To calculate the total volume of the solid we will divide it into n shell cyline thickness of [tex]$\Delta y$.[/tex]
The internal shell cylinder circumference C shall be[tex]$2 \pi r_i$[/tex]
where r is the in The shell cylinder internal area A shall be C.
where h is the height
The volume of the shell cylinder[tex]$\Delta v$[/tex] shall be[tex]$A \cdot \Delta y$.[/tex]
Then we will have:
[tex]\Delta v=A \cdot \Delta y=C \cdot h \cdot \Delta y=2 \pi r_i h \cdot \Delta y \Rightarrow d v=2 \pi r_i h \cdot d y$$[/tex]
The total volume of the solid shall be the summation of all n shell cylinders
[tex]\int d v=\int 2 \pi r_i h d y \Rightarrow V=\int_a^b 2 \pi r_i h d y \Rightarrow V=2 \pi \int_a^b r_i h d y$$[/tex]
[tex]\int d v=\int 2 \pi r_i h d y \Rightarrow V=\int_a^b 2 \pi r_i h d y \Rightarrow V=2 \pi \int_a^b r_i h d y[/tex]
[tex]The value $a$ is the value of $y$ at the point where $x=0$ then:For $x=0 \Rightarrow y=0^3 \Rightarrow y=0$ then $a=0$.The value $b$ is the value of $y$ at the point where $x=1$ then:[/tex]
[tex]For $x=1 \Rightarrow y=1^3 \Rightarrow y=1$ then $b=1$.Since the integration will be with variable $y$ we have to express the function $y=x^3$ in terms of $y$ :$$[/tex]
y=x^3 \[tex]Rightarrow x=\sqrt[3]{y}[/tex]
Substituting in V, we will have:
[tex]V= & 2 \pi \int_a^b r_i h d y=2 \pi \int_0^1(1-y)(1-\sqrt[3]{y}) d y=2 \pi[/tex]\[tex]int_0^1(1-y)\left(1-y^{\frac{1}{3}}\right) d y \\[/tex]
&[tex]1-y \\& \frac{1-y^{\frac{1}{3}}}{1-y} \\& \frac{-y^{\frac{1}{3}}+y^{\frac{1}{3}} \cdot y^{\frac{1}{1}}}{} \\[/tex]
[tex]& 1-y-y^{\frac{1}{3}}+y^{\frac{1}{3}+\frac{1}{1}}= \\& =1-y-y^{\frac{1}{3}}+y^{\frac{1+3}{3}}= \\[/tex]
[tex]& =1-y-y^{\frac{1}{3}}+y^{\frac{4}{3}}= \\& =y^{\frac{4}{3}}-y^{\frac{1}{3}}-y+1[/tex]
Substituting in V:
[tex]V=2 \pi \int_0^1\left(y^{\frac{4}{3}}-y^{\frac{1}{3}}-y+1\right) d y=2[/tex][tex]{1}{3}+1}}{\frac{1}{3}+1}-\frac{y^{1+1}}[/tex]{[tex]1+1}+y\right]_0^1=2[/tex] [tex]\pi\left[\frac{y^{\frac{4+3}{3}}}{\frac{4+3}{3}}[/tex]-[tex]\frac{y^{\frac{1+3}{3}}}{\frac{1+3}{3}}-\frac{y^2}{2}+y\right]_0^1$$[/tex]
Calculating the definite integral:
By definition[tex], $\int_a^b f(x) d x=\left.F(x)\right|_a ^b=F(b)-F(a)$[/tex] then:
[tex]& V=\frac{\pi}{14}\left[\left(12 \cdot 1^{\frac{7}{3}}-21 \cdot 1^{\frac{4}{3}}-14[/tex][tex]\cdot 1^2+28 \cdot 1\right)-\left(12 \cdot 0^{\frac{7}{3}}-21 \cdot[/tex] [tex]0^{\frac{4}{3}}-14 \cdot 0^2+28 \cdot 0\right)\right] \\[/tex]
& V=[tex]\frac{\pi}{14}[(12 \cdot 1-21 \cdot 1-14 \cdot 1+28 \cdot 1)-\underbrace{(12 \cdot 0-21 \cdot 0-14 \cdot 0+28 \cdot 0)}_0][/tex]
[tex]& V=\frac{\pi}{14}(12-21-14+28)=\frac{\pi}{14}(12+28-21-14)=\frac{\pi}{14}(40-35)=\frac{5 \pi}{14} \approx \frac{5 \cdot 3,1416}{14} \approx \frac{15,708}{14} \\& V \approx 1,122[/tex]
Answer:[tex]$V \approx 1,122 u . v$. (units of volume)[/tex]
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I need help guys can someone help me out?
