The simplified expression of integral 3 (x-2 ] 3 +4 dx is (A/3) + 12tanh[tex](sinh^{(-1)}[/tex](3(x-2))) + B
How to find the integral ∫3(x-2)³+4 dx using the substitution sinh(y) = 3(x-2)?To find the integral ∫3(x-2)³+4 dx using the substitution sinh(y) = 3(x-2), we can start by differentiating both sides of the equation with respect to x to find the differential of y:
d(sinh(y))/dx = d(3(x-2))/dx
cosh(y) * dy/dx = 3
dy/dx = 3/cosh(y)
Now, let's solve for dx in terms of dy:
dx = (cosh(y)/3) dy
Substituting this value of dx in the integral:
∫3(x-2)³+4 dx = ∫(3/cosh(y)) * (3(x-2)³+4) dy
Now, we need to substitute the expression for x in terms of y using the given substitution:
3(x-2) = sinh(y)
x - 2 = sinh(y)/3
x = sinh(y)/3 + 2
Substituting this in the integral:
∫(3/cosh(y)) * (3((sinh(y)/3 + 2) - 2)³+4) dy
Simplifying:
∫(3/cosh(y)) * (sinh(y)³+4) dy
To integrate the expression ∫(3/cosh(y)) * (sinh(y)³+4) dy, we can simplify it first:
∫(3/cosh(y)) * (sinh(y)³+4) dy = 3∫(sinh(y)³/cosh(y)) dy + 12∫(1/cosh(y)) dy
To integrate the first term, we can use the substitution u = cosh(y), which implies du = sinh(y) dy:
3∫(sinh(y)³/cosh(y)) dy = 3∫(u³/u) du = 3∫(u²) du = u³/3 + C
For the second term, we can directly integrate 1/cosh(y) using the identity sech²(y) = 1/cosh²(y):
12∫(1/cosh(y)) dy = 12∫sech²(y) dy = 12tanh(y) + D
Now, substituting back y = [tex]sinh^{(-1)}(3(x-2))[/tex]:
u = cosh(y) = cosh[tex](sinh^{(-1)}(3(x-2))[/tex]) = √(3(x-2)² + 1)
Thus, the integral becomes:
∫(3/cosh(y)) * (sinh(y)³+4) dy = (u³/3 + C) + 12tanh(y) + D
Substituting back u = √(3(x-2)² + 1):
= (√(3(x-2)² + 1)³/3 + C) + 12tanh(y) + D
= (√(3(x-2)² + 1)³ + 3C)/3 + 12tanh(y) + D
= (√(3(x-2)² + 1)³ + 3C)/3 + 12tanh[tex](sinh^{(-1)}(3(x-2)))[/tex] + D
To simplify the expression and combine constants, let's assume (√(3(x-2)² + 1)³ + 3C)/3 = A, and 12D = B.
The simplified expression becomes:
(A/3) + 12tanh[tex](sinh^{(-1)}[/tex](3(x-2))) + B
Since [tex]sinh^{(-1)}(3(x-2))[/tex] is the inverse hyperbolic sine function, we can simplify it using the identity sinh[tex](sinh^{(-1)}(x))[/tex] = x:
(A/3) + 12tanh(3(x-2)) + B
This is the simplified form of the integral ∫(3/cosh(y)) * (sinh(y)³+4) dy after combining constants and simplifying the expression.
Learn more about integral
brainly.com/question/31433890
#SPJ11
Solve the linear system X 1 X1 + 2x2 3.21 + 4.02 IL || -1 -1 = via Cramer's rule if possible.
The linear system X₁ + 2X₂ = 3.21 and 4.02X₁ + IL || -1 = -1 using Cramer's rule, we need to find the values of X₁ and X₂.
To apply Cramer's rule, we first need to calculate the determinant of the coefficient matrix and the determinants of the matrices obtained by replacing each column of the coefficient matrix with the constant terms.
