Answer:
32π²
Step-by-step explanation:
To solve this, we can use the washer method.
We can start by dividing the donut into horizontal sections. Each horizontal section is a hollow cylinder, with the inner cylinder's radius being the difference between x=4 and x²+y²=4, which can also be written as x = √(4-y²) (we do not need the negative x values as the inner cylinder does not encompass any part of the circle). This is (4-√(4-y²)) .
Similarly, the outer radius will be equal to the difference between 4 and the other side of the circle. This encompasses the circle and the extra area in between, so x will be negative here, making it (4-(-√(4-y²)) = 4+√(4-y²).
The area of the cylinder for a given Δy can be represented by
(4+√(4-y²))²πΔy - (4-√(4-y²))²πΔy. This space represents a cylinder in the donut created with the height being Δy. Simplifying the equation, we get
(4+√(4-y²))²πΔy - (4-√(4-y²))²πΔy = πΔy((4+√(4-y²))²- (4-√(4-y²))²)
(4+√(4-y²))²- (4-√(4-y²))² = 16+8√(4-y²) + 4-y² - (16-8√(4-y²)+4-y²)
= 16√(4-y²)
(4+√(4-y²))²πΔy - (4-√(4-y²))²πΔy = 16√(4-y²)πΔy
The donut ranges from all y values from -2 to 2, and to add all the areas of the cylinders up between these y values, we can make this an integral,
[tex]\int\limits^2_{-2} {16\pi \sqrt{4-y^2} \, dy\\[/tex]
perform u substitution, make y = 2sin(u). If y=2sin(u), then y/2 = sin(u) and arcsin(y/2) = u
[tex]\int\limits^2_{-2} {16\pi* 2cos(u) \sqrt{4-4sin^2(u)} \, du\\[/tex]
take the 32π out of the integral
[tex]32\pi\int\limits^2_{-2} {cos(u) \sqrt{4-4sin^2(u)} \, du\\\\= 32\pi\int\limits^2_{-2} {cos(u) *2cos(u)} \, du\\\\\\= 32\pi\int\limits^2_{-2} {2cos&^2(u)} \, du\\\\\\\\\\[/tex]
take the 2 out of the integral
[tex]64\pi\int\limits^2_{-2} {cos&^2(u)} \, du\\[/tex]
One reduction formula we can use here is that
[tex]\int\limits {cos^nx} \, dx = \frac{1}{n} cos^{n-1}(x) sin(x) +\frac{n-1}{n} \int\limits {cos^{n-2} (x)} \, dx[/tex]
Applying that here, we get
[tex]\int\limits {cos^2u} \, du = 0.5cos(u)sin(u) + 0.5 \int\limits {1} \, du\\= 0.5cos(arcsin(y/2))sin(arcsin(y/2)) + 0.5(arcsin(y/2))[/tex]
take that cos(arcsin(x)) = √(1-x²) and sin(arcsin(x)) = x and we get
[tex]\int\limits {cos^2u} \, du = 0.5\sqrt{1-y^2/4}*(y/2) + 0.5arcsin(y/2)\\= 0.25y\sqrt{1-y^2/4} + 0.5arcsin(y/2)\\[/tex]
multiply this back with the 64π to get
[tex]16\pi y\sqrt{1-y^2/4} + 32\pi arcsin(y/2)[/tex]
as our integral. Applying this to the bounds of [-2, 2], we get
16π(2)√(1-4/4) + 32πarcsin(1) - (16π(-2)√(1-4/4) + 32πarcsin(-1))
= 0 + 32π(π/2) - 32π(-π/2)
= 16π²+16π²
= 32π²
as our answer
Yoda Soda is the intergalactic party drink that will have all your friends saying, "Mmmmmm, good this is!"
You are throwing a party, and you need 555 liters of Yoda Soda for every 12 guests.
If you have 363636 guests, how many liters of Yoda Soda do you need?
Answer:
16818165
Step-by-step explanation:
12 guests equal 555 liters
total guests 363636
divide 363636 by 12
result is 30303
multiply 30303 times 555
total is 16818165
At the end of March, Jerry bought a plant that was $16.2 inches tall. During April, the plant grew 2] inches,
and during May, the plant grew another 2 inches. How tall was the plant at the beginning of June?