Answer:
Below
Step-by-step explanation:
$ 5 for the first one <=== a fixed constant ....admission charge
.25 per minute <===== charges dependent on the number of minutes
Answer:
pls rate brainliest
Step-by-step explanation:
the entrance fee to the skating rink=$5
each minute the skater uses the rink costs=$0.25
HOPE IT HELPED YOU
Consider the following solid s. The base of S is a circular disk with radius r. Parallel cross-sections perpendicular to the base are squares. Set up an integral that can be used to determine the volume V of the solid. V = dx LIO 26TO = 2 dx Find the volume V of the solid. V=
The volume of the described solid is V = 16r³/3
Given,
A solid line S should be drawn between z=a and z=b. If S has a cross-sectional area of Px through x and is perpendicular to the x-axis, then A (x)
Volume of S = limₙ→∝ ∑i=₁ⁿ A(xi) Δx = [tex]\int\limits^b_a {A(x)} \, dx[/tex]
The equation of circle x² + y² = r² where r is the radius of the circle.
Equation of upper semicircle yu= √(r²-x²) and
Equation of lower semicircle yl = -√(r²-x²)
Area of cross-section A(x) = (yu² - yl²)
Substitute yu and yl values
A(x) = [√(r²-x²) - (-√(r²-x²) )]² = 4(r² -x²) (1)
The limit for volume v varies from -r to r
Thus Volume v = ₋[tex]\int\limits^r_r {A(x)} \, dx[/tex]
Substitute A(x) from (1)
V = ₋[tex]\int\limits^r_r {4(r^{2}-x^{2} ) } \, dx[/tex]
⇒V = 4[r²x - x³/3 ]
⇒V = 4(r³ - r³/3) - 4(-r³ + r³/3) = 4[2 (r³ - r³/3)] = 16r³/3
The volume of the described solid S where the base is a circular disk or radius r and parallel cross sections perpendicular to the base are squares is V = 16r³/3
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Find the requested value.
f(4) for f(x) =
(7x+7, if x ≤0
2-3x, if 0
X, if x ≥ 3
A)4
B)3
C)35
D)-10
The requested value of f(4) is -10.
What is an expression?
An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles.
Here, we have
Given: f(x) =(7x+7, if x ≤ 0
2-3x, if 0 =X, if x ≥ 3
f(4) means the value of f(x) when x = 4.
Which expression on the right applies when x = 4?
The first one because it applies only when x ≤ 0 and
The last one because it applies when x ≥ 3 and 4 is greater than 3. The answer is the last expression.
since 4 lies between 0 and greater than
f(4) = 2 - 3(4)
f(4) = -10
Hence, the requested value of f(4) is -10.
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It's Pre-calculus subject
1) The growth rate is 40%
2) The number of bacteria initially present is 990
3) The number after four hours is 4904
4) The function in terms of time is written as F(t) = 120e^-1.6t
What is the number of bacteria?We know that the number of the bacteria can be obtained on the basis of the function that we have. We know that the process of exponential increase of decrease would have to be modelled by the use of an exponential equation. In this case, we have the equation; n(t) = 990e^0.4t
We know that in this case, the time of the multiplication of the bacterial is t, the percent growth rate of the bacteria is 40% and the initial number of the bacteria is 990.
