Let R be the region enclosed by the graph of f(x) x^4-2.3x^3+4 and the horizontal line y=4, as shown in the figure above.
a. Find the volume of the solid generated when R is rotated about the horzontal line y=-2
b. Region r is the base of a solid. for this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in r. find the volume of the solid.
c. The vertical line x = k divides R into two regions with equal areas. Write but do not solve, an equation involving integral expressions whose solution gives the value k
Answers
The volume of the solid generated when R is rotated about the horizontal line y=-2 is 98.868.
The volume of the solid:3.574
The equation involving integral expressions:
[tex]\int\limits^k_0 {(4-f(x))} \, dx =\int\limits^2_0 {(4-f(x))} \, dx[/tex]
The given equation is f(x)=x^4-2.3x^3+4
f(x)=4 ⇒x=0,x=2.3
Now, substitute the values in the equation:
⇒ Volume= [tex]\pi \int\limits^2_0 {(4-(-2)^2)} \, dx -(f(x)-(-2)^2)dx[/tex]
By solving the equation we get:
volume=98.868
b) consider the point r in the graph which is perpendicular to the x-axis that an isosceles right triangle with a leg in r.
Area of cross-section:
A=1/2(leg)(leg)=1/2(4-f(x))^2
Volume=∫(1/2(4-f(x))^2)dx ∴with limits (0 to 2.3)
Solving the function we get:
Volume=3.574
c)given the vertical line x=k divides R into two regions with equal areas.
∫(4-f(x))dx=∫(4-f(x))dx
2∫(4-f(x))dx=∫(4-f(x))dx ∴with limits 0 to 2.3 and 0 to k
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Related Questions
On a coordinate plane, triangle A B C has points (negative 1, negative 1), (negative 2, negative 1) and (negative 2, negative 4). Triangle A prime B prime C prime has points (1, negative 1), (2, negative 1), and (2, negative 4). What is the name of the line of reflection that maps the pre-image to the image? y = x y-axis x-axis x = 1
Answers
Triangle ABC is reflected across the line y = x.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.
Given is that on a coordinate plane, triangle ABC has points (negative 1, negative 1), (negative 2, negative 1) and (negative 2, negative 4). Triangle A'B'C' has points (1, negative 1), (2, negative 1), and (2, negative 4).
The name of the line across which the reflection of the triangle ABC is done is : y = x.
Therefore, triangle ABC is reflected across the line y = x.
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Answer:
Y-axis
Step-by-step explanation:
What is a number that when you add 5 to it and then multiply the sum by 3 will equal 27?
Answers
So the answer would be 9 I’m pretty sure.
K=ºC+273.15
Slope
Y-intercept
Answers
The slope of the straight line equation as required to be determined in the task content is; 1.
The y-intercept of the straight line equation as represented in the task content is; 273.15.
What values correctly represent the slope and y-intercept of the given equation?It follows from the task content that the values which correctly represents the slope and y-intercept of the given equation be determined.
Since the given equation is;
K = ºC + 273.15
By comparison with the slope-intercept equation of a straight line; y = mx + c where y = K and x = °C.
It follows that the slope, m = 1 and the y-intercept, c = 273.15.
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Solve the system of equations using elimination: 7x + 4y = -9 and
-5x - 3y = 7.
Answers
For the provided functions 7x + 4y = -9 and -5x - 3y = 7, the values of x and y are 1 and -4, respectively.
What is a linear equation?A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."
Given the system of equations using elimination: 7x + 4y = -9 and
-5x - 3y = 7.
From equation 2
-5x - 3y = 7
calculate the value of x in the function of y
-5x = 7 + 3y
x = (-7 - 3y)/ 5
Substituting this value in equation 1
7/5(-7 - 3y) + 4y = -9
y = -4
substituting this value in equation 1
7x + 4(-4) = -9
x = 1
Therefore, the values of x and y the given functions 7x + 4y = -9 and
-5x - 3y = 7 are 1 and -4 respectively.
