Integral for the surface area obtained by rotating the curve about the x-axis is given by [tex]S = \int[1,2] 2\pi xe^(^-^x^) \sqrt{(1 + (e^{(-x)} - xe^{(-x)})^2)} dx[/tex] and about y-axis is given by [tex]S = \int[c,d] 2\pi y \sqrt{(1 + (1/y)^2)} dy[/tex].
What is meant by integral ?
Integral is used to calculate the total or net value of a function over a given interval or to find the area between a function and the x-axis.
(a) To set up the integral for the area of the surface obtained by rotating the curve [tex]y = xe^{(-x)}[/tex] about the x-axis, we can use the formula for the surface area of revolution:
[tex]S = \int[a,b] 2\pi y \sqrt{(1 + (dy/dx)^2)} dx[/tex]
In this case, the curve is given by [tex]y = xe^{(-x)}[/tex], so we need to find [tex]dy/dx[/tex]:
[tex]dy/dx = d/dx (xe^{(-x)})[/tex]
[tex]= e^{(-x)} - xe^{(-x)}[/tex]
Now, we can substitute [tex]y = xe^{(-x)}[/tex] and [tex]dy/dx[/tex] into the formula for surface area:
[tex]S = \int[a,b] 2\pi xe^{(-x)} \sqrt{(1 + (e^{(-x)} - xe^{(-x))^2})} dx[/tex]
Since the bounds of integration are given as 1 ≤ x ≤ 2, the integral becomes:
[tex]S = \int[1,2] 2\pi xe^(^-^x^) \sqrt{(1 + (e^{(-x)} - xe^{(-x)})^2)} dx[/tex]
(b) To set up the integral for the area of the surface obtained by rotating the curve [tex]y = xe^{(-x)}[/tex] about the y-axis, we can use a similar formula:
[tex]S = \int[c,d] 2\pi x \sqrt{(1 + (dx/dy)^2)} dy[/tex]
To find [tex]dx/dy[/tex], we can rearrange the equation [tex]y = xe^{(-x)}[/tex] and solve for x:
[tex]x = y / e^(^-^x^)[/tex]
[tex]x = ye^x[/tex]
Taking the natural logarithm of both sides:
[tex]ln(x) = ln(y) + x[/tex]
[tex]x - ln(x) = ln(y)[/tex]
Differentiating both sides with respect to y:
[tex]dx/dy - (1/x) = 1/y * dy/dy[/tex]
[tex]dx/dy - (1/x) = 1/y[/tex]
Now, we can substitute [tex]x = ye^x[/tex] and [tex]dx/dy[/tex] into the formula for surface area:
[tex]S = \int\dx [c,d] 2 \pi y \sqrt{(1 + (1/y)^2)} dy[/tex]
Since the bounds of integration are not specified in this case, we can leave them as c and d until further information is provided. The integral becomes:
[tex]S = \int[c,d] 2\pi y \sqrt{(1 + (1/y)^2)} dy[/tex]
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The volume of this cylinder is 4,939.22 cubic millimeters. What is the height? Use a 3.14 and round your answer to the nearest hundredth.
Answer:
13 mm
Step-by-step explanation:
V = πr²h
4,939.22 mm³ = 3.14 × (11 mm)²h
4,939.22 mm³ = 3.14 × (11 mm)²h
h = 13 mm
Answer: 13 mm
help with questions 10-13 plz!!
Answer:
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The price of fuel may increase due to demand and decrease due to overproduction. Marco is studying the change in the price of two types of fuel. A and B. over time.
The price in dollars of fuel A after x months is represented by the function below.
10 = 2.961.04)
Part A: is the price of the increasing or decreasing and by what percentage per month? Justify your answer. (5 points)
Part 8: The table below shows the price 9m) in dollars of fuels afer m months
64
m (number of months) 1
3
sim) (price in dollars) 3.04 3.22 3.41 3.61
Which ope of fuel recorded a greater percentage change in pace over the previous month? Justity your answer. (5 points)
Answer:
A. Increasing by 4%.
Step-by-step explanation:
Answer:
A) The price of product A is increasing by 3% per year.