After 4 hours we have;
n(4) = 990e^0.4(4)
n(4) = 4904
b) Given that the initial temperature if the system is 120°F and the rate of the cooling is 1.6, the formula is written as;
F(t) = 120e^-1.6t
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The width and length of a rectangle are 12 units and 16 units. What is the length of the diagonal of the rectangle
The diagonal of a rectangle with a width of 12 units and a length of 16 units is 20 units
Calculating the diagonal of a rectangleFrom the question, we are to determine the length of the diagonal of the rectangle
From the given information,
The width of the rectangle is 12 units
and
The length of the rectangle is 16 units,
The diagonal, D, of a rectangle is given by
D = √(l² + w²)
Where l is the length of the rectangle
and w is the width of the rectangle
l = 16
w = 12
Thus,
The diagonal of the rectangle is
D = √(16² + 12²)
D = √(256 + 144)
D = √400
D = 20
Hence, the diagonal is 20 units
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Directions: Choose the best answer to the question and fill in the corresponding oval on the Multiple Choice section of the Scantron sheet provided.
1. The population of Rochester has been increasing by 5% each year for the past 7 years. What is the growth factor?
A. 7
B. 1.05
C. 50
D. 5
2. Which of the following is not a point on the graph of the function y=4 x+1.
A. (0,4)
B. (-1,1)
C. (1,16)
D. (-2,0)
Answer:
1= 50
2=1,16
Step-by-step explanation:
because it not have graph
Lena sold 4,832 copies of her book. Elias sold 4,832 times one-fourth copies of his book. Which statement compares the number of books sold?
Elias sold four times the number of books Lena sold.
Elias sold one-fourth the number of books Lena sold.
Lena sold twice the number of books Elias sold.
Lena sold one-fourth the number of books Elias sold.
Answer:
B) Elias sold one-fourth the number of books Lena sold.
Step-by-step explanation:
How is this the answer? Well after I looked at the problem again, I realized what I needed to do. I needed to do 4,832*1/4 which is 1208.
I got this from the line that says "Elias sold 4,832 times one-fourth copies"
I hope this helps!!!!
Parallel lines are cut by the transversal shown. Determine the measures of the requested angles.
Cameron invests money in a bank account
which gathers compound interest each year.
After 4 years there is $736.80 in the account.
After 7 years there is $788.82 in the account.
Work out the annual interest rate of the bank
account.
Give your answer as a percentage to 1 d.p.
first off, let's check the equation for each year.
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$736.80\\ P=\textit{original amount deposited}\\ r=rate\to r\%\to \frac{r}{100}\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \end{array}\dotfill &1\\ t=years\dotfill &4 \end{cases} \\\\\\ 736.80=P\left(1+\frac{\frac{r}{100}}{1}\right)^{1\cdot 4}\implies 736.80=P\left(1+\frac{r}{100} \right)^4 \\\\[-0.35em] ~\dotfill[/tex]
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$788.82\\ P=\textit{original amount deposited}\\ r=rate\to r\%\to \frac{r}{100}\\ n= \begin{array}{llll} \textit{times it compounds per year} \end{array}\dotfill &1\\ t=years\dotfill &7 \end{cases} \\\\\\ 788.82=P\left(1+\frac{\frac{r}{100}}{1}\right)^{1\cdot 7}\implies 788.82=P\left(1+\frac{r}{100} \right)^7 \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{using the 1st equation}}{736.80=P\left(1+\frac{r}{100} \right)^4}\implies \cfrac{736.80}{\left(1+\frac{r}{100} \right)^4}=P \\\\\\ \stackrel{\textit{using the 2nd equation}}{788.82=P\left(1+\frac{r}{100} \right)^7}\implies \stackrel{\textit{substituting on the 2nd equation}}{788.82=\cfrac{736.80}{\left(1+\frac{r}{100} \right)^4}\left(1+\frac{r}{100} \right)^7}[/tex]
