Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 8 − 8x2, y = 0
Answers
Answer: V = \(\frac{1024}{15}\pi\)
Step-by-step explanation: The graph for function \(y=8-8x^{2}\) shows the maximum height is 8and x-axis limits are -1 and 1, which will also be the limits of the integration.
For a volume of a solid created by revolution of a single function, it is appropriate to use the Disk Method, i.e.:
\(V=\pi\int\limits^a_b {[f(x)]^{2}} \, dx\)
where
f(x) is radius
dx is representative of height.
For the function above, volume is:
\(V=\pi\int\limits^a_b {[(8-8x^{2})]^{2}} \, dx\)
\(V=\pi\int\limits^a_b {8^{2}(1-x^{2})^{2}} \, dx\)
\(V=64\pi\int\limits^a_b {(1-2x^{2}+x^{4})} \, dx\)
\(V=64.\pi[x-\frac{2}{3}x^{3}+\frac{x^{5}}{5} ]\)
Using limits -1 to 1:
\(V=64\pi(\frac{16}{15} )\)
\(V=\frac{1024}{15}\pi\)
Volume of the rotated solid created by \(y=8-8x^{2}\) about the x-axis is \(\frac{1024}{15}\pi\)
cubic units.
You can use the fact that when the given curve is revolved around x axis, each height y will contribute as radius at that point x and there will be a circle. Multiply that area of circle with dx to get infinitesimal volume and integrate it.
The volume of the solid obtained by revolving the given curve and bounded by y = 0 (x axis) is given by:
\(V = \dfrac{1024\pi}{15} \: \rm unit^3\)
How to get the volume for solid of revolution of given curve?Since the solid will be bounded by y = 0, thus, the curve will be used from x = -1 to x = +1 (since outside that limit, the curve goes down the y =0(x axis) line.(see the graph attached).
Since at each input x in [-1,1], we have the output as radius of a vertical circle standing there with center at (x,0), thus, integrating the area times dx from x = -1 to x = +1, we get:
\(V = \int_{-1}^{+1} \pi y^2 dx = \pi\int_{-1}^{1} (8-8x^2)^2dx\\\\V = 64\pi\int_{-1}^{1} (1-x^2)^2dx = 84\pi\int_{-1}^{1} (1+ x^4 - 2x^2)dx \\\\V = 64\pi[x+ x^5/5 - 2x^3/3]_{x=-1}^{x=1} \\\\V = 128\pi \times \dfrac{8}{15} = \dfrac{1024\pi}{15} \: \rm unit^3\)
Thus,
The volume of the solid obtained by revolving the given curve and bounded by y = 0 (x axis) is given by:
\(V = \dfrac{1024\pi}{15} \: \rm unit^3\)
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Related Questions
I need help with this question please.
Answers
Answer:
It's B. 2/5
Step-by-step explanation:
Because: line G to E is 2/5 of G to F.
4x+7 (3x-3)<-9
Please show steps
Answers
Answer: 3x-3 needs to be done first, and then 4x+7, and then multiply and whatever you get will be your answer
Step-by-step explanation:
Consider the function below, which has a relative minimum located at (-3, -18) and a relative maximum located at 1/3, 14/27). f(x) = -x3 - 4x2 + 3x. Select all ordered pairs in the table which are located where the graph of f(x) is decreasing: Ordered pairs: (-1, -6), (2, -18), (0, 0),(1 , -2), (-3 , -18), (-4. , -12)
Answers
The ordered pairs (-1, -6), (2, -18), (0, 0), and (-4, -12) do not correspond to the intervals where the graph of f(x) is decreasing. The pairs (1, -2) and (-3, -18) are the correct ones.
To determine where the graph of f(x) is decreasing, we need to examine the intervals where the function's derivative is negative. The derivative of f(x) is given by f'(x) = -3x^2 - 8x + 3.
Now, let's evaluate f'(x) for each of the given x-values:
f'(-1) = -3(-1)^2 - 8(-1) + 3 = -3 + 8 + 3 = 8
f'(2) = -3(2)^2 - 8(2) + 3 = -12 - 16 + 3 = -25
f'(0) = -3(0)^2 - 8(0) + 3 = 3
f'(1) = -3(1)^2 - 8(1) + 3 = -3 - 8 + 3 = -8
f'(-3) = -3(-3)^2 - 8(-3) + 3 = -27 + 24 + 3 = 0
f'(-4) = -3(-4)^2 - 8(-4) + 3 = -48 + 32 + 3 = -13
From the values above, we can determine the intervals where f(x) is decreasing:
f(x) is decreasing for x in the interval (-∞, -3).
f(x) is decreasing for x in the interval (1, 2).