(B) The product A recorded a greater percentage change in price over the previous year.
Step-by-step explanation:
(A)
The function representing the price, in dollars, of product A after x years is:
FA(x)=0.69*(1.03)x
The function FA(x)can be written as:
FA(x)=0.69*1+(0.03)x
The function FA(x) is similar to the exponential growth function, y=a(1+r)x .
Here r is the growth rate.
Thus, it can be said that the price of product A is increasing by 3% per year.
(B)
Consider the data of product B for the year 3 and 4.
The price of product B for year 3 and 4 are 10,303.01 and 10,406.04.
Compute the percent price change from year 3 to 4 as follows:
10406.04-10303.01/10303.01*100
which is 0.999%
~1%
The price of product B is increasing by 1% per year.
Thus, the product A recorded a greater percentage change in price over the previous year.
1. Identify the Parent function related to the given function. Choose the correct answer from the choices below:
f(x)=1/2x-9 4/5
Linear Function
Not a Function
Absolute Value Function
Quadratic Function
A triangle has three angles that measure 15 degrees, 85 degrees and (4x - 20). What is the value of x?
A Susan B. Anthony dollar coin has a diameter of 26.5 millimeters. Round your final answers to the nearest hundredth. Show your steps. 7. What is the radius? 8. What is the circumference? 9. What is the area?
Answer:
7. 13.25mm
8. 83.25 mm
9. 551.55 mm²
Step-by-step explanation:
The shape of a coin is circular.
A Susan B. Anthony dollar coin has a diameter of 26.5 millimeters. Round your final answers to the nearest hundredth. Show your steps.
7. What is the radius?
The formula for radius = Diameter/2
= 26.5 mm/2
= 13.25 mm
8. What is the circumference?
The formula for the circumference of a circle = 2πr
= 2× π × 13.25mm
= 83.25220532 mm
Approximately to the nearest hundredth = 83.25 mm
9. What is the area?
The formula for the area of a circle = πr²
= π × (13.25mm)²
= 551.54586025 mm²
Approximately to the nearest hundredth = 551.55 mm²
2. Find the area of a circle with a diameter of 10
feet.
Step-by-step explanation:
d=2r
πr^2
10/2=r
5=r
5^2π
25π=78.5398
Hope that helps :)
y=Ax^2 + Bx + C is the solution of the DEQ: y'=87x. Determine A,B. A 'C' is the constant of integration.
The exact value of A in the general solution is 87/2 and B is 0
How to determine the value of A and B in the general solutionFrom the question, we have the following parameters that can be used in our computation:
y = Ax² + Bx + C
The differential equation is given as
y' = 87x
When y = Ax² + Bx + C is differentiated, we have
y' = 2Ax + B
So, we have
87x = 2Ax + B
By comparing both sides of the equation, we have
2Ax = 87x
B = 0
So, we have
2A = 87
B = 0
Divide both sides of 2A = 87 by 2
A = 87/2
B = 0
Hence, the value of A in the general solution is 87/2 and B is 0
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ms.Rivera went to dinner for new year's eve the meal and sodas cost a total of $138 the sales tax in new york state is about 8% and ms Rivera wanted to leave a 20% tip because the service was good. What was the total cost of the meal
Answer:
138 + 27.6 = 165.6
Step-by-step explanation:
Please give brainliest
Find all possible trigonometric ratios given the following:
tan θ = -7/24 and cos θ > 0
The given information allows us to find the values of trigonometric ratios involving angle θ. Given that tan θ = -7/24 and cos θ > 0, we can determine the following trigonometric ratios: sin θ, csc θ, sec θ, and cot θ
We are given that tan θ = -7/24. Using this information, we can determine the values of sin θ and csc θ.
