write out the form of the partial fraction decomposition of the function (as in this example). do not determine the numerical values of the coefficients. (a) x − 30 x2 x − 30 (x−5)b (x 6)a (x−5)(x 6)​ (b) x2 x2 x 30

Answers

Answer 1

The partial fraction decomposition of the given expression can be written as

\(\frac{(x-30)}{x2+x-30} = \frac{A}{(x-5)} +\frac{B}{(x+6)}\)

What is the partial fraction and quadratic equation?

The expression which is in the form of a rational equation can be converted to multiple fractions by performing factors of the denominator equation. And later split the fraction expression into two or more simpler fractions which form a partial fraction.

A quadratic expression is a second degree polynomial equation whose coefficients are real numbers. It comes from the word “quad” which means “square”. And that is why the degree of a quadratic equation is two.

According to the question, the factors of the denominator expression which is quadratic in nature:

\((x^{2} +x-30)=(x-5)(x+6)\)

Now, the factors of the given expression are possible. The the partial fraction can be written as:

\(\frac{(x-30)}{x2+x-30} = \frac{A}{(x-5)} +\frac{B}{(x+6)}\)

The factors of the second equation are not possible.

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Related Questions

Calculate 6 ÷ (2-3)³

Answers

Answer:

-6

Step-by-step explanation:

What is the solution to the system of equations?

What is the solution to the system of equations?

Answers

Plug the second equation (the one already solved for y) into the first equation

2x - (2x + 3) = 7

2x - 2x - 3 = 7

-3 = 7

Since the 2x's canceled out and left a false statement, this system of equations has no solution.

Hope this helps!

isiah determined that 5a2 is the gcf of the polynomial a3 – 25a2b5 – 35b4. is he correct? explain.

Answers

No, Isaiah is incorrect. The greatest common factor (GCF) of the polynomial a^3 - 25a^2b^5 - 35b^4 is not 5a^2.


To determine the GCF of a polynomial, we need to find the highest power of each variable that is common to all terms. In this case, the polynomial consists of three terms: a^3, -25a^2b^5, and -35b^4.

To find the GCF, we identify the highest power of each variable that appears in all terms. In this polynomial, the highest power of 'a' is a^3, and the highest power of 'b' is b^5. However, the coefficient -25 in the second term does not contain a common factor of 5 with the other terms. Therefore, 5a^2 is not the GCF of the polynomial.

To determine the GCF, we need to find the common factors among all terms. In this case, both 'a' and 'b' are common factors among all terms. The highest power of 'a' that appears in all terms is a^2, and the highest power of 'b' that appears in all terms is b^4. Thus, the GCF of the polynomial a^3 - 25a^2b^5 - 35b^4 is a^2b^4.

In summary, Isaiah is incorrect in identifying the GCF as 5a^2. The correct GCF of the polynomial a^3 - 25a^2b^5 - 35b^4 is a^2b^4.

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You may use your reference chart.


1. Select all statements that are true about the triangles.


a. Triangles ABC and ADC are congruent by AA


b. Triangles ABC and ADC are congruent by

SAS


C. Triangles ABC and ADC are congruent by

SSS


d. Triangles ABC and CDA are congruent by

ASA


e. Triangles ABC and CDA are congruent by

SSS


f. There is not enough information to determine if the

triangles are congruent

Answers

The given statements about two triangles, ABC and ADC, are as follows:

a. Triangles ABC and ADC are congruent by AA.

d. Triangles ABC and CDA are congruent by ASA.

f. There is not enough information to determine if the triangles are congruent. (option a, d, and f)

After considering additional information, it can be concluded that statement a and f are true. Statement a states that triangles ABC and ADC are congruent by the Angle-Angle (AA) criterion, which means they have two corresponding angles that are equal. Statement f indicates that there is not enough information provided to determine if the triangles are congruent.

In summary, based on the given information, the only true statements about the triangles are:

a. Triangles ABC and ADC are congruent by AA.

f. There is not enough information to determine if the triangles are congruent.

a. Triangles ABC and ADC are congruent by AA (Angle-Angle) if they have two corresponding angles that are congruent. However, since the information provided does not specify any angle measures, we cannot determine if the triangles are congruent based on AA alone.

f. There is not enough information to determine if the triangles are congruent. Without additional information about the side lengths or angle measures, we cannot conclusively determine if the triangles are congruent based on the given information.

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In what ratio is the line segment joining (- 1/3 and 4 7 divided at the point?

Answers

The ratio in which line segment joining (-1, 3) and (4, -7) is is 3: 2 and at a point (2,-3).

What is line segment?

A line segment in geometry has two different points on it that define its boundaries. A line segment is sometimes referred to as a section of a line that links two places. The difference between a line and a line segment is that a line has no endpoints and can go on forever in either direction.

What is segment?

a piece of a geometric figure that has been removed by one or more points, lines, or planes is known as segment.

The formula that we used is:

\(P(x,y)=\frac{mx2+nx1}{m+n},\frac{my2+ny1}{m+n}\)

x = m(x₂) + n(x₁)/(m + n)

⇒ 2 = m(-1) + n(3)/(m + n)

⇒ 2(m + n) = 3n - m

⇒ 2m + 2n = 3n - m

⇒ m/n = 3/2

Therefore, The ratio in which line segment joining (-1, 3) and (4, -7) is is 3: 2 and at a point (2,-3).