The coefficient matrix is:
| 1 2 |
| 4.02 IL || |
The determinant of the coefficient matrix, denoted as D, is given by:
D = (1 * IL ||) - (2 * 4.02)
= IL || - 8.04
The matrix obtained by replacing the first column with the constant terms is:
| 3.21 2 |
| -1 IL || |
The determinant of this matrix, denoted as D₁, is given by:
D₁ = (3.21 * IL ||) - (-1 * 2)
= 3.21IL || + 2
The matrix obtained by replacing the second column with the constant terms is:
| 1 3.21 |
| 4.02 -1 |
The determinant of this matrix, denoted as D₂, is given by:
D₂ = (1 * -1) - (4.02 * 3.21)
= -1 - 12.9042
= -13.9042
Now, we can find the values of X₁ and X₂ using the formulas:
X₁ = D₁ / D
X₂ = D₂ / D
Substituting the values we calculated earlier, we have:
X₁ = (3.21IL || + 2) / (IL || - 8.04)
X₂ = (-13.9042) / (IL || - 8.04)
This gives us the solution to the linear system.
Solve the linear system X 1 X1 + 2x2 3.21 + 4.02 IL || -1 -1 = via Cramer's rule if possible.
To know more about Cramer's rule, refer here:
https://brainly.com/question/30682863#
#SPJ11
Sao can text 1500 words per hour. He needs to text a message with 85 words. He only has 5 minutes between classes to complete the text. Can he do it in 5 minutes?
Chapter 07, Problem 075 (See Fluids in the News article titled Ice engineering.) A model study is to be developed to determine the force exerted on bridge piers due to floating chunks of ice in a river. The piers of interest have square cross sections. Assume that the force, R, is a function of the pier width, b, the depth of the ice, d, the velocity of the ice, V, the acceleration of gravity, g, the density of the ice, rho, and a measure of the strength of the ice, Ei, where Ei has the dimension FL-2. (a) Based on these variables determine a suitable set of dimensionless variables for this problem 2 d gd E,b, b, gb、 E, (b) The prototype conditions of interest include an ice thickness of 12 in. and an ice velocity of 11 ft/s. What model ice thickness and velocity would be required if the length scale is to be 1/21? in FRI (c) If the model and prototype ice have the same density can the model ice have the same strength properties as that of the prototype ice? nswer- the tolerance is +/-2% Answer * 2: the tolerance is +/-2%
(a) Using these variables, the dimensionless variables that can be used for this problem are as follows:2d/gdgb/EiE/b
(b) If the density is the same, the model ice can have the same strength properties as that of the prototype ice.
(a) The suitable set of dimensionless variables for this problem includes: g, the acceleration of gravityd, depth of the iceb, pier width Ei, the strength of the ice ρ, density of the ice
Using these variables, the dimensionless variables that can be used for this problem are as follows:2d/gdgb/EiE/b
(b) Given that the length scale is 1/21, and prototype ice thickness and velocity are 12 in. and 11 ft/s, respectively.
We can determine the model ice thickness and velocity using the length scale as follows:
For ice thickness, Model ice thickness = Prototype ice thickness × Length scale= 12 in. × (1/21)= 0.571 in.
For ice velocity, Model ice velocity = Prototype ice velocity × (Length scale)-0.5= 11 ft/s × (1/21)-0.5= 0.765 ft/s≈ 0.77 ft/s(c)
The model ice can have the same strength properties as the prototype ice if their strengths are identical, although the densities are the same.
Therefore, if the density is the same, the model ice can have the same strength properties as that of the prototype ice.
Visit here to learn more about density brainly.com/question/29775886
#SPJ11
Suppose that an urn contains 3 different types of balls: red, green and blue. Let pi denote the proportion of red balls, p2 denote the proportion of green balls and p3 denote the proportion of blue balls. Here ₁-1 Pi = 1. Suppose also that 100 balls are selected with replacement, and there are exactly 38 red, 29 green and 33 blue. Find the M.L.E. p; of p₁, i = 1, 2, 3. Warning: No credit for answers only! P₁=__ S=____ P3 =_____
Let's start solving the given problem. Suppose that an urn contains 3 different types of balls: red, green, and blue. Let pi denote the proportion of red balls, p2 denote the proportion of green balls, and p3 denote the proportion of blue balls.
Here Pi = 1.
Suppose also that 100 balls are selected with replacement, and there are exactly 38 red, 29 green and 33 blue.
We need to find the M.L.E. p; of p1, i = 1, 2, 3.
The probability of obtaining red ball from the urn = pi. The probability of obtaining green ball from the urn = p2. The probability of obtaining blue ball from the urn = p3
Given that, 100 balls are selected with replacement.