Since tan θ = sin θ / cos θ, we can write -7/24 = sin θ / cos θ. Rearranging the equation, sin θ = -7 and cos θ = 24.
Now, we can find the values of csc θ, sec θ, and cot θ.
csc θ is the reciprocal of sin θ, so csc θ = 1 / sin θ = 1 / (-7) = -1/7.
To find sec θ, we use the fact that sec θ = 1 / cos θ. So, sec θ = 1 / (24) = 1/24.
Lastly, to calculate cot θ, we know that cot θ = 1 / tan θ. Thus, cot θ = 1 / (-7/24) = -24/7.
In summary, given tan θ = -7/24 and cos θ > 0, we have sin θ = -7, csc θ = -1/7, sec θ = 1/24, and cot θ = -24/7.
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please someone answer fast!!! I'm so confused and this is due today
Answer:
From greatest to least it would be 3.66666,[tex]\sqrt{11}[/tex],2(1/4),-2.5,-3.97621
Step-by-step explanation:
Just type in a calc
Answer:
[tex]\sqrt{11}=3.31[/tex], -2.5, [tex]2\frac{1}{2}= 2.25[/tex], 3.6, -3.97621...
Step-by-step explanation:
Greatest to least would be:
3.6, [tex]\sqrt{11}[/tex], [tex]2\frac{1}{4}[/tex], -2.5, -3.97621...
Least to greatest would be:
-3.97621, -2.5, [tex]2\frac{1}{4}[/tex], [tex]\sqrt{11}[/tex], 3.6
Hopefully, that helps.
Could someone please help me with this !
And also show work
Answer: C. K= 2.5
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
Answer:
[tex]\Huge\boxed{k=2.5}[/tex]
Step-by-step explanation:
Given -2.1k + 13 + 6.5k = 24 we need to isolate the variable using inverse operations
step 1 combine any like terms
sometimes there are not like terms but in this case there are. When there are like terms (must be on the same side of the = ) you add them together
-2.1k + 6.5k = 4.4k
now we have
4.4k + 13 = 24
Now we want to get rid of the 13
To do so we subtract 13 from each side
13 - 13 cancels out
24 - 13 = 11
now we have 4.4k=11
now we want to get rid of the 4.4
To do so we divide each side by 4.4
4.4k/4=k
11/4.4=2.5
we're left with k - 2.5
What is the lyrics for its good to be alive by among us?
Answer:
HEHEHEHEHEH
I've been waiting for this moment
Feels good to be alive right about now
Good, good, good, good to be alive right about now
Good, good, good, good to be alive right about now
Hallelujah, let that bass line move ya, say hey
Step-by-step explanation:
Have fun
NOLINKS ..................
Answer:
The answer is in the link
Step-by-step explanation:
quntyfcjb/crown!.com :))))
Find the surface area of the prism.
7 cm
13 cm
5 cm
12 cm
The surface area is
Answer:
please give more information
Step-by-step explanation:
[tex]\left \{ {{x=3} \atop {y+1=0}} \right.[/tex] solve graphically this linear system of equations
Answer:
The solution is the point (3, -1)
Step-by-step explanation:
We have the system of equations:
x = 3
y + 1 = 0
To solve this graphically, we need to graph these two lines and see in which point the lines intersect.
To graph the line x = 3, we need to draw a vertical line that passes through x = 3.
To graph y + 1 = 0
First we should isolate y.
y = -1
This is graphed as a horizontal line that passes through y = -1
The graph of these two lines can be seen in the image below.
Where the green line is x = 3, and the blue line is y = -1
Now, looking at the graph we can see that the lines do intersect in the point (3, -1)
Then the solution of the system is the point (3, -1)
Find the flux of the given vector field F across the upper hemisphere x^2 + y^2 + z^2 = a^2, z >= 0. Orient the hemisphere with an upward-pointing normal.
19. F= yj
20. F = yi - xj
21. F= -yi+xj-k
22. F = x^2i + xyj+xzk
6πa² is the flux of F across the upper hemisphere.
