Using the given transformation, the integral can be evaluated over the triangular region R by changing to the u-v coordinate system and we get:
∫0^1∫0^(1-2v/3) (2u + v)^3 du dv + ∫0^(2/3)∫0^(2u/3) (u + 2v)^3 dv du.
The transformation given is x = 2u + v and y = u + 2v. To find the limits of integration in the u-v coordinate system, we need to determine the images of the three vertices of the triangular region R under this transformation.
When x = 0 and y = 0, we have u = v = 0. Thus, the origin (0,0) in the x-y plane corresponds to the point (0,0) in the u-v plane.
When x = 2 and y = 1, we have 2u + v = 2 and u + 2v = 1. Solving these equations simultaneously, we get u = 1/3 and v = 1/3. Thus, the point (2,1) in the x-y plane corresponds to the point (1/3,1/3) in the u-v plane.
Similarly, when x = 1 and y = 2, we get u = 2/3 and v = 4/3. Thus, the point (1,2) in the x-y plane corresponds to the point (2/3,4/3) in the u-v plane.
Therefore, the integral over the triangular region R can be written as an integral over the corresponding region R' in the u-v plane:
∫∫(x^3 + y^3) dA = ∫∫((2u + v)^3 + (u + 2v)^3) |J| du dv
where J is the Jacobian of the transformation, which can be computed as follows:
J = ∂(x,y)/∂(u,v) = det([2 1],[1 2]) = 3
Thus, we have:
∫∫(x^3 + y^3) dA = 3∫∫((2u + v)^3 + (u + 2v)^3) du dv
Now, we can evaluate the integral over R' by changing the order of integration:
∫∫(2u + v)^3 du dv + ∫∫(u + 2v)^3 du dv
Using the limits of integration in the u-v plane, we get:
∫0^1∫0^(1-2v/3) (2u + v)^3 du dv + ∫0^(2/3)∫0^(2u/3) (u + 2v)^3 dv du
Evaluating these integrals gives the final answer.
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what is the function rule for the line ?
The function rule for the line is,
f(x) = -2/3x -2.
The correct option is B.
What is a slope?In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.
If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,
then the equation of line is
y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)
To find the slope;
m = (y₂- y₁) / (x₂ - x₁)
From the diagram,
the function has y-intercept at -2 and x-intercept at -3.
To find the slope:
First, we interpreted two points (-3, 0) and (0, -2).
Then the slope,
m = (y₂- y₁) / (x₂ - x₁)
m = (-2, 0)/(0 + 3)
m = -2/3
Now, the equation is,
f(x) = -2x/3 - 2
Therefore, the equation is,
f(x) = -2x/3 - 2
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find each measure using the information given
G is the incenter of ABC
a)GD
b)BG
c)FC
d)BF
if possible can you show the equation break down if some are needed
If G is the incenter of triangle ABC the values of the unknown sides are
a) GD = 12
b) BG = 13.42
c) FC = 35
d) BF = 6
How to find the unknown dimensions
Considering that G is the incenter of triangle ABC, we have that
GE = GD = GFUsing the right triangle CDG and applying Pythagoras theorem given by the formula
hypotenuse² = opposite² + adjacent²
plugging the values as in the problem
GC² = GD² + DC²
37² = GD² + 35²
GD² = 37² - 35²
GD² = 144
take square root of both sides
GD = √144
GD = 12
Using the right triangle BEG to solve for BG
BG² = EB² + GE²
BG² = 6² + 12²
BG² = 180
take square root of both sides
BG = √180
BG = 13.42
Using the right triangle CFG to solve for FC
GC² = GF² + FC²
rearranging the equation
FC² = GC² - GF²
FC² = 37² - 12²
FC² = 1225
take square root of both sides
FC = √1225
FC = 35
Using the right triangle BFG to solve for BF
BG² = GF² + BF²
rearranging the equation
BF² = BG² - GF²
BF² = 13.42² - 12²
BF² = 36.0964
take square root of both sides
BF = √36.0964
BF = 6
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BRAINLIEST ASAP Please help
a rectangular piece of marble is cut such that the width is triple the length and the height is double the width. write a polynomial function v (x) in standard form for the volume of the piece of marble, where a is the length of the marble. v(x) =
A polynomial function v (x) in standard form for the volume of the piece of marble is v(x) = 18a^3.
