It looks like the boundaries of [tex]R[/tex] are the lines [tex]y=\dfrac23x [/tex] and [tex]y=3x[/tex], as well as the hyperbolas [tex]xy=\frac23[/tex] and [tex]xy=3[/tex]. Naturally, the domain of integration is the set
[tex]R = \left\{(x,y) ~:~ \dfrac{2x}3 \le y \le 3x \text{ and } \dfrac23 \le xy \le 3 \right\}[/tex]
By substituting [tex]x=\frac uv[/tex] and [tex]y=v[/tex], so [tex]xy=u[/tex], we have
[tex]\dfrac23 \le xy \le 3 \implies \dfrac23 \le u \le 3[/tex]
and
[tex]\dfrac{2x}3 \le y \le 3x \implies \dfrac{2u}{3v} \le v \le \dfrac{3u}v \implies \dfrac{2u}3 \le v^2 \le 3u \implies \sqrt{\dfrac{2u}3} \le v \le \sqrt{3u}[/tex]
so that
[tex]R = \left\{(u,v) ~:~ \dfrac23 \le u \le 3 \text{ and } \sqrt{\dfrac{2u}3 \le v \le \sqrt{3u}\right\}[/tex]
Compute the Jacobian for this transformation and its determinant.
[tex]J = \begin{bmatrix}x_u & x_v \\ y_u & y_v\end{bmatrix} = \begin{bmatrix}\dfrac1v & -\dfrac u{v^2} \\\\ 0 & 1 \end{bmatrix} \implies \det(J) = \dfrac1v[/tex]
Then the area element under this change of variables is
[tex]dA = dx\,dy = \dfrac{du\,dv}v[/tex]
and the integral transforms to
[tex]\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \int_{\sqrt{2u/3}}^{\sqrt{3u}} \frac{dv\,du}v[/tex]
Now compute it.
[tex]\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \ln|v|\bigg|_{v=\sqrt{2u/3}}^{v=\sqrt{3u}} \,du \\\\ ~~~~~~~~ = \int_{2/3}^3 \ln\left(\sqrt{3u}\right) - \ln\left(\sqrt{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln(3u) - \ln\left(\frac{2u}3\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln\left(\frac{3u}{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \int_{2/3}^3 du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \left(3-\frac23\right) = \boxed{\frac76 \ln\left(\frac92\right)}[/tex]
Simplify b^10/b^2
A. b^5
B. b^-5
C. b^8
D. b^-8
Answer:
C. b^8.
Step-by-step explanation:
b^10/b^2 We subtract the exponents:-
= b^(10-2)
= b^8.
b^8
Step-by-step explanation:
b^10/b^2 ,. b^10-2 ;. b^8
A legal document contains pages numbered 1 through 50. What is the probability that the lawyer will flip to a page number that is not a multiple of 4
Using its concept, it is found that the probability that the lawyer will flip to a page number that is not a multiple of 4 is given by:
p=19/25
Probability is a field of mathematics that deals with the numerical description of the likelihood that an event will occur or that a statement is true. The probability of an event is a number between 0 and 1, with approximately 0 indicating the impossibility of the event and 1 indicating certainty.
A probability is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, there are 50 numbers, of which 50 - (48/4) = 38 are not multiples of 4, hence the probability is given by:
p=38/50 = 19/25
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Surface area=
Help me please thanks so much :))
Answer:
324[tex]\pi[/tex] units³
Step-by-step explanation:
Surface Area of Sphere = 4[tex]\pi[/tex]r²; where 'r' is the radius of the sphere.
4[tex]\pi[/tex]r² =
4[tex]\pi[/tex]9² =
4[tex]\pi[/tex]81 =
324[tex]\pi[/tex] units³
Answer:
324[tex]\pi[/tex] unit^2
Step-by-step explanation:
The formula for finding the surface area of a sphere is: [tex]A = 4\pi r^2[/tex]
--> A = Surface area of the sphere
--> r = Radius of the sphere
The image shows that the radius of the sphere is 9.
All we need to do is to replace r with 9.
--> A = 4[tex]\pi[/tex]9^2
--> A = 4[tex]\pi[/tex]81
--> A = 324[tex]\pi[/tex]
So that is your final answer! 324[tex]\pi[/tex] unit^2
--> It is unit^2 since it is an area, not a volume.