Now let's check the ordered pairs in the table:
(-1, -6): Not in a decreasing interval.
(2, -18): Not in a decreasing interval.
(0, 0): Not in a decreasing interval.
(1, -2): In a decreasing interval.
(-3, -18): In a decreasing interval.
(-4, -12): Not in a decreasing interval.
Therefore, the ordered pairs (-1, -6), (2, -18), (0, 0), and (-4, -12) are not located in the intervals where the graph of f(x) is decreasing. The correct answer is: (1, -2), (-3, -18).
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Note the complete and the correct question is
Q- Consider the function below, which has a relative minimum located at (-3, -18) and a relative maximum located at 1/3, 14/27).
\(f(x) = -x^3 - 4x^2 + 3x\).
Select all ordered pairs in the table which are located where the graph of f(x) is decreasing: Ordered pairs: (-1, -6), (2, -18), (0, 0),(1 , -2), (-3 , -18), (-4. , -12)
3353 8. A survey showed that 1/3 of the students in the class liked blue 1/4 of the remainder liked red. The remaining 24 students liked yellow. How many students were in the class?
Answers
Answer: there are 48 students in the class
Step-by-step explanation:
let x = the total number of students in the class
1 - 1/3 = 2/3
1/4 * 2/3 = 2/12 = 1/6
1/3x + 1/6x + 24 = x
2/6x + 1/6x + 24 = x
3/6x + 24 = x
1/2x + 24 = x
24 = x - 1/2x
24 = 1/2x
24 * 2 = x
x = 48
there are 48 students in the class
A box contains 5 red marbles, 6 white marbles, and 8 blue. If a Marble is randomly selected from the box, what is the likely hood that it is white? Express it in a fraction.
Answers
Likelihood of white: 6/19
HELP ASAP Which set of statements explains how to plot a point at the location (Negative 3 and one-half, negative 2)?
Start at the origin. Move 3 and one-half units right because the x-coordinate is Negative 3 and one-half. Negative 3 and one-half is between 3 and 4. Move 2 units down because the y-coordinate is -2.
Start at the origin. Move 3 and one-half units down because the x-coordinate is Negative 3 and one-half. Negative 3 and one-half is between -3 and -4. Move 2 units left because the y-coordinate is -2.
Start at the origin. Move 3 and one-half units down because the x-coordinate is Negative 3 and one-half. Negative 3 and one-half is between -3 and -4. Move 2 units right because the y-coordinate is -2.
Start at the origin. Move 3 and one-half units left because the x-coordinate is Negative 3 and one-half. Negative 3 and one-half is between -3 and -4. Move 2 units down because the y-coordinate is -2.
Answers
Answer:
The last one: Start at the origin. Move 3 and one-half units left because the x-coordinate is Negative 3 and one-half. Negative 3 and one-half is between -3 and -4. Move 2 units down because the y-coordinate is -2.
Step-by-step explanation:
To get to (-3 1/2, -2), you would have to go left 3 1/2 units, and down 2 units.
Triangle ABC has vertices at A(−3, 3), B(0, 7), and C(−3, 0). Determine the coordinates of the vertices for the image if the preimage is translated 3 units up.
A′(−3, 0), B′(0, 4), C′(−3, −3)
A′(−3, 6), B′(0, 10), C′(−3, 3)
A′(−6, 3), B′(−3, 7), C′(0, 0)
A′(0, 3), B′(3, 5), C′(0, 0)
Answers
Answer:
To translate triangle ABC 3 units up, we need to add 3 to the y-coordinate of each vertex:
A' = (-3, 3 + 3) = (-3, 6)
B' = (0, 7 + 3) = (0, 10)
C' = (-3, 0 + 3) = (-3, 3)
Therefore, the coordinates of the vertices for the image triangle A'B'C' are A'(-3, 6), B'(0, 10), and C'(-3, 3).
So the correct answer is: A′(−3, 6), B′(0, 10), C′(−3, 3).
The width of a rectangle is 9 less than twice its length. If the area of the rectangle is 96
cm ^2 , what is the length of the diagonal?
Answers
Thee diagonal of the rectangle is 13.86 cm long.