Let's calculate the probability of getting 38 red, 29 green, and 33 blue balls from the urn,
P(38 Red and 29 Green and 33 Blue) = P(Red)38 x P(Green)29 x P(Blue)33 = p₁³⁸ x p₂²⁹ x p₃³³
Therefore, the likelihood of obtaining the 38 red, 29 green, and 33 blue balls is L(p1,p2,p3) = p₁³⁸ x p₂²⁹ x p₃³³
Since, L(p1,p2,p3) is a continuous function, so we have to find the critical points of the function to obtain the minimum value of p₁. Let's take natural logarithm on both sides of the function, we get ln
L(p1,p2,p3) = 38 ln p₁ + 29 ln p₂ + 33 ln p₃.
So, taking partial differentiation with respect to p1, p2 and p3 and equating them to 0. We get the following equations,∂ ln L / ∂p1 = 38/p1 = 0∂ ln L / ∂p2 = 29/p2 = 0∂ ln L / ∂p3 = 33/p3 = 0On
solving the above three equations, we get p1 = 38/100 = 0.38p2 = 29/100 = 0.29p3 = 33/100 = 0.33
Therefore, P₁= 0.38, S= 0.29, P3 = 0.33. Hence, the required solution is P₁= 0.38, S= 0.29, P3 = 0.33.
To know more about probability refer to:
https://brainly.com/question/29660226
#SPJ11
A board game has a spinner divided into sections of equal size. Each section is labeled with a number between 1 and 5.
Spinner
Which number is a reasonable estimate of the number of times the spinner will land on a section labeled 5 over the course of 150 spins?
Answer:
30 times
Step-by-step explanation:
1. find the possibility of the spinner landing on each section
1 / 5 = 0.2
2. multiply # of spins by the possibility
150 * 0.2 = 30 times
The Massachusetts state lottery game, Cash WinFall, used to have a way that anyone with enough money and time could stand a good chance of getting rich, and it is reported that an MIT computer scientist did just that. In this game, a player picks 6 numbers from the range from 1 to 46. If he matches all 6, then he could win as much as $2 million, but the odds of that payout don't justify a bet, so let us ignore the possibility of winning this jackpot. Nevertheless, there were times when matching just 5 of the 6 numbers in a $2 lottery ticket would pay $100,000. Suppose in this scenario that you were able to bet $600,000.
(a) What is the expected amount that you would win?
(b) Derive a bound on the probability that you would lose $300,000 or more in this scenario, that is, that you would have 3 or fewer of the 5 of the 6 winning tickets.
The expected amount that you would win is approximately $0.334891, and the probability of losing $300,000 or more is approximately 0.00030986749 (or 0.030986749%).
To calculate the expected amount that you would win in this scenario, we need to consider the probabilities of various outcomes and their corresponding winnings.
(a) Expected Amount of Winnings:
Let's calculate the expected amount by considering the probabilities and winnings for different outcomes:
- Probability of matching exactly 5 out of 6 numbers:
The probability of matching 5 numbers correctly is given by the combination formula:
P(5) = C(6, 5) * C(40, 1) / C(46, 6) = 0.00000096739
The corresponding winnings are $100,000.
- Probability of matching exactly 4 out of 6 numbers:
The probability of matching 4 numbers correctly is given by the combination formula:
P(4) = C(6, 4) * C(40, 2) / C(46, 6) = 0.0000186101
The corresponding winnings are $5,000.
- Probability of matching exactly 3 out of 6 numbers:
The probability of matching 3 numbers correctly is given by the combination formula:
P(3) = C(6, 3) * C(40, 3) / C(46, 6) = 0.000290201
The corresponding winnings are $500.
Now, let's calculate the expected amount:
Expected Amount = (P(5) * $100,000) + (P(4) * $5,000) + (P(3) * $500)
Expected Amount = (0.00000096739 * $100,000) + (0.0000186101 * $5,000) + (0.000290201 * $500)
Expected Amount = $0.096739 + $0.093051 + $0.145101
Expected Amount = $0.334891
Therefore, the expected amount that you would win is approximately $0.334891.
(b) Probability of Losing $300,000 or More:
To derive a bound on the probability of losing $300,000 or more, we need to calculate the cumulative probability of having 3 or fewer of the 5 winning tickets.
