The volume of the solid obtained by rotating about the x-axis the region enclosed by the curves is 32π/5 cubic units.
In this problem, we are asked to find the volume of a solid obtained by rotating a region enclosed by curves about the x-axis. We will use integration to calculate the volume.
First, let's sketch the region and the solid we need to find the volume of.
The region is enclosed by the curves
=> y = 16 /(x² + 16), y = 0, x = 0, and x = 4.
When we rotate this region about the x-axis, we get a solid with a hole in the center, shaped like a donut.
To find the volume of this solid, we can use the method of cylindrical shells.
We start by taking a thin strip of the region, parallel to the y-axis, and of width dy. This strip has height y, and we rotate it about the x-axis to get a cylindrical shell of thickness dy and radius x.
The volume of this cylindrical shell is given by the formula V = 2πxy dy, where x is the distance from the y-axis to the edge of the shell.
To express x in terms of y, we can solve for x in the equation
y = 16 /(x² + 16).
y(x² + 16) = 16
x² = 16/y - 16
x = √(16/y - 16)
Now we can substitute x into the formula for V to get:
V = 2πx(16/(x² + 16)) dy
[tex]= 2\pi(16/y - 16)^{1/2}(16/y) dy \\\\= 32\pi(y - y^{3/2}) dy[/tex]
To find the total volume, we integrate this expression from y = 0 to y = 1 (since the maximum value of y is 16/16 = 1):
[tex]\int_0^1 32\pi(y - y{(3/2)}) dy = 32\pi/5[/tex]
Therefore, the volume of the solid obtained by rotating the region about the x-axis is 32π/5 cubic units.
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Examine the right triangle below. Let a = 15, b = 20, and c = 25. What is the
sin B=, rounded to the nearest hundredth?
The value of sin B rounded to the nearest hundredth is 0.80.
What are Trigonometric Functions?Trigonometric functions are defined as the real functions which are simply the functions of an angle of a triangle. They are basically the periodic functions which relate an angle in a right angled triangle to the ratios of the length of two sides.
The right triangle represented in the question is given below.
We know that sine of an angle in a right angled triangle is the ratio of it's opposite side to hypotenuse.
In the triangle ABC, AB is the hypotenuse and AC is the opposite side of B.
Sin B = AC / AB
Sin B = b / c
Sin B = 20 / 25
Sin B = 0.8
Hence the value of sine of B is 0.8.
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convert 31,750 feet into miles
To convert 31,750 feet to miles, divide the number of feet by the number of feet per mile:
31,750 feet / 5,280 feet/mile = 6 miles
Samir made a scale drawing of a city. He used the scale 5 inches = 1 yard. What scale factor
does the drawing use?
Simplify your answer and write it as a fraction.
Answer:
the scale factor that Samir used in his drawing is 5/1.
Step-by-step explanation:
Here's a step by step explanation of how to find the scale factor of a drawing:
Identify the scale used in the drawing. In this case, the scale used in Samir's drawing is 5 inches = 1 yard.
Express the scale as a ratio between the two sizes. In this case, the ratio is 5 inches : 1 yard.
Simplify the ratio, if possible. In this case, the ratio 5 inches : 1 yard can be simplified to 5/1.
The simplified ratio, 5/1, is the scale factor of the drawing. This means that for every 5 inches on the drawing, the actual length in the real world is 1 yard.
So, the scale factor that Samir used in his drawing is 5/1.
please help me with this.
Following are the values of the functions -
1) f(a) = 8a - 4
2) f(a+h) = 8a+ 8h - 4
3)[tex]\frac{f (a+h)-(a) }{h}= 8[/tex]
Describe function.An expression or relationship with one or more variables is called a function. Both its inputs and outputs are numerous. There is just one output per input. The function explains how the inputs and outputs are related.