+ I'm leaving this as [tex]\pi[/tex], since the question is asking for the exact value, not the ones that are rounded up.
Can someone help me?
Answer:
discriminant = b^2 - 4ac
Step-by-step explanation:
discriminant = b^2 - 4ac
2^2 - 4x 3x2 = 4-24 = 20
how many 2/7 are 150
Answer:
42.8571
Step-by-step explanation:
We would need to divide 150 by 7 and multiply it by 2 to get the answer.
150/7 = 21.42857
21.42857*2 = 42.8571
division of 3 fraction
(2/5 ÷ 3/8 ) ÷ -3/5
Answer:
-16/9
Step-by-step explanation:
( [tex]\frac{2}{5}[/tex] ÷ [tex]\frac{3}{8}[/tex] ) ÷ [tex]-\frac{3}{5}[/tex]
= ( [tex]\frac{2}{5} * \frac{8}{3}[/tex] ) ÷ [tex]-\frac{3}{5}[/tex]
= [tex]\frac{16}{15}[/tex] ÷ [tex]-\frac{3}{5}[/tex]
= [tex]\frac{16}{15} * -\frac{5}{3}[/tex]
= -16/9
If a polynomial f(x) is divided by (x+a) and leaves the reminder is a and b are constants then
f(-a) is the remainder when f(x) is divided by (x+a). This can be obtained by remainder theorem for polynomials.
What is the required remainder?Given that f(x) is divided by (x+a) and leaves a reminder
Using the remainder theorem for polynomials we get,
f(x) = (x+a)·g(x) + r, where g(x) is the quotient and r is the remainder.
Put x = -a, then
f(-a) = (-a+a)·g(-a) + r
f(-a) = (0)·g(x) + r
f(-a) = r
f(-a) is the remainder.
Hence f(-a) is the remainder when f(x) is divided by (x+a).
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May someone please explain to me how exponents can be used to represent fractions between 0 and 1???
The fraction m/n that lies between 0 and 1 is equivalent to the real number m · n⁻¹, where m, n > 0 and m < n.
How can rational numbers between 0 and 1 be represented in terms of exponents?
By algebra, we know that fractions, a kind of rational numbers, are real numbers of the form m/n, where m and n are integers and n ≠ 0.
In addition, the operator "/" means division, an operator derived from the existence of multiplicative inverse and fundamental operation of multiplication in the real field.
Thus, the rational number m/n is equivalent to the number m · n⁻¹. If the rational number is between 0 and 1, then m, n > 0 and m < n.
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please help im in a rush
Answer:
a)
Surface Area: 172 cm^2
Volume: 112 cm^3
b) Surface Area
c) Volume
Step-by-step explanation:
find the number set which satifies each of the problems If 20 is added to the number, the absolute value of the result is 6
Answer:
{-14,-26}
Step-by-step explanation:
lx+20l=6
So x+20 is either 6 or -6
x=6-20=-14
x=-6-20=-26
In the context of data models, relationships among records in a hierarchical model are called _____.
In the context of data models, relationships among records in a hierarchical model are called Branches. This is further explained below.
What are data models,?Generally, The act of generating a visual representation of either a whole information system or portions of it in order to express the linkages between different data points and structures is known as data modeling.
In conclusion, When discussing data models, the connections between different records in a hierarchical structure are referred to as "Branches."
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Let x represent the number of short-sleeved shirts ordered and let y represent the number of long-sleeved shirts ordered. which system of linear equations represents the situation? 5x 10y = 100 and 5 x plus 10 y equals 100 and startfraction one-third endfraction x plus startfraction one-half endfraction y equals 1,750.x startfraction one-half endfractiony = 1,750 5x 10y = 1,750 and 5 x plus 10 y equals 1,750 and startfraction one-third endfraction x plus startfraction one-half endfraction y equals 100.x y = 100 x y = 100 and x plus y equals 100 and startfraction one-third endfraction x plus startfraction one-half endfraction y equals 1,750.x y = 1,750 x y = 1,750 and x plus y equals 1,750 and startfraction one-third endfraction x plus startfraction one-half endfraction y equals 100.x y = 100
The system of linear equations that represent the given situation is:
5x + 10y = 1750 and 1/3x + 1/2y = 100.
What is a linear equation?A linear equation is an equation with the highest degree of the variable is one. There may be one or more variables but their degree must be 1 to become a linear equation.