What is the length of the diagonal of rectangle?We are given that;
Width of a rectangle is 9 less than twice its length. Thus, if length is L, then;
W = 2L - 9
Area formula for a rectangle is;
A = Length * Width
We are given area of rectangle = 96 cm²
Thus;
L(2L - 9) = 96
2L² - 9L = 96
2L² - 9L - 96 = 0
Using online quadratic equation calculator gives;
L = 9.53 cm
Thus;
W = 2(9.53) - 9
W = 19.06 - 9
W = 10.06 cm
The diagonal of the triangle will be gotten from Pythagoras theorem;
D = √(9.53² + 10.06²)
D = √192.0245
D = 13.86 cm
Thus, we conclude that the diagonal of the rectangle is 13.86 cm long.
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why do the hands on the clock form an angle?
Answers
Answer:
The entire clock measures 360 degrees. As the clock is divided into 12 sections. The distance between each number is equivalent to 30 degrees (360/12)
I hope this helps you!
√(4/7)^2 =
equations with square roots and cube roots
Answers
Answer:
See below
Step-by-step explanation:
If you are squaring a number, and then taking the square root of it, you are essentially undoing the original operation:
\(\displaystyle \sqrt{\biggr(\frac{4}{7}\biggr)^2}=\biggr[\biggr(\frac{4}{7}\biggr)^2\biggr]^{\frac{1}{2}}=\biggr(\frac{4}{7}\biggr)^{2*\frac{1}{2}}=\frac{4}{7}\)
Hence, we are back starting with the original number
Decimal: 0.571 (Not Rounded)
Two more than three times a number
Answers
Answer:
2 + 3x
Step-by-step explanation:
Let x equal the number
2 + 3x
Answer:
2+3x
Step-by-step explanation:
x= the number
Express the given expanded numeral as a Hindu-Arabic numeral.
(1x 10-1) + (4x10-2) + (9x10-3) + (7x10-4)
(1x 10-1) + (4x10-2) + (9x10 -3) + (7x10-4)=Type an integer or a decimal.)
Answers
Answer:
0.1497
Step-by-step explanation:
1×.1 +4×.01 +9×.001 +7×.0001 =
.1 +.04 +.009 +.0007 = 0.1497
The positive variables p and c change with respect to time 1. The relationship between p and c is given by the equation p^2 = (20-c)^3. At the instant when dp/dt = 41 and c = 15, what is the value of dc/dtï¼
Answers
The value of dc/dt due to changes in p and c variables is 0.364.
The positive variables p and c change with respect to time 1. The relationship between p and c is given by the equation p^2 = (20-c)^3. Now the dp/dt value is 41 and c is 15.
Differentiate the given equation to obtain the following:
2p dp/dt = 3(20-c)^2 dc/dt
Substitute the given values in the equation to obtain the following:
2p (41) = 3(20 - 15)^2 dc/dt
Simplify the equation to obtain the following:
82 = 225 dc/dt
Divide both sides of the equation by 225 to obtain the following:
dc/dt = 82/225
Simplify the equation to obtain the following:
dc/dt = 0.364
Hence, the value of dc/dt is 0.364.
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A teacher asks her students to find an expression for the number of tiles needed to
surround such a square pool, and sees the following responses from her students:
4(s+1)
s²
4s+4
2s+2(s+2)
4s
Is each mathematical model correct or incorrect? How do you know?
Answers
The correct mathematical expression to find the number of tiles needed to surround a square pool would depend on how the tiles are arranged and how the dimensions of the pool and the tiles are related.
4(s+1): This expression appears to be correct if we assume that each side of the square pool is surrounded by a row of tiles, and each corner requires an additional tile.
So, the expression can be simplified to 4s+4. This is a valid expression for the number of tiles needed to surround the pool.
s²: This expression does not appear to be correct, as it only gives the area of the square pool, and does not take into account the dimensions of the tiles or the need to surround the pool.
4s+4: This expression is the same as the first expression, 4(s+1), and is a valid expression for the number of tiles needed to surround the pool.
2s+2(s+2): This expression appears to be incorrect, as it gives the total perimeter of the pool plus an additional 4 units, but does not take into account the size of the tiles or the need to surround the pool.
4s: This expression is incorrect, as it only gives the perimeter of the pool and does not take into account the need to surround the pool with tiles.
