, of course, but it is also possible to love someone but not like them all that much (think of a couple that fights a great deal but cant think of life without the other person, or siblings who dont get along but feel a sense of familial love). A group of researchers wanted to examine the degree to which ones liking for his or her partner was correlated with his or her love for that partner.1.What is Variable #1 and how would you operationally define it?2.What is Variable #2 and how would you operationally define it?Scenario G: A researcher believes that the number of "daily hassles" that people experience affects the number of physical and psychological symptoms they have.1.What is Variable #1 and how would you operationally define it?2.What is Variable #2 and how would you operationally define it?Scenario H: A researcher believes that there is a relationship between minority womens experience of racism and the womens psychological distress.1.What is Variable #1 and how would you operationally define it?2.What is Variable #2 and how would you operationally define it?Scenario I: A researcher is interested in looking at the correlation between skin disease severity (psoriasis) and the fear of negative evaluation that psoriasis patients experience.1.What is Variable #1 and how would you operationally define it?2.What is Variable #2 and how would you operationally define it?Scenario J: A researcher believes that a managers sense of humor impacts the productivity of the employees he supervises.1.What is Variable #1 and how would you operationally define it?2.What is Variable #2 and how would you operationally define it? 6. If I took a 10 mL sample from 2 litres of a 100 mM solution of NaCl (sodium chloride or common table salt), what would be the concentration of NaCl in my 10 mL sample?Give an example of when you would record experimental data in a table and explain why this is more appropriate than listing or describing the results.8. Name 2 common functions that you would use on your calculator (not the simple operators addition, subtraction, division, and multiplication).9. If you saw the scientific term 560 nm, what topic do you think might being discussed? Explain why you think this. Production of Renewable Ammonia In recent years, significant interest has been paid to developing fuel and chemicals from renewable feedstocks, In this regard, you are requested to design a plant to produce 150 000 metric tons per annum of Ammonia (at least 99.5 wt. %). The hydrogen to nitrogen feed ratio is 3:1. The feed also contains 0.5 % argon. The feed is available at 40C and 20 atm. The plant should operate for 330 days in a year, in order to allow for shutdown and maintenance. The plant is to be built in Nelson Mandela Bay. In this assessment, you need to assess the feasibility of such a process by conducting a conceptual design, that covers the following topics: 1.1. Design basis 1.2. Literature Survey 1.3. Process Description 1.4. Preliminary block flow diagram (BFD) and process flow diagram (PFD) 1.4.1. Block diagram of the entire process 1.4.2. Process flow diagram for ammonia synthesis 1.5. Preliminary major equipment list The Magnetic Field of a plane wave traveling in the air is given by H=X 50 sin (2 x 10 ^7 -ky) (ma/m) determine the power density carried by the wave Which of the following could be considered an unethical use of evaluation research results?A. Commissioning an evaluation on a state prison with the intention of providing evidence of poor performance to justify cutting fundingB. All of theseC. None of theseD. Demonstrating the success of a federally funded social support program to lobby Congress for additional moneyE. A program director asking a researcher to use neutral, non-biased language in a report that will present negative findings about their program's effectivenessWhich of the following qualitative data analysis methods relies on the use of signs and symbols and their associated social meanings?A. Conversation analysisB. SemioticsC. Cross-case analysisD. Grounded Theory Method PLEASE HELPP: 2.11.2 Project: Performance Task: The Parallax Problem (For San Francisco)The Scenario: Youre looking for a sponsor to pay for you to participate in a sailboat race. Now that youve solved the parallax problem, use the same skills you used there to write a proposal that shows that you can win the race. The Project: Use the information provided in the performance task to estimate your travel costs and to calculate your average speed and the speed of last years winner. Use the questions below to help you gather information to write your proposal3. What is the distance between buoy A and B? (5 points) 4. What are the lengths of the other two triangle legs? (4 points: 2 points each)Remember what you know about the shape of the Race Course.5. What is the total length of the race course? (4 points: 3 for calculation, 1 for answer)Part VIII: Calculate the winners speed. (10 points)1. What was the winners speed during last years race? (5 points: 3 points for speed. 2 points for conversion to knots).2. How does the winners speed compare with your average speed? How much faster or slower are you? (5 points)Part IX: Write your proposal. (8 points)Now its time to make your proposal to the sponsor. Your sponsor will have their logo on your boat, so they want to be sure its likely to do well. The sponsor also needs to know what the expenses and risks are, so they know how much their investment in you will cost.1. Complete the table to summarize the results of your study. (4 points)Category:Race:Risk Analysis:Itemized Travel CostSafety hazardsCompetitive Analysis:My time and speedLast year's winning time and speedReward Analysis:My chances of winning2. Write a summary paragraph explaining why the sponsor should accept your proposal. (4 points) The following is a list of shoe sizes for a group of 13 people.4.5, 9.5, 8, 6.5, 10, 7, 8.5, 6, 7.5, 9, 6, 7, 11Which of the following box plots best represents the numerical data? A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe. A box plot using a number line from 3 to 11.25 with tick marks every one-fourth unit. The box extends from 6.25 to 8.75 on the number line. A line in the box is at 7.25. The lines outside the box end at 4.5 and 10. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe. A box plot using a number line from 3 to 13 with tick marks every one-half unit. The box extends from 6.5 to 9 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 12. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe. A box plot using a number line from 3 to 12.5 with tick marks every one-fourth unit. The box extends from 6.25 to 8.75 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 10.5. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe. Diamagnets have the property that they "dampen" the effects of an external magnetic field by creating an opposing magnetic field. The diamagnet thus has an induced dipole moment that is anti-aligned, such that the induced north pole is closer to the north pole creating the external field. An application of this is that diamagnets can be levitated (Links to an external site.).Now, the mathematics of generally describing a force by a non-uniform field on a dipole is a little beyond the scope of this course, but we can still work through an approximation based on energy. Essentially, whenever the theoretical loss of gravitational potential energy from "falling" no longer can "pay the cost" of increasing the magnetic potential energy, the object no longer wants to fall.Suppose a diamagnetic object floats above the levitator where the magnitude of the magnetic field is 18 T, which is inducing* a magnetic dipole moment of 3.2 Am2 in the object. The magnetic field 2.0 mm below the object is stronger with a magnitude of 33 T. What is the approximate mass of the floating object?Give your answer in units of g (i.e., x10-3 kg), and use g = 9.81 m/s2. You may assume the object's size is negligible. Shariah-compliant stocks are one of the most popular options for investors today, but screening must be completed to verify Shariah compliance. Determine the parameters that must be followed to achieve Shariah conformityislamic banking anf finance