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Write the equation of the line that passes through the points (-3,-1) and (5, -1).
Put your answer in fully simplified point-slope form, unless it is a vertical or
horizontal line
Answers
The equation of line passes through the points (-3, -1) and (5, -1) will be;
⇒ y = - 1
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
Two points on the line are (-3, -1) and (5, -1).
Now,
Since, The equation of line passes through the points (-3, -1) and (5, -1)
So, We need to find the slope of the line.
Hence, Slope of the line is,
m = (y₂ - y₁) / (x₂ - x₁)
m = (- 1 - (-1)) / (5 - (-3))
m = 0/8
m = 0
Thus, The equation of line with slope 0 is,
⇒ y - (-1)= 0 (x - (-3))
⇒ y + 1 = 0
⇒ y = - 1
Therefore, The equation of line passes through the points (-3, -1) and (5, -1) in slope intercept form is,
⇒ y = - 1
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Use strategic thinking to find the solutions to these questions.
a. How much is 50 % of 10 liters of milk?
b. How far is 50 % of a 2,000- kilometer trip?
c. How long is 50 % of a 24 hour day?
Answers
Using strategic thinking, the solutions to the following questions are:
a) 5 liters of milkb) 1,000 kilometersc) 12 hours.What is strategic thinking?Strategic thinking involves using rational thought processes to focus on workable solutions to business problems.
Strategic thinking concentrates on the future based on its changing environments.
In this situation, 50% of a number represents one-half of the given value. We can divide each number by 2 to find the solutions.
a) 50% of 10 liters of milk = 5 liters (10 x 50%)
b) 50% of 2,000 kilometers = 1,000 kilometers (2,000 x 50%)
c) 50% of 24 hours = 12 hours (24 x 50%)
Thus, strategically thinking, half of each value represents its 50%.
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local salesman receives a base salary of $875 monthly. He also receives a commission of 6% on all sales over $1450. How much would he have to sell in a month if he needed to have a monthly income of $2800?
Answers
The total sales required to earn the desired income of $2800 is calculated to be $6,375.
What is sale?Sale is a business transaction involving the exchange of goods or services for money. It can refer to the act of selling, or the thing that is sold. Sales are typically the main source of revenue for businesses. The process of selling involves finding potential customers, negotiating a price, and delivering the product or service. The ultimate goal is to make a profit.
In order to earn a monthly income of $2800, a local salesman would need to sell $6,375 worth of goods in a month. This is calculated by subtracting the base salary of $875 from the desired income of $2800, which leaves $2,925. Then, 6% of $1450 is $87, so the remaining $2,838 must be made up of 6% of the sales above $1450. By dividing this amount by 6%, the total sales required to earn the desired income of $2800 is calculated to be $6,375.
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Consider a standard deck of 52 playing cards with 4 suits.
If A is the event of drawing a 6 from the deck, and B is the event of drawing a black playing card from the deck, what is the intersection of A and B?
(Remember that the black cards are spades and clubs.)
A) drawing a 6, a heart, or a diamond
B) drawing a 6, a club, or a spade
C) drawing the 6 of clubs or the 6 of spades
Answers
The intersection between A and B in a deck of 52 playing cards is option B, drawing a 6, a club, or a spade.
What is the intersection of two sets?In a standard deck of 52 playing cards with 4 suits
Event A is drawing a 6 from the deck
Event B is drawing a black card from the deck
Therefore A will have 4 possibilities of cards with 6 on them
B will have a total of 26 possibilities of black playing cards with both clubs and spade.
The intersection of A and B will be drawing a 6 or a black card(a club, or a spade) as the common elements in the two sets. Therefore, drawing a 6, a club, or a spade.
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Ivanna needs 201 programs for the school play on Thursday. How many boxes of programs will she need, given that each box contains 46 programs?
boxes
31
E
Answers
The number of boxes of programs that she needed will be 5.