Cumulative Probability = P(3) + P(4) + P(5)
Cumulative Probability = 0.000290201 + 0.0000186101 + 0.00000096739
Cumulative Probability = 0.00030986749
Therefore, the bound on the probability of losing $300,000 or more is approximately 0.00030986749, which is equivalent to 0.030986749%.
To know more about probability, refer here:
https://brainly.com/question/30034780
#SPJ4
Ma Bernier has 52 cats and dogs in her house the number of cats is three times the number of dogs how many cats and dogs are in the house
Answer:
dogs - 13
cats - 39
Step-by-step explanation:
Let the number of dogs Ma Bernier have be represented with d
She has 3 times as many cats as dogs, the number of cats she has :
cats = 3d
The sum of the cats and dogs is 52
d + 3d = 52
4d = 52
divide both sides of the equation by 4
d = 13
She has 13 dogs
Cats = 3d
= 3 x 13 = 39
Can someone help me with this. Will Mark brainliest. Need answer and explanation/work. Thank you.
Answer:
-2x+2
Step-by-step explanation:
it is directly underneath the original equation and they never intersect
I need this answer I’ll give brainliest
Can someone help please!!
Answer:
75.39 in^2 (round to the nearest hundreds)
Step-by-step explanation:
I think that we have to find the lateral area
circumference = 2 x radius x pi = 2 x 4 x pi = 25,13 in^2 (round to the nearest hundreds)
lateral surface = (circumference x slant height)/2 = (25,13 x 6)/2 = 75,39 in^2 (round to the nearest hundreds)
PLS HELP MEEEEEEEE I WILL GIVE BRAINLIEST :D
Answer:
A is true and f is true
Step-by-step explanation:
Answer:
a, d, e, and g
Step-by-step explanation:
if you multiply two negatives, you get a positive
anyone know this or can help pls
Answer:
A pyramid
Step-by-step explanation:
Which of the following equations is represented by a line passing through (-9, 7) and (3, 3) in the
standard (x, y) coordinate plane?
Answer:
y = -1/3x + 4
Step-by-step explanation:
(-9, 7) and (3, 3)
Slope:
m=(y2-y1)/(x2-x1)
m=(3-7)/(3+9)
m=(-4)/12
m= -1/3
Slope-intercept:
y - y1 = m(x - x1)
y - 7 = -1/3(x + 9)
y - 7 = -1/3x - 3
y = -1/3x + 4
Step-by-step explanation:
[tex]soln \\ \frac{3 - 7 }{3 - - 9} \\ = \frac{ - 4}{ - 12} \\ = \frac{1}{3 } \\ \frac{y - 7}{x - 3} = \frac{1}{3} \\ then \: you \: cross \: multiplication. \\ 3(y - 7) = 1(x - 3) \\ 3y - 7 = x - 3 \\ 3y = x - 3 + 7 \\ 3y = x + 4 \\ \frac{3y}{3} = \frac{x}{3} + \frac{4}{3} \\ y = \frac{x}{3} + \frac{4}{3} [/tex]
Question 4 - (5+5=10 marks) [For probabilities keep 4 decimal places]
Fifty boxes labeled with numbers from 1 to 50 are laid on a table. In each box there is a blue ball and a red ball. Since a blue ball is bigger than a red ball, we should assume the chance of randomly drawing a blue ball from any box is twice that of a red ball. From each box that you randomly choose, you draw only one ball randomly, without looking into the box or at the drawn ball. Right after a ball is drawn, its corresponding box is moved away from the table to avoid choosing the same box again. You continue this process until 25 boxes are chosen.
a) What is the probability of drawing 17 red balls and 8 blue balls from boxes with even numbered labels? (5 marks)
b) If accidentally you see the fifth ball after being drawn is red, what would be the probability of drawing 17 red balls and 8 blue balls, everything else being the same as mentioned above in the statement of problem. (5 marks)
a) The probability of drawing 17 red balls and 8 blue balls from boxes with even-numbered labels is 0.0051.
b) The probability of drawing 17 red balls and 8 blue balls from boxes with even-numbered labels, given that the fifth ball drawn was red, is 0.1926.
a) The probability of drawing 17 red balls and 8 blue balls from boxes with even-numbered labels can be calculated as follows:
P(drawing 17 red balls and 8 blue balls from boxes with even-numbered labels) = (C(25, 17) × C(25, 8)) / C(50, 25)
= (12620256 / 2462624626080)
= 0.0051 (approx)
Therefore, the probability of drawing 17 red balls and 8 blue balls from boxes with even-numbered labels is 0.0051 (approx).
b) If accidentally, you see the fifth ball after being drawn is red, then the corresponding box will be removed, and there are now 49 boxes remaining on the table.