O 18.2 inches
O 19 inches
0 21 inches
21 inches
Answer:
20.2 inches
Step-by-step explanation:
The plant starts at 16.2 inches
The first month it grows 2 inches
16.2+2 =18.2 inches
Then the next month it grows 2 inches
18.2+2 = 20.2 inches
In the data set shown below, what is the value of the quartiles?
{4.3, 4.5, 4.7, 5, 5.5, 5.7, 5.9, 6, 6.1}
A. Q1 = 4.6; Q2 = 5.5; Q3 = 5.95
B. Q1 = 4.7; Q2 = 5.5; Q3 = 6
C. Q1 = 4.7; Q2 = 5.5; Q3 = 5.9
D. Q1 = 4.6; Q2 = 5.5; Q3 = 5.92
Answer:
A. Q1 = 4.6; Q2 = 5.5; Q3 = 5.95
Step-by-step explanation:
{4.3, 4.5, 4.7, 5, 5.5, 5.7, 5.9, 6, 6.1}
First find the median or the 2nd quartile
There are 9 data points so the middle is the 5th
4.3, 4.5, 4.7, 5, 5.5, 5.7, 5.9, 6, 6.1}
Q2 = 5.5
Now looking at the data on the left, we need to find the middle, which is Q1 or the first quartile
4.3, 4.5 , 4.7, 5,
It is between 4.5 and 4.7 so we average
(4.5+4.7)/2 = 9.2/2 = 4.6
Q1 is 4.6
We do the same for the data on the right, which is the third quartile or Q3
5.7, 5.9, 6, 6.1
(5.9+6)/2 = 11.9/2 = 5.95
Q3 = 5.95
Answer: IT'S A !!
Step-by-step explanation:
f=((-1,1),(1,-2),(3,-4)) g=((5,0),(-3,4),(1,1),(-4,1)) find (f/g)(1)
9514 1404 393
Answer:
-2
Step-by-step explanation:
(f/g)(1) = f(1)/g(1) = -2/1 = -2
__
The value of f(1) is the second number in the ordered pair (1, -2) that is part of the definition of function f. Similarly, for g, we look for the ordered pair that has 1 as its first value. The second value is g(1).
find the equation of straight line passes through a point (0 ,- 3 )which makes an angle tan^-1(1/3) with the line 3x- 2Y + 13 =0
Answer:
Step-by-step explanation:
Edit:
y = (7/9)x = 3
9514 1404 393
Answer:
y = 11/3x -3
Step-by-step explanation:
The slope of a line is the tangent of the angle it makes with the x-axis. The slope of the given line is 3/2, so the angle it makes with the x-axis is arctan(3/2) ≈ 56.310°. We want a line that makes an angle of arctan(1/3) ≈ 18.435° with the given line, so its slope will be ...
tan(56.310° +18.435°) = tan(74.745°) = 11/3
The y-intercept of the desired line is given as (0, -3), so the equation of the line we want is ...
y = 11/3x -3
_____
Additional comment
The desired slope can be found using the formula for the tangent of the sum of angles. However, simply adding the angles on a calculator saves a lot of arithmetic. (Full precision values must be used.)
The line we have found is at the desired angle measured CCW from the point of intersection with the given line. If you allow the line to have that angle measured CW from the point of intersection, then the slope will be 7/9, and the equation will be ...
y = 7/9x -3
how would I solve this and what is the answer.
Answer:
b is -6 or -10
Step-by-step explanation:
[tex] \frac{1}{b} = \frac{1}{b - 12} + \frac{ {b}^{2} + 16b + 48}{ {b}^{2} - 12b} \\ \\ \frac{1}{b} = \frac{b + ( {b}^{2} + 16b + 48)}{b(b - 12)} \\ \\ b - 12= b + ( {b}^{2} + 16b + 48) \\ {b}^{2} + 16b + 60= 0 \\ (b + 6)(b + 10) = 0 \\ b = - 6 \\ and \\ b = - 10[/tex]
The graph of the function f(x)= x2 − 4x + 6 is shown here. What is its axis of symmetry? A. x = 0 B. x = 2 C. x = 6 D. x = -2
9514 1404 393
Answer:
B. x = 2
Step-by-step explanation:
The graph is symmetrical about the vertical line through its vertex. The x-value of that vertex is 2, so the line of symmetry is ...
x = 2
round to the nearest Ten-thousand: 849,708
Answer:
850,000
Step-by-step explanation:
Answer: 850,000
Concept:
Here, we need to know the order and name of each place value.