What is volume?
Volume is a term used in mathematics to describe the volume of three-dimensional space occupied by an object or a closed surface. The measurement of volume is done in cubic units, like m^3, cm^3, in^3, etc. Volume is also referred to as capacity on occasion.
Given:
A rectangular piece of marble is cut such that the width is triple the length and the height is double the width.
We have to write a polynomial function v (x) in standard form for the volume of the piece of marble.
Let 'a' is the length of the marble.
⇒ Width of marble = 3a (width is triple the length)
⇒ Height of marble = 2(3a) = 6a (height is double the width)
⇒ Volume of marble = length x width x height
= a x 3a x 6a
Volume of marble = 18a^3
Hence, a polynomial function v (x) in standard form for the volume of the piece of marble is v(x) = 18a^3.
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{41,57,77,83,95}
what is the set of even numbers that correspond to the given set of odd numbers that satisfy the function is
A. {42,58,78,84,96}
B. {44,60,80,86,98}
C. {82,114,154,166,190}
D. {84,116,156,168,192}
E. {86,118,158,170,194}
Eric invests an amount in a bank that pays compound interest at a rate of 2.16% per year at the end of 5 years , the value of his investment is $6999.31 calculate the amount eric invests
The amount of eric investment would be $ 6290.
What is meant by compound interest?Investors benefit from compound interest, although the term "investors" can indicate many different things. For instance, banks profit from compound interest when they make loans and then reinvest the interest they earn to make more loans. When holders of deposits receive interest on their bank accounts, bonds, or other investments, they also profit from compound interest.
The concept of compound interest applies outside of contexts where the phrase is normally employed, such as bank accounts and loans, despite the fact that the term compound interest includes the word interest.
Let the rate of interest be 2.16 % per year
Time n = 5 years
Amount A = $6999.31
A = P(1 + R/100)ⁿ
substitute the values in the above equation, we get
⇒ 6999.31 = P (1 + 2.16/100)⁵
simplifying the equation, we get
⇒ P = 6290
Therefore, the amount of investment would be $ 6290.
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Consider the following frequency table representing the scores on a test. . Scores on a Test Class Frequency 20-38 7 39-57 5 58-76 9 77-95 5 96-114 3
The answer is :
a) The lower class boundary for the fifth class is 96.
b) The upper-class boundary for the second class is 57.
c) The class width is 18.
d) Number of scores between 19.5 and 95.5 is 26.
a) Fifth class is 96-114.
The lower class boundary for the fifth class is 96.
b) Second class is 39-57.
The upper-class boundary for the second class is 57.
c) Class width = 38- 20 = 18, 57-39 = 18, 76- 58 = 18, 95-76 = 18, 114 - 93=18
So, the class width is 18.
d) Number of scores between 19.5 and 95.5 is = 7 + 5 +9 +5 = 26
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The complete question is:
Consider the following frequency table representing the scores on a test. . Scores on a Test Class Frequency 20-38 7 39-57 5 58-76 9 77-95 5 96-114 3
Required:
a. Determine the lower class boundary for the fifth class.
b. Determine the upper-class boundary for the second class.
c. Determine the class width of each class.
d. Determine the number of scores between 19.5 and 95.5
Is a scale factor of 1/3 an enlargement?
A scale factor of 1/3 is not an enlargement. A scale factor of less than 1 is considered a reduction and not an enlargement.
An enlargement is a transformation that increases the size of a figure, it is characterized by a scale factor greater than 1. A dilation with a scale factor greater than 1 will increase the size of a figure and thus is considered an enlargement.