The number of even-numbered boxes among these 49 boxes is 24.
Therefore, the probability of drawing 17 red balls and 8 blue balls from boxes with even-numbered labels, given that the fifth ball drawn was red, can be calculated as follows:
P(drawing 17 red balls and 8 blue balls from boxes with even-numbered labels, given that the fifth ball drawn was red)
= (C(24, 16) × C(25, 7)) / C(49, 23)
= (8751600 / 45379690908)
= 0.1926 (approx)
Therefore, the probability of drawing 17 red balls and 8 blue balls from boxes with even-numbered labels, given that the fifth ball drawn was red, is 0.1926 (approx).
To know more about probability, visit:
https://brainly.com/question/32004014
#SPJ11
(a) Bob wants to open a new fastfood shop. He estimates he needs to spend at least $100,000 on renovation and buying commercial kitchen appliances. The average running cost per customer (for food, staff salary, etc.) is $7. Each customer spends $15 on average.
(i) If the number of customers is n, write down the cost function and revenue function as functions of n.
(ii) Determine the minimum number of customers for the business to breakeven.
(iii) Bob has to borrow $100,000 from a bank with interest rate of 3% p.a. with interest payable monthly. If Bob does not repay a single cent, how much does Bob owe the bank after 3 years?
i) The cost function is x = 100,000 + 7n while the revenue function is y = 15n.
ii) The minimum number of customers for the business to break even is 12,500.
iii) The amount that Bob owes the bank after 3 years at 3% p.a. interest payable monthly, but without any repayment, is $109,272.70.
What is a function?A function is a mathematical equation showing the equality of two or more algebraic expressions.
a) Renovation and appliances costs = $100,000
Average running cost per customer = $7
Selling price per customer = $15
i) Let the number of customers = n
Cost function, x = 100,000 + 7n
Revenue function, y = 15n
ii) For the business to break even, y must be equal to x:
15n = 100,000 + 7n
8n = 100,000
n = 12,500
iii) Bank loan = $100,000
Interest rate = 3% p.a.
Interest payment = monthly
Amount after 3 years without any repayment = $100,000 x 1.03^3
= $109,272.70 ($100,000 x 1.092727)
Learn more about cost and revenue functions at https://brainly.com/question/30566291.
#SPJ4
What is the measure of the other acute angle
Answer:
77 degrees
Step-by-step explanation:
All triangle add up to 180 degrees and right triangle all have one angle that is 90 degrees so 90 + 14 is 104 and 180 - 104 is 77.
Answer:
76 degrees
Step-by-step explanation:
What we know:
1. the triangle is a right triangle, meaning on angle has to be 90 degrees
2. One angle is 14 degrees
3. A triangle's angles add up to 180 degrees.
----------------------------------------------------------------------------------------------------
90+14+x=180
104+x=180
x=76
Find the sum. 5+ 8 + 11 + ... + 122 The sum is 1. a ) (Type an integer or a simplified fraction.)
The sum of the Arithmetic Progression 5 + 8 + 11 + ... + 122 is 2540.
Here we are given to find the sum of 5 + 8 + 11 + ... + 122
Here we can see that this is an Arithmetic Progression with the first term as 5 and the common difference as
second term - first term
= 8 - 5
= 3
Here we need to find the sum from 5 to 122
now, we know that for any term
aₙ = a + (n - 1)d
where,
a = first term
d = common difference
hence putting aₙ as 122 we get
5 + (n - 1)3 = 122
or, 5 + 3n - 3 = 122
or, 3n + 2 = 122
or, 3n = 120
or, n = 40
Hence, we need to find the sum of the first n terms
we know that the sum S is
S = 0.5n(a + aₙ)
here we know that n = 122, a = 5 and aₙ = 122
hence we get
S = 0.5 X 40(5 + 122)
= 20(127)
= 2540
Hence the sum is 2540.