The general form of a linear equation is y = mx + c.
Calculation:
In the given situation,
Consider x as the number of short-sleeved shirts ordered and y as the number of long-sleeved shirts ordered.
It is given that,
Price for short-sleeved shirts = $5
Price long-sleeved shirts =$10
The total price for 100 shirts ordered = $1,750
So, we can write the linear equation as
5x + 10y = 1750 ...(i)
It is given that,
After the first week, they sold 1/3 of the short-sleeved shirts and 1/2 of the long-sleeved shirts. I.e.,
1/3x + 1/2y = 100 ...(ii)
Thus, from equations (i) and (ii), the system of linear equations that represent the given situation is
5x + 10y = 1750
1/3x + 1/2y = 100
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Disclaimer: The given question is incomplete. Here is the complete question.
Question:
The drama club is selling short-sleeved shirts for $5 each, and long-sleeved shirts for $10 each. They hope to sell all of the shirts they ordered, to earn a total of shirts ordered, and to earn a total of $1,750. After the first week of the fundraiser, they sold 1/3 of the short-sleeved shirts and 1/2 of the long-sleeved shirts, for a total of 100 shirts.
Let x represent the number of short-sleeved shirts ordered and let y represent the number of long-sleeved shirts ordered. which system of linear equations represents the situation?
Answer:
c
Step-by-step explanation:
Which of the following sets is equal to {1, 2, 3, ...}?
{x | x R, x ≥ 1}
{x | x R, x > 1}
{x | x N, x ≥ 1}
From the list of options, the set that is equal to the set {1, 2, 3, ...} is {x | x N, x ≥ 1}
How to determine the set?The set is given as:
{1, 2, 3, ...}
The three dots (...) after 3 implies that the elements of the set include other values such as 4, 5, 6 and so on
Using the above scenario, we can see that the set {1, 2, 3, ...} contains only positive integers.
All positive integers are natural numbers and they are denoted by N
From the list of options, we have:
{x | x R, x ≥ 1}
This represents the set of all real numbers greater than or equal to 1.
Note that this set includes decimal numbers i.e. 1, 1.5, 1.89 and so on
This does not represent the set {1, 2, 3, ...}
{x | x R, x > 1}
This represents the set of all real numbers greater than 1.
Note that this set includes decimal numbers i.e. 1.5, 1.89 and so on
This does not represent the set {1, 2, 3, ...}
{x | x N, x ≥ 1}
This represents the set of all natural numbers greater than or equal to 1.
Note that this set does not include decimal numbers i.e. 1, 2, 3, 4 and so on
This does represent the set {1, 2, 3, ...}
Hence, the set that is equal to the set {1, 2, 3, ...} is {x | x N, x ≥ 1}
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Given a polynomial function f(x) = 2x2 – 3x 5 and an exponential function g(x) = 2x – 5, what key features do f(x) and g(x) have in common? both f(x) and g(x) have the same domain of ([infinity], -[infinity]). both f(x) and g(x) have the same range of [0, -[infinity]). both f(x) and g(x) have the same x-intercept of (2, 0). both f(x) and g(x) increase over the interval of [-4 , [infinity]).
Both f(x) and g(x) have the same domain of (9,-infinity).
What is a function?A function is an expression that shows the relationship between two or more numbers and variables.
Plotting the functions f(x) = 2x² – 3x + 5 and g(x) = 2ˣ - 5:
Both f(x) and g(x) have the same domain of (9,-infinity).
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On a coordinate plane, a line goes through (negative 4, 0) and (4, negative 4). a point is at (2, 3). what is the equation of the line that is parallel to the given line and passes through the point (2, 3)? x 2y = 4 x 2y = 8 2x y = 4 2x y = 8
The equation of the line that is parallel and passes through the point (2,3) is; x +2y =8.
What is the equation of the line that passes through (2,3)?If follows from the task content that the slope of the first line is; (-4-0)/(4-(-4)) = -1/2.
Hence, since the lines are parallel, they have the same slope and the equation of the second line is;
-1/2 = (y-3)/(x-2)
-x+2 = 2y -6
x +2y =8.
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Answer: b on edg
Step-by-step explanation:
Suppose . has 4 critical points. List them in increasing lexographic order. By that we mean that (x, y) comes before (z, w) if or if and . Also, determine whether the critical point a local maximum, a local minimim, or a saddle point.