A dilation with a scale factor of 1/3 will decrease the size of a figure, as each point on the figure will be moved one-third as far from the origin as it was before the dilation. This means that the dilated figure will be smaller than the original figure and will have a scale factor of 1/3.
It's important to note that if the scale factor is less than 1, the dilation will reduce the size of the figure and if the scale factor is greater than 1, the dilation will increase the size of the figure.
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The formula for the area A of a triangle with base length b and height / is as follows.
A
4= 1/2bh
Suppose the height of the triangle is 6 units longer than the base length. Rewrite A in terms of b only.
It is not necessary to simplify.
A =
Answer:
A = 1/2b^2 + 3b
Step-by-step explanation:
If the height of the triangle is 6 units longer than the base length, we can represent the height with the variable h and the base length with the variable b.
h = b + 6
We can substitute h with b+6 in the formula for the area of the triangle:
A = 1/2bh
A = 1/2b(b+6)
So, the area of the triangle can be represented in terms of b only, as
A = 1/2b^2 + 3b
The volume of a cylinder is 90pie cm³. If the radius is 3 cm, what is the height
of the cylinder?
Answer:
The height of the cylinder is 10 cm.
Step-by-step explanation:
1. The volume of a cylinder is given by
-V = pi r^2 h where r is the radius and h is the height
- 90 pi = pi r^2 h
2. We know the radius is 3
-90 pi = pi (3)^2 h
-90 pi = 9 pi h
3. Divide each side by 9 pi
- 90 pi /9 pi = 9 pi h / 9pi
- 10 =h
The height of the cylinder is 10 cm.
The graph of a piecewise function is shown.
The graph shows a piecewise function in 3 pieces. The first piece is curved and increasing from the left through negative 3 comma negative 7 and ends at a closed point negative 2 comma negative 2. The second piece is linear with an open point at negative 2 comma negative 1 and another endpoint at 2 comma 1. The third piece is curved with an endpoint at 2 comma 1, and it goes through the point 3 comma 6 and continues to increase to the right.
What is the domain of the function?
(−∞,∞)
[−∞,∞]
(−∞,0)
(0,∞)
The domain of the piecewise function as described in the task content is; ( −∞ , ∞ ).
What is the domain of the described piecewise function?It follows from the task content that the domain of the piecewise function which is as described in the task content be determined.
The domain of a function is the set of all possible input values of that function.
For the first piece, since the curve increases from left ( -∞ ) through a point and ends at a closed point (-2, -2). Hence, the set of inputs for the first piece in interval notation is; ( -∞, -2 ].
Also, for the second piece which is linear with an open point at (-2, -1) and ends at an endpoint (2, 1). Hence, the domain of the second piece is; [-2, 2].
Finally, the third piece is curved with an endpoint at (2, 1) and continues to increase tonthe right. Hence, the domain of the third piece is; [2, ∞)
Ultimately, by combining all sets of input values as determined; ( -∞, -2 ], [-2, 2] and [2, ∞); the domain of the whole piecewise function is; (−∞,∞).
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Katrina runs 15 miles in 2 1/2 hours what is the average number of minutes it takes her to run 1 mile
Answer: 45 min
Step-by-step explanation:
State the slope and x-intercept of the graph of y = 1/5x – 7.
Answer
slope: 1/5
x-intercept= 35
Step-by-step explanation:
0=1/5(X)-7
7=1/5x
7/1/5= 35
so
x-intercept is: (35,0)
If UY= 3x - 6 and YV = x + 2, find UV.
(WY INTERSECTS UV AT Y)
WY intersects UV at Y and hence the value of UV is 8.
What is point of intersection?They are referred to be intersecting lines when two or more lines in a plane cross each other. The point of intersection, which can be found on all intersecting lines, is the common point shared by the intersecting lines.
Given that WY intersects UV at Y.
Hence, the value of UY = YV.