Please refer to the attachment below for the specified names.
Solve:
8 = Hundred thousands
4 = Ten thousands
9 = One thousands
7 = Hundreds
0 = Tens
9 = Ones
Since the values before the ten thousands place, which would be the one thousands place, is greater than 5, then we should round up.
Therefore, the rounded value would be [tex]\boxed{850,000}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
2(4×+2)=10
[tex]6 \times + 4 = 10 \\ \\ 6 \times = 10 - 4 \\ \\ 6 \times = 6 \\ \\ [/tex]
that is the answer
Answer:
x = 3/4
Step-by-step explanation:
2(4x + 2) = 10 Remove the brackets
8x + 4 = 10 Subtract 4 from both sides
8x = 6 Divide by 8
x = 6/8
x = 3/4
Check
2(4*3/4 + 2) =?10
2( 3 + 2) = 10
2*5 = 10
10 = 10
Look at the image to receive the answers
Answer:
345.4 using the 3.14 for pi
345.6 using the pi button
Step-by-step explanation:
The surface area of a cylinder is given by
SA=2 pi rh+2 pir^2
The radius is 5 and the height is 6
SA = 2·pi·5·6+2·pi·5^2
SA = 60pi + 50pi
110pi
Letting pi = 3.14
SA =345.4
Letting pi = pi button
= 345.57519 = 345.6
The measures of the exterior angles of a pentagon are in a ratio of 1:23:4:5. The measure of the largest exterior angle is
Select one:
a. 120
b. 48
c. 24
d. 72
Answer:
A) 120
Step-by-step explanation:
1:2:3:4:5
SUm of exterior angles of ANY polygon is equal to 360
x+2x+3x+4x+5x=360
15x=360
x=24
the largest angle is 5x, 5(24)=120 degrees
I need a Long equation for 30
if x = -5
x2 + 3
x - 2
Answer:
Step-by-step explanation:
Which of the following illustrates the truth value of the given statements? A triangle has four sides, and a rectangle has three sides. T F → F F T → F T T → T F F → F
Answer:
F F → F
Step-by-step explanation:
The statement is given as:
A triangle has four sides, and a rectangle has three sides.
It is again a false statement since a triangle is a polygon with three sides and a rectangle is a polygon with four sides.
Hence, the answer is:
F F → F
write the expression as a decimal , 6 x 1 + 9 x 1/10 + 8 x 1/100 + 6 x 1/1000 =__
Answer:
6.986.
Step-by-step explanation:
6 x 1 + 9 x 1/10 + 8 x 1/100 + 6 x 1/1000
We do the multiplications first ( according to PEMDAS):-
= 6 + 9 * 0.1 + 8 * 0.01 + 6 * 0.001
= 6 + 0.9 + 0.08 + 0006
= 6.9 + 0.086
= 6 986.
The value of the equation in the decimal form is A = 6.986
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
A = 6 x 1 + 9 x 1/10 + 8 x 1/100 + 6 x 1/1000
On simplifying the equation , we get
The value of 6 x 1 = 6
The value of 9 x 1/10 = 0.9
The value of 9 x 1/100 = 0.08
The value of 6 x 1/1000 = 0.006
So , substituting the values in the equation A , we get
A = 6 + 0.9 + 0.08 + 0.006
On simplifying the equation , we get
A = 6.986
Therefore , the value of A is 6.986
Hence , the value of the equation is 6.986
To learn more about equations click :
https://brainly.com/question/19297665
#SPJ2
It need the answer to these please
hopefully this answer can help you to answer the next question
You are playing a version of the roulette game, wherethe pockets arefrom 0 to 12 and even numbers are red and odd numbersare black (0 isgreen). You spin 3 times and add up the values yousee. What is theprobability that you get a total of 17 given on thefirst spin youspin a 2
Using the concept of probability, it is found that there is a 0.0592 = 5.92% probability that you get a total of 17 given on the first spin you spin a 2.