To evaluate f(a), we simply substitute the value of "a" into the function:
1) f(a) = 8a - 4
To evaluate f(a+h), we substitute the value of "a+h" into the function:
2) f(a+h) = 8(a+h) - 4
f(a+h) = 8a+ 8h-4
To find the expression for [tex]\frac{f (a+h)-(a) }{h}[/tex], we divide the result by h: L(a+h)-f(a) h we can subtract f(a) from f(a+h) and then
3) [tex]\frac{f (a+h)-(a) }{h}[/tex]
[tex]= \frac{ (8a+8h-4-(8a-4)}{h} \\= \frac{8h}{h}= 8[/tex]
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PLEASEE HELPPP!!! 3 QUESTIONS
Consider the function below.
f(x)=8/x−6
What would be the output if the input is 8?
responses:
-1/4
-4
-8/14
-8/6
What is the average rate of change of the function f(x)=3x^2−5 over the interval [2, 6]?
responses:
-3
-48
-12
-24
The average mass of a certain type of microorganism is 2.4×10^−6
grams. What is the approximate total mass of 5,000 of these microorganisms?
responses:
-0.00012 g
-0.0012 g
-0.012 g
-0.12 g
A. The value of the function at x = 8 will be 4. Then the correct option is B.
B. The average rate of change of the function in the interval of [2, 6] will be 24. Then the correct option is D.
C. The total mass of 5,000 of these microorganisms will be 0.012 g. Then the correct option is C.
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
A. The function is f(x) = 8 / (x - 6), then the value of the function at x = 8 is given as,
f(8) = 8 / (8 - 6)
f(8) = 4
Then the correct option is B.
B. The function is f(x) = 3x² - 5, then the average rate of change of the function in the interval of [2, 6] is given as,
Average rate = [3(6² - 4²)] / (6 - 2)
Average rate = 24
Thus, the correct option is D.
C. The average mass of a certain type of microorganism is 2.4 × 10⁻⁶ grams. Then the total mass of 5,000 of these microorganisms is given as,
⇒ 2.4 × 10⁻⁶ × 5,000
⇒ 0.012 grams
Thus, the correct option is C.
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3. If an angle measures 2π radians, what is the measure of the angle in degrees?
Answer:
360 degrees
2 pi radians, means you have 2 pi of something and they are radians. It is the same 360 degrees. A radius is the line from the side of the circle to the center.
Reason: pi radians is 180 degrees. Double each value to find that 2pi radians is 360 degrees. It's a full revolution of a circle. Use a unit circle if necessary.
LMNP is a rectangle. Find the value of x and the length of each diagonal.
LN = 4x+10 and MP = 9x-5
X =
Answer:
x=5 and LN= 30, MP= 40
Step-by-step explanation:
Add like terms to both sides of the equation and you will get your answer
You randomly survey middle school students and high school students about whether they prefer spring or winter. The results
are shown in the two-way table. Find the percent of middle school students that prefer each season and the percent of high school students that prefer each season
The percentage of middle school students that prefer each season and the percentage of high school students that prefer each season is 57.93% and 42.07% respectively.
What is the percentage?The percentage is the ratio of the composition of value to the overall composition of value multiplied by 100.
Here,
Total number of students = 93 + 31 + 36 + 54 = 214
The percentage of students in middle school prefer spring,
= number of students in middle school prefer spring / total student × 100%
= 93/214 × 100%
= 43.45%
Similarly,
The percentage of students in middle school prefer winter,
= number of students in middle school who prefer winter/ total students × 100%
= 31/214 × 100%
= 14.48%
Thus, the percentage of middle school students that prefer each season and the percentage of high school students that prefer each season are 43.45 + 14.48 = 57.93% and 100-57.93% = 42.07%.
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Help, Solve it by using Elmination, Linear Equation
Answer:
The mistake they made was that they subtracted [tex]2y[/tex] from [tex]2y[/tex] and got [tex]0y[/tex]. Instead of subtracting, they should have added it.
Step-by-step explanation:
Given:
[tex]8x+2y=44\\5x+2y=35[/tex]
We can combine the system of equations into a single equation:
[tex]13x+4y=79[/tex]
From this step, we can conclude that the answer for part b is incorrect. The mistake they made was that they subtracted [tex]2y[/tex] from [tex]2y[/tex] and got 0y. Instead of subtracting, they should have added it.