Substitute the value of UY and YV as follows:
UY = YV
3x- 6 = x + 2
3x - x = 2 + 6
2x = 8
x= 2
Now, the value of UV = UY + YV
Substitute the values:
UV = UY + YV
UV = 3x - 6 + x + 2
UV = 4x - 4
Substitute the value of x as follows:
UV = (4)(2) - 4
UV =8 -4
UV = 4
Hence, the value of UV is 8.
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The Expression (t^6) (t^-14) is equivalent to t^n . What is the value of n
The equation (t6)(t-14) is identical to tn when n = 8. A mathematical expression is composed of numbers .
what is expressions ?You can multiply, divided, add, or take something away in mathematics. An expression is put together as follows: Expression, numeric value, and math operator A mathematical expression is composed of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) Contrasting phrases and emotions is possible. An algebraic expression is any mathematical statement that has elements, quantities, and an arithmetic operation. As an illustration, the phrase 4m + 5 consists of the terms 4m and 5, as well as the given equation's variable m, which are all split by the algebraic sign +.
given
(t^6) (t^-14) = t^n
= 6 -14 = n
n = 8
The equation (t6)(t-14) is identical to tn when n = 8.
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Solve the system of linear equations by graphing please solve all on graph and where both points intercept
Therefore , the solution of the given problem of linear equation comes out to be x = 4 and y =4.
How a linear equation is defined.A model of linear regression is one that applies the formula y=mx+b. B is the slope, and m is the y-intercept. Ignoring the fact that while both y and y are distinct components, the previous sentence is frequently referred as a "mathematical formula with two variables." Bivariate linear equations only have two variables. The application domains of linear equations have zero solutions in all cases. The equation for a single system of equations is Y=mx+b, where m stands for the gradient & b for the y-intercept. Y=mx+b, where m stands for slopes and b for the y-intercept, is how the equation is expressed.
Here,
Given :
=> y = 2x -4
=> x + 2y = 12
and
=> x + 2(2x -4) = 12
=> x + 4x - 8 =12
=> 5x = 20
=> x = 4
=> y = 2(4) - 4
=> y = 8 -4
=> y =4
Therefore , the solution of the given problem of equation comes out to be x = 4 and y =4.
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The function f is defined by f(x) = x^2 +
3x – 10. If f(x - 2) = x^2 + bx + c, what
are the values of b and c?
(Please explain too, I want to know how to do this)
The value of c is -12
What are functions?Function, in mathematics, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given: f(x) = x² +3x – 10 and f(x - 2) = x²+ bx + c
f(x - 2)= (x-2)²+3x-6-10
= x² -4x+4+3x-16
=x²-x-12
Thus comparing it to the given equation we get c=-12
Hence, the value of c is -12
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Aurelia is buying a new computer the shipping cost is 8% of the purchase price if Aurelia’s computer costs $585 how much will it cost to have it shipped show your work
It will cost Aurelia $631.80 to have her new computer shipped.
To calculate the shipping cost, we need to multiply the purchase price of the computer by the percentage of shipping cost, which is 8%.
To do this, we can use the following formula:
Shipping cost = Purchase price x (Shipping percentage/100)
In this case, the purchase price is $585 and the shipping percentage is 8, so we can plug in these values into the formula:
Shipping cost = $585 x (8/100)
Now we can solve the shipping cost:
Shipping cost = $46.80
So the total cost for Aurelia to have her new computer shipped would be
$585 + $46.80 = $631.80.
Thus, it will cost $631.80 for Aurelia to have her new computer shipped.
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A family wants to rent a car to go on vacation. Company A charges $55.50 and 15¢ per mile. Company B
charges $50.50 and 9¢ per mile. How much more does Company A charge for x miles than Company B?
Company A charges $5.00 and 6 cents much more for x miles than Company B. The solution is obtained using arithmetic operations.