[tex]788.82=736.80\cfrac{ ~~ \left(1+\frac{r}{100} \right)^7 ~~ }{\left(1+\frac{r}{100} \right)^4}\implies 788.82=736.80\left(1+\frac{r}{100} \right)^3 \\\\\\ \cfrac{788.82}{736.80}=\left(1+\frac{r}{100} \right)^3\implies \sqrt[3]{\cfrac{788.82}{736.80}}=1+\cfrac{r}{100}\implies \sqrt[3]{\cfrac{788.82}{736.80}}=\cfrac{100+r}{100} \\\\\\ 100\sqrt[3]{\cfrac{788.82}{736.80}}=100+r\implies 100\sqrt[3]{\cfrac{788.82}{736.80}}-100=r\implies {\Large \begin{array}{llll} \stackrel{\%}{2.3}\approx r \end{array}}[/tex]
please help i need to turn in tomorrow
Answer:
(I will only answer one and I think my explanation can help you understand so you do it yourself) For number 8: the slope-intercept form is: y=-1/3x-11
Step-by-step explanation:
x+3y=-33
3y=-x-33(take away x from both sides of the = sign)
y=-1/3x-11(divide 3 on both sides of the = sign)
(7) The sum of five consecutive even integers is at least 400 . What is the minimum value of the first even integer? Choose from the following solutions:
A 80
B 70
C 76
D 82
Answer:
C 76
Step-by-step explanation:
let x = 1st number
let x+ 2 = 2nd number
let x + 4 = 3rd number
let x + 6 = 4th number
Let x + 8 = 5th number
x + x + 2 + x+ 4 + x + 6 + x + 8 = 400 Combine like terms
5x + 20 = 400 Subtract 20 from each side
5x = 380 Divide both sides by 5
x = 76
Check:
76 + 78 + 80 + 82 + 84 = 400
Find an equation for the graph sketched below:
8
6
-5
f
on
N
-5-4-3-2-1
5 6
1 2 3 4 5
Answer:
[tex]f(x)=3^{x+1}-3[/tex]
Step-by-step explanation:
The curve of the graph shows it is Exponential: [tex]f(x)=b^{x-h}+k[/tex]
To find b, h, and k, we need to "un-transform" the graph from [tex]f(x)=b^{x}[/tex]
1. h = horizontal shift/translation; k = vertical translation:
(0,1) ⇒ (-1,-2), so h = -1 and k = -3 and [tex]f(x)=b^{x+1}-3[/tex]
2. Given the points (-1,-2), (0,0), and (1,6), use one of them to find b:
[tex]f(x)=b^{x+1}-3\\0=b^{0+1}-3\\0=b^{1}-3\\3=b^1\\3=b[/tex]
3. Place b, h, and k in their correct places in the function.
perpendicular bisector of AB.
Which diagram shows a valid approach to this construction?
The diagram that shows a valid approach to the given construction is option a for the perpendicular bisector.
What is meant by perpendicular bisector?A perpendicular bisector is a section of a line that divides a straight segment into two congruent or equal segments. At exactly its midpoint, they split the line section. With the line segment it divides, a perpendicular bisector produces a 90° angle. A line segment that bisects another line segment at an angle of 90 degrees is referred to as a perpendicular bisector. In other terms, a perpendicular bisector separates a line segment into two equal pieces by intersecting it at a 90° angle. The term "perpendicular bisector" refers to a precise drawing in which a line is accurately divided in half by another line that is perpendicular to the first line. To divide into two equal portions, or bisect, is a verb.
Therefore, The diagram that shows a valid approach to the f=given construction is option a.
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A base of a right solid, which is 3 centimeters high, is shown in the picture below. Find
the volume of the right solid. All dimensions are in centimeters.