We can continue solving the system of equations by writing y in terms of x.
[tex]8x+2y=44\\2y=44-8x\\y=22-4x[/tex]
now substitute this into the single equation:
[tex]13x+4y=79\\13x+4(22-4x)=79\\13x+88-16x=79\\-3x+88=79\\-3x=-9\\x=3[/tex]
Since we now know that [tex]x=3\\[/tex], we can substitute this into the equation we made for y.
[tex]y=22-4x\\y=22-4(3)\\y=22-12\\y=10[/tex]
The solution for the system of equations is [tex]x=3[/tex] and [tex]y=10\\[/tex].
Write an equation to describe the sequence below. Use n to represent the position of a term
In the sequence, where n = 1 for the first term.
-25, -50, -100,...
Write your answer using decimals and integers.
a,
Blank 1:
Blank 2:
Submit
C
C
An equation to show the sequence is -25n.
What is a geometric series?When all the terms of the geometric sequence are added, than that expression is known as geometric series.
The sum of terms of a geometric sequence;
Now, lets suppose its initial term is , multiplication factor will be r
and let it has total n terms, then, its sum is given as:
[tex]S_n = \dfrac{a(r^n-1)}{r-1}[/tex]
(sum till nth term)
We are given that;
The first term of sequence=-25
D= -50-(-25)
=-25
Now,
an = a + (n-1)d
=-25+(n-1)-25
=-25-25n+25
=-25n
Therefore, the answer of the sequence will be -25n.
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Question is attached below!
Thanks! :)
The parameters for the home mortgage, and the monthly payment, PMT, for the mortgage are as follows;
Part 1;
P = $175,000, APR = 7%, n = 12, Y = 30
Part 2;
The monthly payment for the mortgage, PMT ≈ $1,164.28
What is a mortgage?A mortgage is a loan agreement, with the possession of the collateral presented for the loan such as a house being bought with the loan amount, being relinquished by the debtor, upon default
Part 1
The value of the mortgage = $175,000
The fixed Annual Percentage Rate, APR = 7%
The duration of the payment = 30 years
The principal amount of the mortgage, P = $175,000The annual percentage rate, APR = 7% = 0.07The number of payment (periods), per year, n = Monthly = 12The number of years of the loan, Y = 30Part 2; The monthly payment PMT loan formula can be presented in the following format;
[tex]PMT = \frac{P\times \frac{APR}{n} }{1-\left(1+\frac{APR}{n}\right)^{(-n\cdot Y)} }[/tex]Therefore;
[tex]PMT = \frac{175000\times \frac{0.07}{12} }{1-\left(1+\frac{0.07}{12}\right)^{(-12\times 30)} } \approx 1164.28[/tex]
The monthly payment for the mortgage is, PMT ≈ $1,164.28
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Solve: 6(x + 5) = -2(x-3)
Answer:
Step-by-step explanation:
6(x+5) = 6x+30
-2(x-3)=-2x+6
6x+30=-2x+6
8x+30=6
8x=-24
x=-3
Answer: x = -3
Step-by-step explanation:
6(x + 5) = -2(x-3)
1. Distribute: 6x + 30 = -2x + 6
2. Add 2 to both sides: 8x + 30 = 6
3. Subtract 30 on both sides: 8x = -24
4. Divide by 8 on both sides: x = -3
Help, please thank you, and have a wonderful day
Students are reading a book that has 280 pages. Nora already read 30% of the book. She is choosing between two options to finish reading the book.
Option 1: read 25% of the remaining pages each night
Option 2: read 28 pages each night
Which option should Nora use to finish the book in the least amount of time?
Explain your reasoning.
Answer:
Step-by-step explanation:
To determine which option is faster, we need to determine the total number of nights required for each option.
First, let's find the number of pages Nora still needs to read by subtracting the number of pages she's already read from the total number of pages: 280 pages * 30% = 84 pages
So, Nora still needs to read 280 pages - 84 pages = 196 pages.