Thus, two of the given expressions (4(s+1) and 4s+4) are correct, one expression (s²) is incomplete, and two expressions (2s+2(s+2) and 4s) are incorrect.
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The domain of this emath equation is
Answers
Answer:
x ≥ 5
Step-by-step explanation:
You want the domain of the equation y = √(x -5) -1.
DomainThe domain of the function is the set of x-values for which it is defined. The square root is not defined for negative values, so the domain is ...
x -5 ≥ 0
x ≥ 5 . . . . . . the domain of this function
__
Additional comment
The range is the set of y-values the function may produce. Here, that set of values is y ≥ -1.
<95141404393>
Solve for x: 2(4-x)-3(x+3)=-11
Answers
Answer:
x=2
Step-by-step explanation:
I don’t really have an explanation, it was all mental math
Answer:
\(\sf x=2\)Step-by-step explanation:
\(\sf 2(4-x)-3(x+3)=-11\)
Expand:-
\(\sf 2\left(4-x\right)-3\left(x+3\right)\)\(\sf 8-2x-3\left(x+3\right)\)\(\sf 8-2x-3x-9\)\(\sf -5x-1\)\(\sf -5x-1=-11\)Now, add 1 to both sides:-
\(\sf -5x-1+1=-11+1\)\(\sf -5x=-10\)Divide both sides by -5:-
\(\sf \cfrac{-5x}{-5}=\cfrac{-10}{-5}\)\(\sf x=2\)Therefore, the value of x is 2!
- - - - - - - - - - - - - - - - - - - - - -
Hope this helps!
3.
Name the image
Line
Line segment
Ray angle
Answers
Answer:
ray
Step-by-step explanation:
VocabularyDifference of two squares
Answers
The difference of two squares refers to the next expression:
\(a^2-b^2=(a+b)(a-b)\)Helppppppp I’ll give brainlyyyyyyyyyyyyy
Answers
The values of a, h and k for the function include the following:
a = 2
h = 3
k = -2
How to write an equation for the transformed logarithm?In Mathematics and Geometry, the general form of a logarithm function is represented by the following mathematical equation:
f(x) = alog(±x + h) + k
Where:
a represent the scale factor of the logarithmic graphh represent a horizontal translation.k represent a vertical translation.x represent the base.Based on the given logarithm function f(x) = 2log₄(x + 3) - 2, we can reasonably infer and logically deduce that the scale factor (a) used is equal to 2.
This ultimately implies that, the values of a, h and k for the function are 2, 3, and -2 respectively.
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what the word form of 500.2
Answers
Answer:
How to Write Out USD 500.2 Dollars in Words: five hundred dollars and twenty cents.
I count on the best
Chad has a rope that is 15 yards long. How many pieces of rope measuring 5/7 of a yard can he divide his rope into?
A. 28
B. 21
C. 35
D. 14
Answers
Answer:
Chad can divide his rope in 21 pieces. (Answer B)
Step-by-step explanation:
To solve we divide.
15 divided by 5/7 is what we need to know.
When dividing fractions it is the same thing as multiplying by the reciprocal (flipped version of the fraction).
\(15\) ÷ \(\frac{5}{7}\) = 15 × \(\frac{7}{5}\)
\(\frac{15}{1}\) × \(\frac{7}{5}\) = \(\frac{105}{5}\)
\(\frac{105}{5}\) = 21
Chad can divide his rope in 21 pieces.
Answer:
21 pieces
Step-by-step explanation:
Divide (5/7 yd / piece) into 15 yds:
15 yds
----------------------- = 21 pieces
(5/7 yd / piece)
16.24% as a simplest fraction
Answers
Answer:
203/1250
Step-by-step explanation:
16.24% is equivalent to 0.1624
I used a calculator to convert.
If x and y vary inversely, and x = 8 when y = 1/4, what is y when x = -4?
Answers
\(y \propto \dfrac 1x\\\\\implies y = \dfrac kx \\\\\implies \dfrac 14 = \dfrac k{8}\\\\\implies k=2\\\\\text{When}~ x =-4\\\\y = \dfrac{k}{x} = \dfrac{2}{-4}= -\dfrac 12\)
Which statement about the zeros of the graphed function is true?
A polynomial function passes through (0.8, 4), (2, minus 4), (4.6, 0.5), (6, 0), and (7.5, 8) also intercepts the x-axis at 1, 4 and 6 units.
A.
The function has three distinct real zeros.
B.