-------------------------------------
A probability is the number of desired outcomes divided by the number of total outcomes.Since you spin a 2 on the first trial, the probability desired is equivalent to the probability of finding a sum of 15 in the next two trials.-------------------------------------
In each trial, there are 13 possible outcomes(pockets numbered from 0 to 12), so the total number of outcomes is:
[tex]T = 13^2 = 169[/tex]
-------------------------------------
The desired outcomes are those who result in a sum of 15. Thus, they are:
(3,12), (4,11), (5,10), (6,9), (7,8), (8,7), (9,7), (10,5), (11,4), (12,3).That is, 10 desired outcomes, so [tex]D = 10[/tex].-------------------------------------
Thus, the probability is:
[tex]p = \frac{D}{T} = \frac{10}{169} = 0.0592[/tex]
0.0592 = 5.92% probability that you get a total of 17 given on the first spin you spin a 2.
A similar problem is given at https://brainly.com/question/21444373
is the multiplicative identity for rational number
Answer:
The rational number 1 is the multiplicative identity for rational numbers.
Step-by-step explanation:
I hope this will help you buddy
Suppose the linear regression line y = 4.009x - 77.531 predicts a pizza
parlor's profits based on the number of pizzas sold. If x represents the
number of pizzas sold, and y represents the pizza parlor's profits in dollars,
about how much can the pizza parlor expect in profits if it sells 325 pizzas?
Answer:
Option C, 1225
Step-by-step explanation:
y=4.009x-77.531
selling 325 pizzas, so x=325, now putting the value of x in the equation,
y=4.009×325-77.531
or, y=1225.394
y = 1225.394, which is the profit, if we approximate it to the nearest value, it's $1225
(5, 4)
11-5-2)
16.nat's the slope-intercept form of the equation of the line graphed in this figure?
O A. y = 5/3x + 1
O B.y=-3x + 1
O C. y = 3x + 1
O D.y = -5/3X - 1
Answer:
The answer is B. y=3/5x+1. Tachycardia means ...................heart rate a) fast b) slow c) irregular d) arrhythmic
Hi,
Answer:
Tachycardia is a fast heart rate
Find an equation of the plane orthogonal to the line
(x,y,z)=(0,9,6)+t(7,−7,−6)
which passes through the point (9, 6, 0).
Give your answer in the form ax+by+cz=d (with a=7).
The given line is orthogonal to the plane you want to find, so the tangent vector of this line can be used as the normal vector for the plane.
The tangent vector for the line is
d/dt (⟨0, 9, 6⟩ + ⟨7, -7, -6⟩t ) = ⟨7, -7, -6⟩
Then the plane that passes through the origin with this as its normal vector has equation
⟨x, y, z⟩ • ⟨7, -7, -6⟩ = 0
We want the plane to pass through the point (9, 6, 0), so we just translate every vector pointing to the plane itself by adding ⟨9, 6, 0⟩,
(⟨x, y, z⟩ - ⟨9, 6, 0⟩) • ⟨7, -7, -6⟩ = 0
Simplifying this expression and writing it standard form gives
⟨x - 9, y - 6, z⟩ • ⟨7, -7, -6⟩ = 0
7 (x - 9) - 7 (y - 6) - 6z = 0
7x - 63 - 7y + 42 - 6z = 0
7x - 7y - 6z = 21
so that
a = 7, b = -7, c = -6, and d = 21
An equation of the plane orthogonal to the line 7x - 7y - 6z = 21.
The given line is orthogonal to the plane you want to find,
So the tangent vector of this line can be used as
The normal vector for the plane.
The tangent vector for the line is,
What is the tangent vector?A tangent vector is a vector that is tangent to a curve or surface at a given point.
d/dt (⟨0, 9, 6⟩ + ⟨7, -7, -6⟩t ) = ⟨7, -7, -6⟩
Then the plane that passes through the origin with this as its normal vector has the equation
⟨x, y, z⟩ • ⟨7, -7, -6⟩ = 0
We want the plane to pass through the point (9, 6, 0), so we just
translate every vector pointing to the plane itself by adding ⟨9, 6, 0⟩,
(⟨x, y, z⟩ - ⟨9, 6, 0⟩) • ⟨7, -7, -6⟩ = 0
Simplifying this expression and writing it in standard form gives
⟨x - 9, y - 6, z⟩ • ⟨7, -7, -6⟩ = 0
7 (x - 9) - 7 (y - 6) - 6z = 0
7x - 63 - 7y + 42 - 6z = 0
7x - 7y - 6z = 21
So that, a = 7, b = -7, c = -6, and d = 21.