What are arithmetic operations?There are four basic operations which are listed as follows: Addition(‘+’): This operation deals with finding the sum. Subtraction(‘-’): This operation deals with finding the difference. Multiplication(‘×’): This operation deals with finding the product. Division(‘÷’): This operation deals with finding the quotient.
In order to find how much more does company A charge than company B, we need to subtract the charges of company B from that of Company A.
So, on subtracting we get,
$55.50 and 15 cents - $50.50 and 9 cents = $5.00 and 6 cents per mile.
For x miles, we multiply $5.00 and 6 cents by x miles.
Also, we could convert dollars to cents by multiplying it by 100.
as $1 =100 cents.
5565 - 5059 cents = 506 cents.
For x miles, 506x cents.
Hence, Company A charges $5.00 and 6 cents much more for x miles than Company B.
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May I have help on this please? thanks if you do!
Answer: 1. linear equation
2. non-linear equation
3. non-linear equation
4. non-linear equation
5. linear equation
Step-by-step explanation:
not sure
Find RS and QR in rectangle PQRS
RS and QR are -8 and -5, respectively, in the rectangle P Q R S. Consequently, v is equal to 23.
What is the value of QR and RS ?here, QR is 3 v - 77, RS is 4 v - 97, and PS is v-23.As a result, PS = v - 23 ⇒ v = 23. As a result, v has a value of 23. So, RS = 4 × 23 - 97 = -5 is used to calculate its value.So, QR = 3 × 23 - 77 = -8 is used to determine its value. A All of the internal angles of a rectangle's four sides are equal to 90 degrees.
Every corner or vertex has a right angle where the two sides meet. It differs from a square in that the length of the opposing sides of the rectangle are equal. First, determine the width w using the area of rectangle formula, which is "w = A/l," to determine the length and width of a rectangle.
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Help? This be mad confusing
Answer:
Angle EFP = 115
Step-by-step explanation:
3x + 2.5 = 4x - 35
2.5 = x - 35
x = 37.5
3(37.5) + 2.5
= 112.5 + 2.5
= 115
Can a triangle be constructed with sides 2.5 cm 3.0 cm and 6.5 cm give reason?
Based on the side-side-side rule, it is possible to construct the triangle.
Let the side a be 2.5 cm, side b be 3.0 cm and side c be 6.5 cm. Now, the sum of any two sides of triangle should be greater than the thrid side of triangle.
Side a + side b > side c
2.5 + 3 > 6.5
Performing addition on Left Hand Side of the equation
5.5 not greater than 6.5
Side b + side c > side a
3 + 6.5 > 2.5
Performing addition on Left Hand Side of the equation
9.5 > 2.5
side c + side a > side b
6.5 + 2.5 > 3
Performing addition on Left Hand Side of the equation
9 > 3
Thus, the right angled triangle can be constructed from sides 2.5 cm 3.0 cm and 6.5 cm.
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Elizabeth has a summer job at the gym. She earns a $30 bonus for each gold membership that she sells and a
$15 bonus for each silver membership that she sells.
Write a polynomial expression that describes Elizabeth’s total bonus
The polynomial expression would be 30G + 15S.
What is the polynomial expression?
A polynomial expression is an algebraic expression that is made up of one or more terms, where each term is the product of a constant coefficient and one or more variables raised to a non-negative integer power.
Elizabeth earns a $30 bonus for each gold membership she sells, and a $15 bonus for each silver membership she sells. Let G be the number of gold memberships she sells and S be the number of silver memberships she sells.
We can express Elizabeth's total bonus as:
$30G + $15S
This is a polynomial expression because it is a sum of two or more terms, each term being the product of a constant coefficient and a non-negative power of a variable.
It is also a linear polynomial expression because the highest power of the variable is 1.
Hence, the polynomial expression would be 30G + 15S.
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Suppose the cost of a new computer is $650 and the sales tax is 7.5%. What is the total cost of this computer, including the tax?