The problem requires us to compute the flux of the given vector field F across the upper hemisphere x² + y² + z² = a², z ≥ 0. We are to orient the hemisphere with an upward-pointing normal. The four vector fields are:
F = yj
F = yi - xj
F = -yi + xj - kz
F = x²i + xyj + xzk
To begin with, we'll make use of the Divergence Theorem, which states that the flux of a vector field F across a closed surface S is equivalent to the volume integral of the divergence of the vector field over the region enclosed by the surface, V, that is:
F · n dS = ∭V (div F) dV
where n is the outward pointing normal unit vector at each point of the surface S, and div F is the divergence of F.
We'll need to write the vector fields in terms of i, j, and k before we can compute their divergence. Let's start with the first vector field:
F = yj
We can rewrite this as:
F = 0i + yj + 0k
Then, we compute the divergence of F:
div F = d/dx (0) + d/dy (y) + d/dz (0)
= 0 + 0 + 0 = 0
So, the flux of F across the upper hemisphere is 0. Now, let's move onto the second vector field:
F = yi - xj
We can rewrite this as:
F = xi + (-xj) + 0k
Then, we compute the divergence of F:
div F = d/dx (x) + d/dy (-x) + d/dz (0)
= 1 - 1 + 0 = 0
So, the flux of F across the upper hemisphere is 0. Let's move onto the third vector field:
F = -yi + xj - kz
We can rewrite this as:
F = xi + y(-1j) + (-1)k
Then, we compute the divergence of F:
div F = d/dx (x) + d/dy (y(-1)) + d/dz (-1)
= 1 - 1 + 0 = 0
So, the flux of F across the upper hemisphere is 0. Lastly, let's consider the fourth vector field:
F = x²i + xyj + xzk
We can compute the divergence of F directly:
div F = d/dx (x²) + d/dy (xy) + d/dz (xz)
= 2x + x + 0 = 3x
Then, we express the surface as a function of spherical coordinates:
r = a, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π/2
Note that the upper hemisphere corresponds to 0 ≤ φ ≤ π/2.
We can compute the flux of F over the hemisphere by computing the volume integral of the divergence of F over the region V that is enclosed by the surface:
r² sin φ dr dφ dθ
= ∫[0,2π] ∫[0,π/2] ∫[0,a] 3r cos φ dr dφ dθ
= ∫[0,2π] ∫[0,π/2] (3a²/2) sin φ dφ dθ
= (3a²/2) ∫[0,2π] ∫[0,π/2] sin φ dφ dθ
= (3a²/2) [2π] [2] = 6πa²
Therefore, the flux of F across the upper hemisphere is 6πa².
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select the correct answer. which expression is equivalent to x 3x2−2x−3÷x2 2x−3x 1 if no denominator equals zero? a. 1x2−2x−3 b. 1x2−4x 3 c. 1x2 2x−3 d. x 3x 1
The correct answer is option c. 1/(x² + 2x - 3). To determine which expression is equivalent to the given expression, let's simplify it step by step:
The given expression is (x³ - 2x - 3) ÷ (x² + 2x - 3).
Option a. 1/(x² - 2x - 3):
This option is not equivalent to the given expression because it represents the reciprocal of the quadratic denominator, which is different from the given expression.
Option b. 1/(x² - 4x + 3):
This option is not equivalent to the given expression because the signs of the quadratic terms are different. The given expression has a positive quadratic term, while this option has a negative quadratic term.
Option c. 1/(x² + 2x - 3):
This option is equivalent to the given expression because it represents the reciprocal of the quadratic denominator with the same signs for the quadratic terms.
Option d. x/(3x - 1):
This option is not equivalent to the given expression because it lacks the term x³ in the numerator.
Therefore, the correct answer is option c. 1/(x² + 2x - 3).