The function has two distinct real zeros and two complex zeros.
C.
The function has four distinct real zeros.
D.
The function has one distinct real zero and two complex zeros.
Answers
The function has three distinct real zeros.
The correct answer is A.
To determine the statement about the zeros of the graphed function, let's analyze the given information.
We have the following points on the graph:
(0.8, 4), (2, -4), (4.6, 0.5), (6, 0), and (7.5, 8)
Additionally, the function intercepts the x-axis at 1, 4, and 6 units.
To find the zeros of the function, we need to identify the x-values where the function intersects the x-axis.
From the given information, we know that the function intersects the x-axis at 1, 4, and 6 units.
These three x-values correspond to three distinct real zeros of the function.
The correct statement about the zeros of the graphed function is:
The function has three distinct real zeros.
The correct answer is A.
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Question 1-2
What is the value of a³ + b (6 + c), when a = 2, b = 3, and c = 4?
Answers
Answer:
\(\huge\boxed{\sf 38}\)
Step-by-step explanation:
Given expression:= a³ + b (6 + c)
Put a = 2, b = 3 and c = 4
= (2)³ + 3 (6 + 4)
= 8 + 3(10)
= 8 + 30
= 38\(\rule[225]{225}{2}\)
Answer:
Step-by-step explanation:
the requied answer is 38.
according to the question the value of a=2,b=3,c=4.
here,
to find the value of a³ + b (6 + c)we have to do it in steps:
step 1: solve the bracket (6+4) =10.
step 2: solve the value of a³ =8.
now put these values ,
=8+3(10)
=38.
solve for x: 5x-4y=24
Answers
Answer:
x=4y+24/5
Step-by-step explanation:
add 4y
Divide by 5
Kindly solve the following with the cirrect method .SOLVE ALL . I'll give brainliest + thanks + follow
Answers
The following percentages are listed below:
12.5 %40 %6.25 %6.667 %41.667 %75 %How to use percentages in real life situationsIn this question we have seven cases of real life situations in which percentages are used. Mathematically speaking, percentages are represented by the following expression:
x = r / r' × 100 (1)
Where:
r - Real quantityr - Maximum quantityNow we proceed to determine quantities related to percentages:
2.8 mm as a per cent of 2.24 cm
x = (2.8 mm / 22.4 mm) × 100 %
x = 12.5 %
What per cent of 1.5 m is 60 cm?
x = (60 cm / 150 cm) × 100 %
x = 40 %
What per cent of 2 kg is 125 g?
x = (125 g / 2000 g) × 100
x = 6.25 %
What per cent of R 6 to 40 p?
x = (40 / 600) × 100
x = 6.667 %
What per cent of a day is 10 h?
x = (10 h / 24 h) × 100
x = 41.667 %
What per cent of 7 1 / 3 m in 5 1 / 2 m ?
x = [(11 / 2) / (22 / 3)] × 100 %
x = 75 %
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Find the measure of AB.
20,
E
61%
D
B
A
21. B
65°
D
C
22. A
B
E
D
91
Answers
To find the measure of angle AB, we can use the fact that the sum of the angles in a triangle is 180 degrees.
Given:
Angle A = 65 degrees
Angle B = 91 degrees
We can subtract the sum of these two angles from 180 degrees to find angle AB:
Angle AB = 180 - (Angle A + Angle B)
Angle AB = 180 - (65 + 91)
Angle AB = 180 - 156
Angle AB = 24 degrees
Therefore, the measure of angle AB is 24 degrees.
Nao and Arban drive to work.
Nao drives 95 miles in 2.5 hours.
Arban drives 128 km in 1 hour 15 min.
Work out the difference between their average speeds in km/h.
1 mile = 1.6 km
Thank You.
Answers
Answer:
41.6 km/h
Step-by-step explanation:
Nao drives 95mi/2.5hr or 38 miles per hour, or 60.8 km/h
1 hr 15 min is the same as 1.25 hours
Arban drives 128km/1.25hr or 102.4 km/h
The difference is 102.4-60.8 = 41.6
Bo kept track of his cookie sales. He
discovered that 3 out of 5 people who
came into his bakery for cookies bought
chocolate chip cookies.
If 400 customers bought cookies at Bo's
Bakery last month, how many of them
bought chocolate chip cookies?
Answers
Answer:240
Step-by-step explanation:
400/5 = 80
80 x 3 = 240
240/400 people bought choc chip