Answer: $698.75
Step-by-step explanation: 7.5% of 650 is $48.75. 650 plus 48.75 = 698.75
Is the statement always, sometimes, or never true? The LCM of 2 numbers is the product of the two numbers. Give at least 2 examples to support your reasoning.
The given statement "The LCM of 2 numbers is the product of the two numbers" is false because the LCM of 2 numbers does not have to be product of the numbers involved.
How to illustrate the LCM?The smallest positive integer that is divisible by both a and b is known as the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b in mathematics and number theory.
The statement is "never" true. The LCM (Least Common Multiple) of two numbers is the smallest multiple that is common to both numbers, whereas the product of two numbers is simply the result of multiplying the two numbers together.
For example, the LCM of 2 and 3 is 6, whereas the product of 2 and 3 is 6. But the LCM of 2 and 4 is 4 , whereas the product of 2 and 4 is 8.
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Suppose your family is going to buy a new air conditioning unit. A more expensive brand cost $2600 to buy but only $50 a month to operate. Let “x” represent time in months.
a. write an equation for the cost of the cheaper brand
b. Write an equation for the cost of the more expensive brand.
c. What is the cost of running the cheaper unit for 36 months?
d. For how many months will the cost of both brands be equal?
An equation for the cost of the cheaper brand is y = 60x + 1,800.
An equation for the cost of the more expensive brand is y = 50x + 2,600.
The cost of running the cheaper unit for 36 months is $3,960.
The number of months for which the cost of both brands would be equal is 8 months.
How to write an equation to model this situation?In order to write an equation that model this situation, we would assign variables to the time and cost respectively, and then translate the word problem into algebraic equation as follows:
Let the variable x represent the time measured in months.Let the variable y represent the cost measured in dollars.Since a brand cost $1800 to buy with a monthly operation fee of $60, an equation for the cost of this cheaper brand is given by:
y = 60x + 1,800.
When x = 36 months, the cost of running the cheaper unit is given by:
y = 60(36) + 1,800.
y = $3,960.
For the more expensive brand, we have:
y = 50x + 2,600.
Next, we would determine the number of months for which the cost of both brands would be equal;
50x + 2,600 = 60x + 1,800
60x - 50x = 2,600 - 1,800
10x = 800
x = 800/10
x = 8 months.
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Complete Question:
Suppose your family is going to buy a new air conditioning unit. One brand costs $1800 to buy and $60 a month to operate. A more expensive brand cost $2600 to buy but only $50 a month to operate. Let “x” represent time in months.
a. write an equation for the cost of the cheaper brand
b. Write an equation for the cost of the more expensive brand.
c. What is the cost of running the cheaper unit for 36 months?
d. For how many months will the cost of both brands be equal?
Calculate the number of full meals a bag of dog food makes. On average, each dog gets 180 g of food and the bag is 50 kg.
Answer:
277.777778 Or about 277 meals
Step-by-step explanation:
We can find this out by converting 50 kg into grams. There are 1000 grams in a kg. So to convert 50 kg into grams we have to multiply by 1000. 50 x 1000 = 50000 g. 50000g / 180 g = 277.777778 or about 277.
Is the linear expression a factor of the polynomial?
Polynomial: 2x³ +15x²-9x-10
Linear Factor. x+8
Missing: 2x³ +15x²- 10Linear
Step-by-step explanation:
This is a two part question. For the first blank, type the letter associated with with
the correct graph of this system. For the second blank, type the correct solution to
the system in point notation: (x,y).
Let
f(x) = (x − 2)² – 3
g(x) = -4x+1
==
A
C
B
D
The solutions to the system of equations in this problem is given as follows:
(0,1) and (0.25, 0).
The graph is given by the image presented at the end of the answer.
How to solve the system of equations?The equations that compose the system in this problem are given as follows:
f(x) = (x - 2)² - 3.g(x) = -4x + 1.We solve the system graphically, hence the solution is given by the point of intersection of the graphs of the two functions.
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