To learn more about Arithmetic Progression visit
https://brainly.com/question/11688485
#SPJ4
HELP IM GONNA GIVE YOU A BRAINLEST!!!
Without graphing, determine whether the function represents exponential
growth or exponential decay. Then find the y-intercept.
y=2/3 (1/2)^x
Answer:
y intercept is (0, 2/3)
Step-by-step explanation:
the y-inetercept is 0
i don’t know the exponential
growth or exponential decay but you can use math.way to help you answer the question
OK IF YOU KNOW THE ANSWER PLEASE TYPE IN COMMENTS!
Timothy has 2 turtles named Buster and Bubbles. Buster is 75 millimeters long and Bubbles is 6 centimeters long. Buster is longer. Is this true or false?
Group of answer choices
1.True
1.False
Answer: True
Explanation:
1 centimeter = 10 millimeters
Buster = 75 millimeters long
Bubbles = 60 millimeters long
EXPLAIN THE DIFFERENCE OF THE CIRCUMFERENCE AND THE AREA OF THE CIRCLE.
PLS ANSWER IT.
circumference is around the circle the border of the circle the area of the circle is the space in the circle
is 0.2857 a rational number?Explain
Answer:
0.2857 is not a rational number.
Step-by-step explanation:
Because rational numbers have to be greater than 1. Decimals are not over 1 so it is not a rational number. Your welcome!
Estimate 103 + 94 by first rounding each number to the nearest ten
Answer:
200
Step-by-step explanation:
103+94
=100+100
=200
Answer:
197 the nearest ten is 200
Step-by-step explanation:
he need to put the nearest ten
For each of the days above, work out how much money would be made by each court if all seats were sold.
Seats:
Centre court has 14 979 seats for sale. No 1 court has 11 429 seats for sale. No 2 court has 4000 seats for sale. No 3 court has 2000 seats for sale.
Prices:
Centre Court: £56 No 1 Court: £45 No 2 Court: £41 No 3 Court: £41 Grounds Admission: £25
The total money made by each court would be as follows:
Centre Court: £838,824, No 1 Court: £514,305
No 2 Court: £164,000, No 3 Court: £82,000
To calculate the amount of money made by each court if all seats were sold, we need to multiply the number of seats by the corresponding ticket price for each court. Here are the calculations for each court:
Centre Court:
Number of seats: 14,979
Ticket price: £56
Total money made: 14,979 seats ×£56/seat = £838,824
No 1 Court:
Number of seats: 11,429
Ticket price: £45
Total money made: 11,429 seats ×£45/seat = £514,305
No 2 Court:
Number of seats: 4,000
Ticket price: £41
Total money made: 4,000 seats ×£41/seat = £164,000
No 3 Court:
Number of seats: 2,000
Ticket price: £41
Total money made: 2,000 seats × £41/seat = £82,000
Therefore, if all seats were sold, the total money made by each court would be as follows:
Centre Court: £838,824
No 1 Court: £514,305
No 2 Court: £164,000
No 3 Court: £82,000
Learn more about numbers here:
https://brainly.com/question/24908711
#SPJ11
MULTIPLICACIÓN Y DIVISIÓN DE NÚMEROS ENTEROS
11. -2115 ÷ -9 =
12. 7854 ÷ -34
13. 3425 × -4 =
14. -7 × 5 × -3 =
15. 12 × -7 × 9 =
I need help with this answer . Please help . (This is timed!!)
Answer: Its A
Step-by-step explanation: Just count the lines and since there is 8 lines it would be ?/8 so we going to count up until the dot which stops at 3 soo it would be 3/8
solve the point A and B A) The region bounded above by the parabolay = 3x-x2 and y = 0 is rotated around a vertical line x=-1 forming a solid, find its volume Note: When performing the step-by-step procedures used and the method used to find the volumen ex B) = Given the following function which is one to one f(x) = ex/1-eX Find its inverse; You must keep in mind the processes of factoring, properties of exponents, logarithmic properties, and so on. Check if it is indeed its inverse, for this you can do it algebraically or graphically
Previous question
Next que
A. The volume of the solid formed by rotating the region bounded above by the parabola y = 3x-x² and y = 0 around the vertical line x = -1 is approximately 9.74 cubic units and B. The inverse function is found to be ln(x/(1 - x)).