To learn more about the equation of plane visit:
https://brainly.com/question/1603217
What is the US TREASURY BONDS pretax expected return?
Answer:
this is the return on an investment that does not include the taxes the investor must pay on this return
Which of the following functions has order 2 rotational symmetry about the same origin?
The answer is "Option B", and the further explanation can be defined as follows:
A rectangle is symmetrical in 2 ways. In particular, it has two-order rotational symmetry (RS2).If an object rotates 360 degrees, it's an ordering of symmetrical is the number of times it appears to be the same.There are three levels of matching: Order 2 if only two times, Order 3 if three matches are made, and so forth.The wrong choice can be defined as follows:
In option a and c, both are wrong because it has no rotational symmetry.In option d, it is wrong because it downs the diagram on the negative (x,y) axis.Therefore, "Option B" is correct and its diagram is defined in the attached file.
Learn more:
2 rotational symmetry: brainly.com/question/1531736
A house was bought two years ago at the price of $130,000. Each year, the
house's value decreased by 2%. What's the house's value this year?
Answer:
124800
Step-by-step explanation13
130,000 * 2% = 2,600
2,600*2= 5,200
130,000-5,200= 124800
Solve the equation for y. Identify the slope and y-intercept then graph the equation.
Y=-3x+1
Y=
M=
B=
Please Include a picture of the graph and show your work if you can
Which is the equation of the line with slope o passing through the point (-3,-1)?
9514 1404 393
Answer:
y = -1
Step-by-step explanation:
We assume you want the line with a slope of zero. That is a horizontal line, so y is a constant. In order to make the line go through the point with y=-1, the equation of the line is ...
y = -1
At what points do these graphs intersect
y=x²-x-6
y=x+2
Answer:
Solve algebraicallyEliminate y:
x² - x - 6 = x + 2x² - 2x = 8x² - 2x + 1 = 9(x - 1)² = 3²1) x - 1 = 3 ⇒ x = 4 ⇒ y = 62) x - 1 = -3 ⇒ x = -2 ⇒ y = 0The solution is:
(-2, 0), (4, 6)Solve graphicallySee the attached
Answer:
[tex]thank \: you[/tex]
The product of two rational numbers is 47/42 and one of them is -11/21, find the other number
answer:
another number is -47/22
explanation:
let one number be x and the other be y
-11/21 × y = 47/42
y = -47/22
Answer:
- [tex]\frac{47}{22}[/tex]
Step-by-step explanation:
let n be the other number , then
- [tex]\frac{11}{21}[/tex] × n = [tex]\frac{47}{42}[/tex] ( divide both sides by - [tex]\frac{11}{21}[/tex] )
n = [tex]\frac{\frac{47}{42} }{-\frac{11}{21} }[/tex]
= [tex]\frac{47}{42}[/tex] × - [tex]\frac{21}{11}[/tex] ( cancel 21 and 42 )
= [tex]\frac{47}{2}[/tex] × - [tex]\frac{1}{11}[/tex]
= - [tex]\frac{47}{22}[/tex]
PLEASE HELP
select all the statistical questions.
A what is the typical amount of rainfall for the month of June in the galapagus islands
B how much did it rain yesterday at the Mexico City International airport
C why do you like to listen to music
D how many songs does the class usually listen to each day
E how many songs did you listen to today
F what is the capital of Canada
G how long does it typically take for 2nd graders to walk a lap around the track
The selected statistical questions are as follows:
A what is the typical amount of rainfall for the month of June in the Galapagos islands
D how many songs does the class usually listen to each day
G how long does it typically take for 2nd graders to walk a lap around the track
Statistical questions are known by these two main characteristics:
they are answered by the collection of datathere is always variability in the data collectedThis means that the data vary. They are not the same data. For example, "How old is Monday?" is not a statistical question because it has one answer.
Thus, statistical questions are not answered by a single number or answer, especially when the correct answer is just one.
Learn more examples of statistical questions here: https://brainly.com/question/15729334