4
5
Answer:
5 i think
Step-by-step explanation:
Suppose that the functions f and g are defined for all real numbers x as follows.
f(x)=4 x
g(x)=2 x-5
By using the concept of function, it can be calculated that-
(g.f)(x) = 8x - 5
(g - f)(x) = -2x - 5
(g + f)(3) = 13
What is a function?
A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.
There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.
Here,
a) The given functions are
f(x) = 4x
g(x)= 2x - 5
(g.f)(x) = g(f(x)) = 2(4x) - 5 = 8x - 5
(g - f)(x) = g(x) - f(x) = 2x - 5 - 4x = -2x - 5
(g + f)(x) = g(x) + f(x) = 2x - 5 + 4x = 6x - 5
(g + f)(3) = 6 [tex]\times[/tex] 3 - 5 = 13
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A person jogs 4 miles in 1/5 hour what’s the persons speed in miles per hour
Answer:
Below
Step-by-step explanation:
WOW! THIS is not 'jogging' :
4 miles / 1/5 hr = 20 m/hr
In what quadrant does the terminal side of -400 degrees lie?
Answer:
1st
Step-by-step explanation:
-400 degrees can also be seen as positive 40 degrees
if p(e)=0.47 find the odds in favor of e
0.53.The elements left over after the given set's components have been eliminated from the universal set are the complement of the set.
How do you determine the likelihood of a compliment?A pair of occurrences that are incompatible with one another are complementary to one another. For instance, the complement of a coin flip is tails when heads is the desired result. The Complement Rule asserts that the total of the probability of an event and its complement must equal 1, or for the event A, P(A) + P(A') = 1.The elements left over after the given set's components have been eliminated from the universal set are the complement of the set. A set's complement is denoted by the letters Ac or just A'. A' = - A is the formula for a complement in this instance.Explanation:
P(event E)=0.47
Consequently, P(not E)=1P(E) [P(E)+P(not E)=1] =10.47 =0.53
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PLEASE HELP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 50 POINTS
Answer:
Step-by-step explanation:
Its A
Answer:A
Step-by-step explanation:A cylinder with 8 units.
Find the value of each variable.
Simplify. your answers as much as possible.
The value of each variable iS
x = 6
y = 4
What is angle bisector theorem?
The angle bisector theorem in geometry is concerned with the proportions of the two segments that a line that bisects the opposite angle divides a triangle's side into. It compares their proportional lengths to the proportional lengths of the triangle's other two sides.
Apply the angle bisector theorem to solve for x and y respectively.
✔️4/(9 - x) = 8/x
Cross multiply
4*x = 8(9 - x)
4x = 72 - 8x
Add 8x to both sides
4x + 8x = 72
12x = 72
12x/12 = 72/12
x = 6
✔️y/2 = 6/3
Cross multiply
3*y = 6*2
3y = 12
3y/3 = 12/3
y = 4
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Fill the missing fraction.
Z - 1 1/2 = 1 1/2
Answer:
Z=3
Step-by-step explanation:
1 1/2+1 1/2= 2+1=3
What’s the answer please thank u
Answer: 1/2
Step-by-step explanation:
slope is rise over run (up and down over left to right)
The student body at Springfield Middle School was surveyed about their preference between math and reading. The data below shows the results of the survey organized by gender. Use this two- way frequency table to answer the questions below.
Answer: 2
Step-by-step explanation:
help simplify this 3^1-1
Answer: 2 or 1?
Step-by-step explanation:
3^1 = 3 then 3 - 1 = 2
or 3^(1 - 1) then 3^0 = 1
Solve the following
(d). m(x)=xcos(-x)
By solving the following trigonometric function of m(x)= x cos(-x) we get m(x) = cos(x).
What is trigonometry ?