What is Algebra?Algebra is the study of algebraic expressions, while logic is the manipulation of those concepts.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This rule is used to answer the problem correctly and precisely.
Ivanna needs 201 programs for the school play on Thursday. If each box contains 46 programs.
Then the number of boxes of programs that she needed will be given as,
⇒ 201 / 46
⇒ 4.369
The number of boxes cannot be in decimal form. Then the number of boxes will be 5.
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Which event will have a sample space of S = {g, b}?
A. Flipping a fair, two-sided coin
B. Rolling a six-sided die
C. Flipping a two-sided coin with one side green and the other side blue
D. Choosing a tile from a pair of tiles, one with the letter A and one with the letter B
Answers
An event that will have a sample space of S = {g, b} include the following: C. Flipping a two-sided coin with one side green and the other side blue.
What is a sample space?In Mathematics and statistics, a sample space can be defined as a set or collection of possible outcomes (results or outputs) that are obtainable from a random experiment. Mathematically, a sample space is represented or denoted by using the symbol, "S."
Additionally, a sample space may contain a number of outcomes (results or outputs) that depends on the experiment. Generally speaking, the subset of possible outcomes (results or outputs) of that are obtainable from an experiment is typically referred to as events.
In this context, we can reasonably infer and logically deduce that "Flipping a two-sided coin with one side green and the other side blue" is an event which would have a sample space that is modeled by this mathematical expression;
S = {g, b}
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Answer: C: Flipping a two-sided coin with one side green and the other side blue.
2. A doctor's office casts a shadow that is 21 feet long. At the same time of day, a nearby tree standing 7 feet tall casts a 3 -foot-long shadow. How tall is the doctor's office?
25 ft.
49 ft.
9 ft.
34 ft.
Answers
The height of the doctor's office is solved using similar triangles to give
49 feetWhat are similar triangles?When the corresponding sides of two triangles are proportionate and their corresponding angles are congruent, The triangles are said to be similar.
As a result, the triangle's side should be proportionate if its corresponding angles are congruent.
Examining the descriptions shows that pair of equivalent sides are
height of tree and shadow of tree,
height of doctor's office and shadow's of doctor's office
The solution is worked out
height of tree / shadow of tree = height of doctor's office / shadow's of doctor's office
7 / 3 = height of doctor's office / 21
cross multiplying
3 * height of doctor's office = 7 * 21
height of doctor's office = 7 * 21 / 3
height of doctor's office = 49 feet
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I need a quick answer please and thank you!
Answers
Information number of triangles
sides of 20 cm, 5cm and 15 cm : no triangle
angles 35, 35, and 110 degrees many triangles
angles 50, 50, and 85 degrees no triangle
sides of 7, 3 and 9 one unique triangle
What are the conditions for formation of triangle?A maximum of one unique triangle is formed when the three sides of the triangle given are in the form that the sum of the two shortest sides will be greater than the third side
In the problem, for a triangle to be formed
15 cm + 5 cm > third side, here the sum is not greater hence no triangle
7 + 3 > third side, here the sum is greater than the third side hence a unique triangle is formed
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how do i answer this question? please help
Answers
Answer:
multiply the second equation by -2(x, y) = (2, 2)Step-by-step explanation:
Apparently, you want to solve the system of equations by multiplying or dividing one of them by a factor that will make the y-coefficients the same.
MultiplicationThe y-coefficients are 2 and -1, so we can multiply the first by -1/2 to make it the same as the second, or we can multiply the second by -2 to make it the same as the first.
We prefer to work with integers, rather than fractions, so we choose to multiply the second equation by -2.