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A medical team randomly selects people in an area, until he finds a person who has a corona virus, Let p is the probability that he succeeds in finding such a person, is 0.2 and X denote the number of people asked until the first success. (i) What is the probability that the team must select 4 people until he finds one who has a corona virus? (ii) What is the probability that the team must select more than 6 people before finding one who who has a corona virus?
Answer : i) The probability of finding the first case in 4 trials is 0.1024 ii) The probability that the team must select more than 6 people before finding one who has a corona virus is 0.4095.
Explanation : Given information:Let p is the probability that he succeeds in finding such a person, is 0.2 and X denote the number of people asked until the first success.
(i) What is the probability that the team must select 4 people until he finds one who has a corona virus?
The number of trials required until the first success follows geometric distribution.
The probability of finding the first case in 4 trials is: P(X = 4) = q^3p, where q = 1 - p. We have p = 0.2 and q = 0.8. So, P(X = 4) = 0.8^3 × 0.2 = 0.1024
(ii) What is the probability that the team must select more than 6 people before finding one who who has a corona virus? P(X > 6) = 1 - P(X ≤ 6) The probability of finding the first case in the first 6 trials is:P(X ≤ 6) = 1 - q^6p= 1 - 0.8^6 × 0.2= 0.59049P(X > 6) = 1 - P(X ≤ 6)= 1 - 0.59049= 0.4095 Therefore, the probability that the team must select more than 6 people before finding one who has a corona virus is 0.4095.
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Which golf ball went higher, and how many feet? (Desmos!)
Answer:
1. 36
2. Second
Step-by-step explanation:
- For the first ball, we can see the given function:
[tex]f(x)=-16(t^{2}-3t )[/tex]
[tex]=-16[t^{2} -3t+(3/2)^{2}-(3/2)^{2} ][/tex]
[tex]=-16(t-\frac{3}{2} )^{2} +(-\frac{3}{2} )^{2} *(-16)[/tex]
[tex]-16(t-\frac{3}{2} )^{2} +36[/tex]
So the vertex is ([tex]\frac{3}{2}[/tex], 36), it means when the ball was hit by the [tex]\frac{3}{2}[/tex] seconds, it arrived at the highest height of 36 feet.
- For the second ball, we can see the given graph: the vertex is (2,64), it means when the ball was hit by the 2 seconds, it arrived at the highest height of 64 feet.
- Compare to the two heights, 36 (first ball) is less than 64 (second ball), so the second ball went higher.
This equation shows how the amount of time that a receptionist named Terrence spends on the phone is related to the number of phone calls he routes to employees.
t = p + 17
The variable p represents the number phone calls he routes, and the variable t represents the number of minutes he is on the phone. In all, how many phone calls does Terrence have to route to spend a total of 20 minutes on the phone?
phone calls
hmmm I dunno sorry ......
can i get an owa owa ?????
owa owa???
owa owa??
OWA OWA??
OWA OWA??
OWA OWA?Select all the figures that are shaded to represent 60% of the whole. Whoever answers correctly and first will be marked as brainliest!!!!
Answer:
the first one
Step-by-step explanation:
PLEASE HELP ME!!!! I WILL MARK!!!
Answer:
6(x - 3 ) = 6x - 18
2(3x - 3) = 6x - 6
Step-by-step explanation:
A solid object has the right triangle with vertices (0, 0), (3, 0), and (0, 4) as its base.
a) Any cross section of the solid, taken parallel to the y-axis and perpendicular to the x-axis, is a square. Find the volume of the solid.
b) Any cross section of the solid, taken parallel to the y-axis and perpendicular to the x -axis, is a smi-circle. Find the volume of the solid.
a. The volume of the solid is 24 cubic units.
b. The volume of the solid is 4π cubic units.
How to calculate tie valuea. Volume = Area of Base * Height
The base is a right triangle with base length of 3 units and height of 4 units. The area of the base can be calculated as:
Area of Base = (1/2) * base * height
= (1/2) * 3 * 4
= 6 square units
The height of the solid is 4 units.