Trigonometry is the branch of mathematics that deals with the relationship between ratios of the sides of a right-angled triangle with its angles. The ratios used to study this relationship are called trigonometric ratios, namely, sine, cosine, tangent, cotangent, secant, cosecant.Given that;
m(x) = x cos(-x)
m(x) = x cos (x) [ we know that cos(-x) = cos(x) ]
m(x)/x = x cos(x)/x [divide both sides by x]
m = cos(x)
Therefore, m(x)=x cos(-x) ⇒cos(x)
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Equals how many CC? Please I need help
1. 177.441177
2. 354.882354
3. 118.294118
4. 473.176472
5. 236.588236
The production costs, P, for a band to have a concert can be modeled by the equation P=850 + 5x. where X is the number of concert tickets sold. The band earns $50 for each ticket sold. The band's goal is for the amount earned on ticket sales minus production costs to equal $5,000 per concert. How many tickets does the band have to sell for each concert to meet the goal?
The band have to sell 76 tickets for each concert to meet the goal
How many tickets does the band have to sell for each concert to meet the goal? From the question, we have the following parameters that can be used in our computation:
P = 850 + 5x
Also, we have
Bonus = $50
This means that
Cost = 850 + 5x + 50x
So, we have
Cost= 850 + 55x
When they earn $5000, we have
850 + 55x = 5000
Evaluate the like terms
55x = 4150
Divide
x = 76
Hence, the number of tickets is 76
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Is 181 prime or composite? Justify your answer. A. 181 is prime because it has a factor of 9. B. 181 is prime because its only factors are 1 and itself. C. 181 is composite because it has a factor of 9. D. 181 is composite because it has a factor of 3.
HELPPPP ME
Answer:
B
Step-by-step explanation:
Assuming these 4 are the only possibilities:
A: This cannot be correct because if 181 were divisible by 9, it would not be prime, as the problem suggests.
C: This cannot be correct because 181/9 is 21.11, which is not a whole number
D: This cannot be correct because 181/3 is 60.333, which is not a whole number
Thus, the answer is B.
FUN FACT:
When seeing whether a number is divisible by 3, add up its digits, and if they are divisible by 3, so is the number. Example: 36 is divisible by 3 because 3 + 6 = 9, and 9 is divisible by 3. 181 is not, however, because 1+8+1=10, and 10 is not divisible by 3.
Same with 9! 459 is divisible by 9 because 4+5+9 = 18, which is divisible by 9. 181 is not, however, because 1+8+1 = 10, which is not divisible by 9.
I hope this helps!
Answer:
B. 181 is prime because its only factors are 1 and itself.
Step-by-step explanation:
Prime number
A whole number greater than 1 that cannot be made by multiplying other whole numbers (its only factors are 1 and itself).
Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.
Composite number
A number that is divisible by a number other than 1 and itself (so it has more than 2 factors).
An example of a composite number is 6 as its factors are 1, 2, 3 and 6.
The factors of 181 are 1 and 181.
Therefore, it is prime because its only factors are 1 and itself.
Positive or negative domains
Answer:
I believe it is positive although I am not 100% sure.
Step-by-step explanation:
The arrows tend to go upwards even though it does have a curve at the bottom. It is still going upwards.
Suppose that balloon owners get to pay lower costs for insurance next year. How would this affect the demand curve for balloon rides? How would this affect the supply curve for balloon rides?
A balloon payment is a one-time, large-scale principal payment made near the end of the loan term.
What impact would this have on the demand curve for hot air balloon rides?Total revenue is the amount of money made from the sale of a specific number of items at a specific price.A demand curve is a graph that shows the amounts of stores consumers want at different prices and contains the demand for that product on the y-axis and the price on the x-axis.If a product's price decreases, consumers are more likely to choose it over alternatives that are suddenly more expensive or to simply purchase more quantities of the product, which raises demand from customers.In a similar vein, demand will decline if price rises, all else being equal.To learn more about demand curve demonstrate refer to:
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