-2(3x -y) = -2(4)
-6x +2y = -8 . . . . . . . y-coefficient is the same as in the first equation
SolutionNow, we can subtract the second equation from the first to eliminate the y-terms:
(3x +2y) -(-6x +2y) = (10) -(-8)
9x = 18 . . . . . . . . . simplify
x = 2 . . . . . . . . . divide by 9
Substituting this into the original second equation gives ...
3(2) -y = 4
6 -4 = y = 2 . . . . . . . add (y-4)
The solution is (x, y) = (2, 2).
__
Additional comments
The picture posted here has no question, so there is no way to tell how to answer it. Any answer must be provided in accordance with whatever instructions are given.
Solving these equations is somewhat easier if you make the y-coefficients opposites, rather than the same. This can be done by multiplying the second equation by 2: 6x -2y = 8. Now the equations can be added to give 9x = 18, as above.
Of course, the x-coefficients are already the same, so we can more easily eliminate the x-terms. Subtracting the second equation from the first gives ... (3x +2y) -(3x -y) = (10) -(4), or 3y = 6.
Solve the system by graphing and write your answer as an ordered pair . y=3x-4 . y=-1/2x+3
Answers
This system of equations has been solved graphically and its solution is equal to the ordered pair (2, 2).
What is a graph?In Mathematics, a graph refers to a type of chart that is typically used for the graphical representation of ordered pairs on both the horizontal and vertical lines of a cartesian coordinate, which are the x-coordinate and y-coordinate respectively.
In this scenario, we would use an online graphing calculator to plot the given system of equations and then take note of the point of intersection of the lines on the graph;
y = 3x - 4 ........equation 1.
y = -1/2x + 3 ........equation 2.
Based on the graph (see attachment), we can logically deduce that the solution to the given system of equations is the point of intersection of the lines on the graph representing each of them, which is given by the ordered pair (2, 2).
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The graph of the system of linear equation is attached below with solution of x, y as (2, 2)
What is Graph of system of linear equationA graph of a system of linear equations is a set of lines (or a line in case of a unique solution) on a coordinate plane that represent the solutions of the system. The graph of a system of linear equations can have different types of solutions depending on the number of lines and the relationship between them.
In this problem, we will use a graphing calculator to plot the equation and find the point of intersection which is the solution to the graph.
y = 3x - 4y = -1/2x + 3The point of intersection which shows the solution (x,y) of the system of linear equation is at (2, 2)
Kindly find the attached graph below as solution.
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Solve for the balloon volume using the
equation below where:
volume = v
circumference = c
V=
Answers
The volume of the balloon is 1441.64cm³.
How to calculate the volume?To calculate the volume these are few ways
Solve for Volume by space. All physical objects occupy space, and you can find the volume of some of them by measuring their physical dimensions. Solve for Volume by Density and Mass. Density is defined as an object's mass per a given unit of volume. Solve for Volume by Displacement.The radius of the balloon is given as 7 cm. Also, the equation to calculate the volume is given as c³/6π²
The circumference c will be:
= 2πr = 2 × 22/7 × 7
= 44
The volume will be:
= c³/6π²
= 44³/(6 × 3.14²)
= 85284/(59.1576)
= 1441.64
Therefore, volume of the balloon will be 1441.64 cm³.
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Hello! Please help asap.
Answers
The line graph of the inequality is represented by an open circle at x = - 9 going towards negative infinity.
What are inequalities and their types?Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.
The symbol a < b indicates that a is smaller than b.
When a > b is used, it indicates that a is bigger than b.
a is less than or equal to b when a notation like a ≤ b.
a is bigger or equal value of an is indicated by the notation a ≥ b.
The given inequality is x + 3 < - 6.
x < - 6 - 3.
x < - 9.
The line graph of the inequality will be represented as an open circle at
x = - 9 extending towards negative infinity because x is strictly smaller
than 9.
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Lana flips a standard coin five times. What is the probability that she flips a minimum of two tails?
Answers
The probability that she flips a minimum of two tails is 13/16.
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
We have,
The sample space of flipping a coin 5 times.