The points are (0,0) Minimum and (1/6, 1/18) is the saddle point.
What is the saddle point?
A saddle point, also known as a minimax point, is a place on the surface of a function's graph where the slopes in orthogonal directions are all zero but the function does not have a local extremum.To solve the question we have:
f(x,y) = xy(1-2x-6y)
Therefore, f(x,y) = xy-2x²y-6xy²
Finding the partial derivative of the above gives:
fₓ = y - 4xy-6y²
= x - 2x²-12xy
Therefore, when the above equations are set to 0 we get:
y - 4xy-6y² = 0.......(1)
x - 2x²-12xy = 0......(2)
Dividing equations (1) and (2) by y and x respectively give:
1 - 4x - 6y = 0
1 - 2x - 12y = 0
Solving gives:
x = 0.167 = 1/6
y = 0.056 = 1/18
When we place x in equation (1) we have:
y - 4(1/6)y-6y² = 0
y = 0 or y = 1/18
Similarly, x = 0 or 1/6
Therefore, the four critical points are:
(0,0) (1/6,1/18)
When x = 0 and y = 0
f(x,y) = 0 Hence this is a minimum
When x = 1/6 and y = 1/18
f (x,y) = 1/324 Which is a saddle point.
Therefore, the points are (0,0) Minimum, and (1/6, 1/18) is the saddle point.
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The question you are looking for is here:
(1 point) The function f(x,y)=xy(1−2x−6y) has 4 critical points. List them and select the type of critical point. Points should be entered as ordered pairs and listed in increasing lexicographic order. By that, we mean that (x,y) comes before (z,w) if x
PLEASE HELP IM SO STUCK
If the graph of the line falls from left to right the slope is negative.
Slope = (y2 - y1)/(x2 - x1)
---> y = -2x + 3
A college professor noted that the grades of his students in an introductory statistics class were normally distributed with a mean of 54.50 and a standard deviation of 9. If 67.66% of his students received grades of C or above, what is the minimum score of those students receiving a grade of at least a C
Students who receive a grade of at least a C must have a minimum score of 50.38.
Calculation of the minimum scoreWe may assume that the cut-off will be below 67.66 % because that percentage is more than 50%.
To get the value of 67.66 - 50 = 17.66, we must search in a regular normal table.
Hence, A z value of 0.458 will cause that value to occur.
Next, increase the standard deviation of 9 by 0.458.
The result is 4.122.
This indicates that the threshold for a student earning a C or higher is the mean minus 4.122.
So, the final result is 54.50-4.122 = 50.38.
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What is the slope of the line that passes through the points ( 8 , − 6 ) (8,−6) and ( 8 , − 9 ) (8,−9)? Write your answer in simplest form
As slope is undefined, then the slope of the line is parallel to the x axis.
How to determine the slope of a line
In this problem we have a line that passes through two points, the slope is determined by definition of secant line:
m = [- 9 - (- 6)]/(8 - 8)
m = - 3/0
As slope is undefined, then the slope of the line is parallel to the x axis.
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8) Find m/A.
Need the answer asap please some1 respond
The measure of angle A in the isosceles trapezoid is: 106°.
What are the Angles in an Isosceles Trapezoids?In an isosceles trapezoid, the base angles are equal while the opposite angles are supplementary or have a sum of 180°.
The diagram given is an isosceles trapezoid. Therefor we would have:
m∠A = 180 - 74 [supplementary angles]
m∠A = 106°
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Faith purchased a new car in 1998 for $32,900. the value of the car has been depreciating exponentially at a constant rate. if the value of the car was $19,900 in the year 2003, then what would be the predicted value of the car in the year 2007, to the nearest dollar?
The predicted value of the car in the year 2007 is 13310
Given the purchased value of new car in 1998 is $ 32,900
The value of car was $ 19,900 in the year 2003
We need to find the value of the car in the year 2007
Let the predicted value be x
The exponentially constant rate in 2003
32900*x^5 = $19900 (∵ x^5 = 5 years difference that is 1998 to 2003 )
Now In year 2007
2007 = 32900*x^9
x = 0.9043
The predicted value of the car is 0.09043
But we want the predicted value in 2007
Therefore ,
In 2007 ≈ x = 13310.02
Rounding off to the nearest dollar is $13310
Therefore the predicted value of the car in 2007 is $13310
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[tex]\mathbb Question[/tex]
Attached above!!