Volume = Area of Base * Height
= 6 * 4
= 24 cubic units
b) Any cross section of the solid, taken parallel to the y-axis and perpendicular to the x-axis, is a semicircle.
Volume = (1/2) * π * radius² × height
Volume = (1/2) * (1/2) * π * 2² * 4
= (1/4) * π * 4 * 4
= π * 4
Therefore, the volume of the solid is 4π cubic units.
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IM GIVING BRAINLIEST!!PLEASE HELP!!
Answer: c
Step-by-step explanation:
I need help on the circled problem please
Answer:
There would be infinite solutions because the equations are exactly the same.
I hope this answered your question
How many permutations of S9, have cycle strucrure 3^3?
There is only 1 permutation in S9 with a cycle structure of [tex]3^3[/tex].
To find the number of permutations of S9 with a cycle structure of [tex]3^3[/tex], we can use the concept of cycle index.
In a permutation with a cycle structure of[tex]3^3[/tex], we have three cycles of length 3. The cycle index of S9 with respect to cycles of length 3 can be determined using the Polya enumeration theorem.
The cycle index of S9 with respect to cycles of length 3 is given by:
[tex]Z(S9, t1, t2, t3) = (t1^3 + t3^3)^3[/tex]
Expanding this expression, we get:
[tex]Z(S9, t1, t2, t3) = (t1^3 + t3^3)^3\\\= (t1^9 + 3t1^6t3^3 + 3t1^3t3^6 + t3^9)[/tex]
To count the number of permutations with the desired cycle structure, we need to find the coefficient of the term [tex]t1^9t3^9[/tex].
From the expanded form, we see that the coefficient [tex]t1^9t3^9[/tex] is 1.
Therefore, there is only one permutation in S9 with a cycle structure of [tex]3^3[/tex]
In summary, there is 1 permutation of S9 that has a cycle structure of [tex]3^3[/tex].
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Suppose that 5 people should be randomly selected from a group of 20 forming couples by 10. What is the probability that the 5 unrelated chosen from related persons (that is, no chosen person be a couple)?
The probability that none of the 5 randomly selected individuals are part of a couple is 0.016.
What is the probability that none of the 5 randomly selected individuals are part of a couple?A probability means the branch of math which deals with finding out the likelihood of the occurrence of an event. Its measures the chance of an event happening.
We will know total number of possible outcomes when selecting 5 individuals from a group of 20. This can be calculated using the combination formula:
C(20, 5) = 20! / (5! * (20 - 5)!)
C(20, 5) = 15,504
We know that when we select an individual, we are removing their corresponding partner from the pool of available choices. This means that for each individual we choose, the number of available choices decreases by 1.
The number of favorable outcomes can be calculated as follows:
= 20 * 18 * 16 * 14 * 12
= 967,680
The probability will be:
= Outcomes / Favorable outcomes
= 15,504 / 967,680
= 0.01602182539
= 0.016.
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A bakery produces five types of bagels, two of which are chocolate chip and cinnamon raisin.
(a) If there are at least 10 bagels of each type, how many different selections of 10 bagels are there?
(b) Suppose there are only 3 chocolate chip and 2 cinnamon raisin bagels, but at least 10 of the other three types. How many different selections of 10 bagels are there?
a) If there are at least 10 bagels of each type, we can calculate the number of different selections of 10 bagels by using the concept of combinations. Since there are 5 types of bagels and we need to select 10 bagels, the calculation can be done as follows:
[tex]\(\binom{10+5-1}{10} = \binom{14}{10}\)[/tex]
b) If there are 3 chocolate chip and 2 cinnamon raisin bagels, and at least 10 of the other three types, we can calculate the number of different selections of 10 bagels using the same concept of combinations. In this case, we have 3 types of bagels (excluding chocolate chip and cinnamon raisin) with at least 10 bagels each. So the calculation becomes:
[tex]\(\binom{10+3-1}{10} = \binom{12}{10}\)[/tex]
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