= {H, T} {H, T} {H, T} {H, T} {H, T}
= {HH, HT, TH, TT} {HH, HT, TH, TT} {H, T}
= {HH, HT, TH, TT} {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
= {HHHHH, HHHHT, HHHTH, HHHTT, HHTHH, HHTHT, HHTTH, HHTTT, HTHHH, HTHHT, HTHTH, HTHTT, HTTHH, HTTHT, HTTTH, HTTTT, THHHH, THHHT, THHTH, THHTT, THTHH, THTHT, THTTH, THTTT, TTHHH, TTHHT, TTHTH, TTHTT, TTTHH, TTTHT, TTTTH, TTTTT}
There are 32 possible outcomes.
There are 26 outcomes with a minimum of two tails.
The probability of flipping a minimum of two tails.
= 26/32
= 13/16
Thus,
The probability of fliping a minimum of two tails is 13/16.
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A construction worker needs 3 2/3 pounds of concrete mix. He has 1 1/2 pounds of concrete mix in one bag, and he has 3/10 pound of concrete mix in another bag.
How many more pounds of concrete mix does the construction worker need?
Enter your answer as a mixed number in simplest form by filling in the boxes
Answers
On solving the provided question, we cam say that - by unitary method the answer is 4/5
What is unitary method ?The unit technique is an approach to problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value. The unit method, to put it simply, is used to extract a single unit value from a supplied multiple. For instance, 40 pens would cost 400 rupees, or the price of one pen. The process for doing this may be standardized. a single country. anything that has an identity element. (mathematics, algebra) (Linear algebra, mathematical analysis, mathematics of matrices or operators) Its adjoint and reciprocal are equivalent.
here, construction worker = 2/3 pounds
he has = 1/2
so the answer is 4/5
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Six friends went to the movies, if the total cost was $27.00, how much was each ticket?
Answers
Answer:
$4.50 each
Step-by-step explanation:
27/6=4.50
Ill give brainliest to the person who answers
Answers
Step-by-step explanation:
They are congruent by the AAS postulate.
ΔKNJ ≅ ΔKLM by AAS. They are congruent by the AAS postulate, because it states that if two angles and the non-included side are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Since they both are tick marked on one side and shown to have the same angle, you must note that they both share the same side. And because they both share the same side; they have the same angle.
Find the slope of the following graph and choose the correct result.
A : 1/4
B : 1/3
C : 1/5
D : 2/3
Answers
Answer:
1/3
Step-by-step explanation:
Slope = Rise/Run
We are rising 1 and running 3 on the graph
PLEASE HELP DONT IGNORE ASAP.
Determine if the situations described below are proportional or not proportional.
Answers
Answer: both are proportional
Step-by-step explanation:
Both. ..............................
This velocity-time graph shows 8 seconds of a car journey.
What was the total distance travelled by the car during this time?
If your answer is a decimal, give it to 1 d.p.
Velocity (m/s)
20
15
10
5
0
Time(s)
Answers
The distance covered by the car in 8 seconds of the journey is 100 meters.
What is the distance?The complete movement of an object, regardless of direction, is referred to as distance. The amount of ground a thing travels from its starting point to its destination is also referred to as distance.
Given:
Velocity (m/s) = 20, 15, 10, 5, 0
Calculate the distance as shown below,
[tex]Distance = velocity \times time[/tex]
Distance = 0 × 20 + 15 × 2 + 10 × 4 + 5 × 6 + 0 × 8
Distance = 0 + 30 + 40 + 30 + 0
Distance = 100 meter
Thus, the distance covered by the car is 100 meters.
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prove that there is a 1-1 correspondence between the set of all positive integers and the set of all rational numbers
Answers
A one-to-one correspondence between the set of all positive integers and the set of all rational numbers exists.
By definition, any rational number can be expressed as a fraction a/b, where a and b are integers and b is nonzero. Since any positive integer can be expressed as the fraction a/1, each positive integer can be associated with a rational number a/1.