Answer:
see explanation
Step-by-step explanation:
(a)
play ends at 9 : 55 pm
15 minutes to get to bus stop , then time is
9 : 55 + 15 min = 10 : 10 pm = 22 : 10
the earliest bus she can catch is 22 : 25
(b)
time to wait 22 : 10 → 22 : 25 = 15 minutes
A company has $36,000 to hire a mathematician and his or her secretary. Out of respect for the mathematician's training, the mathematician will be paid $8000 more than the secretary. How much will each be paid
The money paid to the secretary is $14000 and the money paid to mathematician is $22000.
What is an equation?A mathematical statement that establishes the equality of two mathematical expressions is the called an equation.
Total money available for hiring = $36000
Let the money given to secretary = X
Money given to the mathematician = X + 8000
X + (X + 8000) = 36000
2X = 28000
X = 14000
Money given to secretary is $14000.
Money given to the mathematician is $22000.
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convert this hight 5,2 into cm
Answer:
5.2 Inch to cm = 13.208 cm
5.2 feet to cm = 158.496 cm
Step-by-step explanation:
Inch: 1 inch = 2.54 cm, so multiply 2.54 * 5.2. We get the answer: 13.208 cm
Feet: 1 foot = 30.48 cm, so multiply 30.48 * 5.2. We get the answer: 158.496 cm
The shallowest section of an underground tunnel is -50 feet. The deepest section of the tunnel is -135 feet. How many feet below the shallowest section is the deepest section?
A. -95 feet
B. -185 feet
C. 85 feet
D. 195 feet
Answer:
c
Step-by-step explanation:
1) eliminate the negative answers since we need the amount of feet below a section and a negative number below a section doesn't make sense in this context and would result in a distance higher than the shallowest section.
Just find the absolute value or the difference between 135 and 50 resulting in 85
Which one of the following foods is the top source of protein chosen by Americans in the United States?
Answer:
The correct answer is beef
Step-by-step explanation:
I took the quiz and got it correct
A plane can cover 9904 km in 8 hours how mucb distance will it cover in 1 hour?
In one hour the plane will cover a distance of 1238 km.
What is a linear equation?
A linear equation has one or two variables. No variable in a linear equation is raised to a power greater than 1.No variable is used as the denominator of a fraction. A linear equation is defined as an equation that is written in the form of ax+by=c. When solving the system of linear equations, we will get the values of the variable, which is called the solution of a linear equation.
solving this we will get the valve of Y is x is given.
calculation:-
today distance = 9904 km.
total time is taken = 8 hours.
∴total distance covered in one hour = 9904/8
= 1238 km.
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Read the following description of a relationship:
Ron is making a banner by stringing letters on a cord. The cord weighs 8
ounces, and each letter weighs 10 ounces.
Let n represent the number of letters and w represent the weight of the banner.
This relationship can also be shown in a table. Complete the equation that represents the
relationship between n and w
The equation that represents the relationship between n and w is w = 10n + 8.
How to illustrate the expression?From the information, Ron is making a banner by stringing letters on a cord and the cord weighs 8
ounces and each letter weighs 10 ounces.
Let represent the number of letters
Let w represent the weight of the banner.
In this case, when n = 6, w = 68 and when n = 7, w = 78.
Therefore, the relationship will be:
w = 10n + 8
To confirm let's use n = 6
10n + 8
10(6) + 8
= 68
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Answer:
Step-by-step explanation:
w=10n+8
Need help understanding how to solve
For the two figures to be similar, the missing length must be 30.
What is the missing length?For the two figures to be similar, the dimension of each figures must be proportional to each other.
Let the missing length be x.
Hence,
x/25 = 24/20
We cross multiply
x × 20 = 24 × 25
x × 20 = 600
x = 600/20
x = 30
Therefore, for the two figures to be similar, the missing length must be 30.
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On a particular vehicle, the front tire makes three revolutions for every one revolution the back tire makes. How many times larger is the radius of the back tire than the radius of the front tire?
Answer:
3 times larger
Step-by-step explanation:
Front tire circumference = pi *d
rear tire is 3 times this = pi 3 d
so rear tire has radius 3x that of the front tire