For Option 1, Nora would read 25% of the remaining pages each night, so the number of pages she reads each night would be: 196 pages * 25% = 49 pages
The total number of nights required would be 196 pages / 49 pages/night = 4 nights.
For Option 2, Nora would read 28 pages each night, so the total number of nights required would be 196 pages / 28 pages/night = 7 nights.
Since 4 nights is less than 7 nights, Nora should choose Option 1 to finish reading the book in the least amount of time.
Factorise:
(a - 2b)^3 + (2a - b)^3
Answer:
Step-by-step explanation:
To factorise the expression, we can use the factor theorem which states that if a polynomial p(x) has a factor (x - r), then p(r) = 0.
We can use this theorem to find the factors of the polynomial expression:
(a - 2b)^3 + (2a - b)^3
We can write (a - 2b)^3 as (a - 2b)(a^2 - 4ab + 4b^2). Using the factor theorem, we see that:
(a - 2b)(a^2 - 4ab + 4b^2) = 0 when a = 2b
Similarly, we can write (2a - b)^3 as (2a - b)(4a^2 - 4ab + b^2). Using the factor theorem, we see that:
(2a - b)(4a^2 - 4ab + b^2) = 0 when a = b/2
Now we have two equations: a = 2b and a = b/2, which we can use to solve for a and b in terms of each other. We see that:
a = 2b = b/2
Multiplying both sides by 2, we get:
2a = b
We can substitute this back into the original expression:
(a - 2b)^3 + (2a - b)^3 = (a - 2(2a))^3 + (2a - (2a))^3 = (-3a)^3 + (0)^3 = -27a^3
So the factorised form of the expression is:
(a - 2b)^3 + (2a - b)^3 = -27a^3.
Work Shown:
Use the sum of cubes factoring rule
x^3 + y^3 = (x+y)(x^2 - xy + y^2)
In this case,
x = a-2by = 2a-bSo,
x^3 + y^3 = (x+y)(x^2 - xy + y^2)
(a-2b)^3 + (2a-b)^3 = (a-2b+2a-b)((a-2b)^2 - (a-2b)(2a-b) + (2a-b)^2)
(a-2b)^3 + (2a-b)^3 = (3a-3b)((a^2-4ab+4b^2) - (2a^2-ab-4ab+2b^2) + (4a^2-4ab+b^2))
(a-2b)^3 + (2a-b)^3 = (3a-3b)((a^2-4ab+4b^2) - (2a^2-5ab+2b^2) + (4a^2-4ab+b^2))
(a-2b)^3 + (2a-b)^3 = (3a-3b)(a^2-4ab+4b^2 - 2a^2+5ab-2b^2 + 4a^2-4ab+b^2)
(a-2b)^3 + (2a-b)^3 = (3a-3b)(3a^2-3ab+3b^2)
(a-2b)^3 + (2a-b)^3 = 3*3(a-b)(a^2-ab+b^2)
(a-2b)^3 + (2a-b)^3 = 9(a-b)(a^2-ab+b^2)
The answer has been fully confirmed with WolframAlpha.
Consider the line 9x-8y=-9.
Find the equation of the line that is parallel to this line and passes through the point (−3, 4).
Find the equation of the line that is perpendicular to this line and passes through the point (-3, 4).
Equation of parallel line:
Equation of perpendicular line:
1
=
X
0=0
5
Determine the more basic function that has been shifted, reflected, stretched, or compressed q(x) = -1/x + 1
The more basic function that has been shifted, reflected, stretched, or compressed to create q(x) = -1/x + 1 is the function f(x) = 1/x.
What do you understand by the team function?The transformation applied to f(x) to obtain q(x) is divided into several steps:
Reflection: The negative sign in front of 1/x reflects the graph of f(x) across the x-axis, causing the y-values to change sign.
Vertical shift: Adding "+1" to -1/x raises the graph of f(x) by one unit.
Horizontal shift: Although there is no explicit horizontal shift in q(x), we can see that the graph of q(x) has a vertical asymptote at x=0, whereas the graph of f(x) does not. This means that the graph of q(x) has been shifted to the right by one unit.