To find the volume of the solid, we can use the method of cylindrical shells. The integral for the volume is given by V = ∫[a,b] 2πxf(x) dx, where f(x) represents the height of the shell at each x-coordinate.
First, we need to find the bounds of integration. The parabola y = 3x - x² intersects the x-axis at x = 0 and x = 3. Therefore, the bounds of integration are [0, 3].
Next, we need to express the height of the shell, f(x), in terms of x.
Evaluating the integral, we get V = ∫[0,3] 2π(x + 1)(3x - x²) dx. After integrating and simplifying, the volume is approximately 9.74 cubic units.
(B) To find the inverse of the function f(x), we swap the roles of x and y and solve for y. So, we start with y = eˣ/(1 - eˣ).
Step 1: Swap x and y: x = eʸ/(1 - eʸ).
Step 2: Solve for y: x(1 - eʸ) = eʸ.
Step 3: Expand and isolate eʸ: x - xeʸ = eʸ.
Step 4: Factor out eʸ: eʸ(x - 1) = x.
Step 5: Divide both sides by (x - 1): eʸ = x/(x - 1).
Step 6: Take the natural logarithm of both sides: y = ln(x/(x - 1)). Thus, the inverse function is g(x) = ln(x/(x - 1)), where x ∈ (0, 1).
To know more about parabola and inverse, visit,
https://brainly.com/question/30190650
#SPJ4
Complete question - A. The region bounded above by the parabola y = 3x-x2² and y = 0 is rotated around a vertical line x=-1 forming a solid, find its volume.
B. Given the following function which is one to one f(x) = eˣ/1-eˣ Find its inverse.
HELP PLEASE!!!
The bear population in Canada was 380,000 in the year 2015, and environmentalists think that the population is increasing at a rate of 2.5% per year.
Consider the function that represents the exponential growth of the bear population in Canada.
Part A: Defines a variable for the function and state what the variable represents.
Part B: What is a reasonable domain for the situation?
Part C: Write the function that represents the exponential growth of the bear population.
Part D: What will the bear population estimated to be in 2050?
Answer:
The population of Bear in 2050 is 4750000
Step-by-step explanation:
A) The exponential growth equation for bear is as follows -
dN/dT = rmax * N
Where dN/dT = change in population
rmax is the maximum rate of change
N = Base population
B) Here the per capita rate of increase (r) will always be a positive value irrespective of the and hence we will assume this population to be growing exponentially.
C) dN/dT = rmax * N
D) dN / 5 = 2.5 * 380,000
dN = 5*2.5 * 380000
= 4750000
⚠️⚠️⚠️⚠️help⚠️⚠️⚠️⚠️
Y Let f:R → R be defined by f(3) = Then x2 +1' a. f is a Borel function. b. f is not a measurable function. c. [f > 3] is not a measurable set. d. None of the above.
Y Let f:R → R be defined by f(3) = Then x2 +1' a is (b) f is not a measurable function.
To determine if f is a measurable function, we need to check if the preimage of any measurable set in the codomain (R) is a measurable set in the domain (R).
In this case, the function f(3) = x^2 + 1 is defined only at x = 3. Since the domain R is continuous and f is only defined at a single point, the preimage of any set in the codomain will either be an empty set or a singleton set containing only the point x = 3.
For example, consider the set [f > 3] in the codomain R. The preimage of [f > 3] under f would be the set of all x in the domain R such that f(x) > 3. However, since f is only defined at x = 3, the preimage would be the singleton set {3}. Since {3} is not a measurable set in the domain R, f is not a measurable function.
Therefore , the correct option is (B) f is not a measurable function.
To know more about measurable functions , refer here:
https://brainly.com/question/11577594#
#SPJ11
Select the graph of the solution. Click until the correct graph appears. {x | x < 4} ∩ {x | x > -2}
Answer:
It converges
Step-by-step explanation:
dx/dy = x∧4 + 9x∧21
f(x) = ∫(x∧4 + 9x∧21)dy 0 > f(x) > ∞
= x∧5/5 + 9x∧22/22 + c
x = ∞ and x = 0
∴ c = 1 /5 + 9/22 = 27/22