Thus, there is a one-to-one correspondence between the set of all positive integers and the set of all rational numbers. Furthermore, since any rational number a/b can be expressed as the product of a and b, it can also be associated with the product of two positive integers a and b.
Thus, there is also a one-to-one correspondence between the set of all rational numbers and the set of all positive integers.
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There are two sets of rainfall data for two different time periods in the same location.
Period 1: 2.3, 2.1, 2.2, 2.2, 2.2, 2.1, 2.4, 2.5, 2.2, 2.0, 1.9, 1.9, 2.1, 2.2, 2.3
Period 2: 2.3, 2.1, 3.3, 1.5, 3.6, 1.6, 3.0, 1.1, 4.7, 2.1, 2.4, 1.9, 2.8, 0.5, 2.3
Draw a conclusion based on the data provided.
has to be a sentence
Answers
Comparing the two sets of rainfall data for two different time periods in the same location, it can be concluded that the amount of rainfall in Period 1 was fairly constant whereas the amount of rainfall in Period 2 varied greatly.
What conclusion can be drawn from the rainfall data?The rainfall data for two different time periods in the same location provides information about the pattern of rainfall over the time periods.
The rainfall data over the period did not vary widely apart. The data shows that the amount of rainfall did not change too much.
However, for Period 2, the amount of rainfall had very different values.
Hence, it seems rainfall was more precise in frequency in Period 1 than in Period 2.
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Please :) !!
Find the value of b in each parallelogram. Then find each side length or
angle measure.
Answers
Answer:
Step-by-step explanation:
help please i don’t understand this!!!
Answers
Each side length of an equilateral triangle is (6x³+9x)/3 units
What is perimeter?
The complete length of a shape's edge serves as its perimeter in geometric terms. Adding the lengths of all the sides and edges that surround a form yields its perimeter. It is calculated using linear length units such centimeters, meters, inches, and feet.
An equilateral triangle has all the sides equal in length
Let each side be y
Perimeter is the sum of all sides
So, Perimeter = y+y+y = 3y
Given, Perimeter = 6x³+9x
=> 3y= 6x³+9x
=> y= (6x³+9x)/3
Thus, each side length of an equilateral triangle is (6x³+9x)/3 units
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TJ Maxx's stock from Tuesday to Wednesday went from $22 to $25. Find the percent increase in the stock
Answers
Jordan is painting her kitchen.
To get the color she wants, she mixes 2 parts red paint and 6 parts yellow paint.
If she needs 18 parts of yellow paint, how much red paint will she need?
Answers
Answer:
1. 1/3 or 1 red paint : 3 yellow paint
2. 3/1 or 3 or 3 yellow paint: 1 red paint
Step-by-step explanation:
In the diagram below, RT intersects QU at point S.
Which postulate should be used to prove that
RQS≈ TUS
Answers
The triangle QRS and TUS are congruent.
What means postulate?
In mathematics, a postulate is a statement that is considered exactly true without showing any proof on it.
What is parallelogram?The quadrilateral whose opposite sides are equal in length and parallel to each other is called parallelogram. The parallelogram that has four right angles is called rectangle. One of the main properties of parallelogram is that the diagonals bisect at a point.
In the given question, two diagonals QU and RT intersect at point S.
If we join the point Q with T and R with U, RQTU parallelogram is formed.
QU and RT are the diagonals of the parallelogram.
According to the properties of parallelogram, RQ = TU and QT =RU
As the diagonals intersect at S, we can discuss about two triangle RQS and TUS
we find side length QS = side length SU and side length RS = side ST
On the other hand, ∠QSR =∠UST because the diagonals bisect at S
Finally, the triangle RQS and TUS has two equal side length and one equal angles. Besides, other side and remaining angles are also similar.
So, ΔRQS≅ΔTUS
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