The more basic function that has been shifted, reflected, stretched, or compressed to create q(x) = -1/x + 1 is the function f(x) = 1/x.
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HELPPP PLEASE I NEED HELP
The rule A(x,y) ⇒ (-y,x) describes a transformation. The correct option is D.
What is a transformation of rotation?The concept of rotation in mathematics has its roots in geometry. Any rotation is a movement of a certain area while maintaining at least one point. It may be used to describe, for instance, how a rigid body moves around a fixed point.
Our point A(x,y) becomes A' when a point is rotated 90 degrees counterclockwise about the origin (-y,x). To put it another way, flip x, and y, making y negative.
The transformation is described by the rule A(x,y) (-y,x). The right answer is D.
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Only using positive exponents, what is the simplified form of the expression p−6q7r/p2q−2
Step-by-step explanation:
Here is it. I hope it will help
Sarah has 275 red beads and 3 times as many blue beads. She uses a total of 156 beads to make a bracelet. How many beads are left?
Answer: 994 beads
Step-by-step explanation:
Red Beads = 275
Blue Beads = 825 (275 x 3)
825 + 275 = 1100 beads (total)
1100 - 156 = 994 beads
Find the equation for the line through the points (2,3) and (7,1)
Answer:
Step-by-step explanation:
m=3-1/2-7= -2/5
c⇒ -2
y=mx+c
y=-2/5x-2
Draw a rectangular fraction model to explain your thinking. 1/2 of 2/3 = 1/2 of_ third(s) =_ third(s)
A rectangle that has been divided into equal parts is referred to as a rectangular fraction. Depending on how many equal pieces you divide the whole into, the equal parts may be divided into halves, thirds, fourths, and so on.
1/2 of 2/3 is equal to 1/3
What is the equation?
The equation is defined as a mathematical statement with at least two terms that contain variables or values that are equal.
To represent the multiplication of 1/2 and 2/3, we can draw a rectangle and divide it into equal parts.
1) First, we draw a rectangle.
2) Then, we divide it into 2 equal parts horizontally.
3) Next, we divide each half vertically into 3 equal parts.
4) Now we can see that each small rectangle represents 1/6 of the whole rectangle.
So, 1/2 of 2/3 is equal to 1/2 of 1/3.
Therefore, 1/2 of 2/3 is equal to 1/3.
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Determine whether the ordered pair is a solution to the system of equations or inequalities y<4x-1; y>x. Yes or no?
(1,3)
(0,0)
Answer:
Step-by-step explanation:
To determine whether an ordered pair is a solution to a system of equations or inequalities, we need to substitute the values for x and y into each equation or inequality and see if the resulting expression is true. If it is true for all of the equations or inequalities, then the ordered pair is a solution to the system.
Here's the step-by-step process for each ordered pair:
(1,3):
Substitute x = 1 and y = 3 into y < 4x - 1: 3 < 4(1) - 1 = 3 < 3, which is true.
Substitute x = 1 and y = 3 into y > x: 3 > 1, which is true.
Since both inequalities are true, the ordered pair (1,3) is a solution to the system y < 4x - 1 and y > x.
(0,0):
Substitute x = 0 and y = 0 into y < 4x - 1: 0 < 4(0) - 1 = -1, which is false.
Substitute x = 0 and y = 0 into y > x: 0 > 0, which is false.
Since at least one inequality is false, the ordered pair (0,0) is not a solution to the system y < 4x - 1 and y > x.
This is a sketch of y = sin x.
The vertex A is at (90°, 1).
Write down the coordinates of the vertex A of the curve with equation
a) y = sin x + 2
b) y=-sin x
Vertex A: (90°, 3) and Vertex B: (90°, -1) of the sine function.
EquationsThe vertex A of the curve y = sin x + 2 is obtained by shifting the vertex of y = sin x upward by 2 units. Since the vertex of y = sin x is (90°, 1), the vertex of y = sin x + 2 will be:
A = (90°, 1 + 2) = (90°, 3)
Therefore, the coordinates of the vertex A of the curve y = sin x + 2 are (90°, 3).
b) The curve y = -sin x is obtained by reflecting the curve y = sin x about the x-axis. Since the vertex of y = sin x is (90°, 1), the vertex of y = -sin x will be:
A = (90°, -1)
Therefore, the coordinates of the vertex A of the curve y = -sin x are
(90°, -1).
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composite functions homework
The composite function has its value to be g o h(x) = -20x^2 - 30x
How to determine the composite functionfrom the question, we have the following parameters that can be used in our computation:
h(x) = 4x + 6
g(x) = -5x
Using the above as a guide, we have the following:
g o h(x) = g(x) * h(x)
substitute the known values in the above equation, so, we have the following representation
g o h(x) = (4x + 6) * -5x
Evaluate
g o h(x) = -20x^2 - 30x
Hence, the value is g o h(x) = -20x^2 - 30x
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3ab2 +1
3a²b-1
(a = 2 ; b = -2)
EVALUATE THE EXPRESSION
Answer:
3ab2+1 = -23
3a^2b-1 =-25
Step-by-step explanation:
a student gets paid to sell raffle tickets at the school basketball game. He earns $15 a day, plus an extra $0.25 for each raffle ticket he sells and $2 for each hour he works at the game. If d= days, r= raffle tickets, and h= hours, what function can he use to calculate his earnings?
The function can be used to calculate his earnings, 15d + 0.25r + 2h
What are functions?Function is a relation between a set of inputs and a set of outputs which are permissible. In a function, for particular values of x we will get only a single image in y. It is denoted by f(x).
The function that the student can use to calculate his earnings is:
Earnings = 15d + 0.25r + 2h
where:
d = number of days worked
r = number of raffle tickets sold
h = number of hours worked at the game
The first term, 15d, represents the base pay of $15 per day worked.
The second term, 0.25r, represents the additional earnings of $0.25 per raffle ticket sold.
The third term, 2h, represents the additional earnings of $2 per hour worked at the game.
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Rui is a professional deep water free diver. His altitude (in meters relative to sea level), x xx seconds after diving, is modeled by: d ( x ) = 1 2 x 2 − 10 x d(x)= 2 1 x 2 −10xd, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 2, end fraction, x, squared, minus, 10, x How many seconds after diving will Rui reach his lowest altitude?
Answer: look at the image for the answer and explanation
Step-by-step explanation:
consider the following: Point (3,5), and Line y-1=0. Write the slope intercept form of the equation of the line through the given point an parallel to the given line.
Step-by-step explanation:
please see the attached picture
A magician shuffles a standard deck of playing cards and allows an audience member to pull out a card, look at it, and replace it in the deck. Three additional people do the same. Find the probability that of the 4 cards drawn, at least 1 is a face card. (Round your answer to one decimal place.
The probability that at least 1 of the 4 cards drawn is a face card is 0.6.
How to find probability of 4 cards drawn?A typical deck contains 52 cards in total. There are 51 cards left in the deck after the first spectator draws a card. The replacement card leaves 52 cards in total, which is the same as before. The number of cards remains 52 for all four draws since each succeeding audience member will draw a card and replace it.
Finding the likelihood that none of the four cards are face cards and subtracting it from one will give us the chance that at least one of the four cards is a face card.
A deck of cards has 12 face cards (4 jacks, 4 queens, and 4 kings). Consequently, the likelihood that a single draw will not provide a face card is:
P(not a face card)=(52 - 12)/(52)=(40/(52)=(10/13).
The likelihood that none of the four draws will produce a face card is:
P(no face cards are present) = (10/13)4 = 0.432
As a result, the likelihood that at least 1 of the four cards revealed is a face card is as follows:
P(none are face cards) = 1 - P(at least one is a face card) = 1 - 0.432 = 0.568
The chance, rounded to one decimal place, is roughly 0.6.
Consequently, there is a 0.6 percent chance that at least one of the four cards will be a face card.
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