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When the sum of the internal angles of a polygon is 10 right angles, then how
many sides does it have?

Answers

Answer:

7 sides

Step-by-step explanation:

formula for sum of interior angles

(n-2)× 180= 10 right angles

(n-2)× 180 = 900

n-2= 900÷180

n= 5+2

n= 7

2. Find the distance between the two points. Round to the nearest tenth if necessary.
(0,9), (-8, -4)
21
15.3
9.4
233​

Answers

Answer:

\(\boxed{D = 15.8\ units}\)

Step-by-step explanation:

The coordinates are (0,9) and (-8,-4)

Distance Formula = \(\sqrt{(x2-x1)^2+(y2-y1)^2}\)

D = \(\sqrt{(-8-0)^2+(-4-9)^2}\)

D = \(\sqrt{(-8)^2+(-13)^2}\)

D = \(\sqrt{81+169}\)

D = \(\sqrt{250}\)

D = 15.8 units

Answer:

15.3

Step-by-step explanation:

The coordinates are (0,9) and (-8,-4)

Distance of two points formula:

\(D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Plug in the values.

\(D=\sqrt{(-8-0)^2+(-4-9)^2}\)

\(D=\sqrt{(-8)^2+(-13)^2}\)

\(D=\sqrt{64+169}\)

\(D=\sqrt{250}\)

\(D= 15.264338...\)

\(D \approx 15.3\)

What is the coefficient of x in 2x + 5 = 8​

Answers

Answer:

2

General Formulas and Concepts:

Algebra I

Leading Coefficients vs Constants

Step-by-step explanation:

The leading coefficient will always be the number multiplied to the highest degree variable.

Our highest degree term is 2x. Our leading coefficient would then be 2.

Adamo saved 13 pictures on his camera and deleted 32 pictures. If there are now 108 pictures, how many pictures had he saved originally

Answers

Answer:

He had saved 127 pictures originally

Explanation:

Parameters:

Pictures saved = 13

Pictures deleted = 32

New total number of pictures = 108

Let old total number of pictures = x

Then

x + 13 - 32 = 108

x - 19 = 108

x = 108 + 19

= 127

He had saved 127 pictures originally

I set z=t=0(x,y,z,t)
and I got a partial solution (0,1,0,0).
I solved two homogeneous matrices once for z=1
and t=0
, then for z=0
and t=1
and I got two solutions (1,1,1,0)
and (1,1,0,1).
Then, I got (0,1,0,0)+a∗(1,1,1,0)+b∗(1,1,0,1
)
Therefore, all possible results are (0,1,0,0),(1,0,1,0),(1,0,0,1),(0,1,1,1)
Would this be correct?

Answers

The correct set of possible results would be (0, 1, 0, 0), (1, 2, 1, 0) and (1, 2, 0, 1).

Your approach seems to be correct, but there seems to be a minor mistake in your final list of possible solutions. Let's go through the steps to clarify.

Given the initial conditions z=t=0, you obtained a partial solution (0,1,0,0).

Next, you solved the homogeneous equations for z=1 and t=0, which resulted in a solution (1,1,1,0).

Similarly, solving the homogeneous equations for z=0 and t=1 gives another solution (1,1,0,1).

To find the general solution, you combine the partial solution with the solutions obtained in the previous step, using parameters a and b.

(0,1,0,0) + a(1,1,1,0) + b(1,1,0,1)

Expanding this expression, you get:

(0+a+b, 1+a+b, 0+a, 0+b)

Simplifying, you obtain the following set of solutions:

(0, 1, 0, 0)

(1, 2, 1, 0)

(1, 2, 0, 1)

Therefore, the correct set of possible results would be:

(0, 1, 0, 0)

(1, 2, 1, 0)

(1, 2, 0, 1)

Note that (0, 1, 1, 1) is not a valid solution in this case, as it does not satisfy the initial condition z = 0.

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a) Looking at the​ histograms, what is the smallest sample size you would be comfortable using the Normal model as an approximation for the sampling​ distribution?

The Normal model would be reasonable to use for a sample size of at least 75.

b) What does the​ Success/Failure Condition say about what sample size should be​ used?

The​ Success/Failure Condition is satisfied for sample sizes of at least 75.

Answers

it is not clear whether the data is related to proportions or not. Therefore, the statement that "the Success/Failure Condition is satisfied for sample sizes of at least 75" may not be applicable in this context.

In statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that is commonly used to approximate the distribution of many random variables. One of the main applications of the normal distribution is to model the sampling distribution of the sample mean. However, the accuracy of the normal approximation depends on the sample size.

To determine the smallest sample size that would be reasonable to use the normal model as an approximation for the sampling distribution, we need to look at the histograms of the data. If the histograms are approximately bell-shaped and symmetric, the normal model is likely to be a good approximation.

In this case, it is stated that the normal model would be reasonable to use for a sample size of at least 75. This means that if the sample size is 75 or larger and the histogram is approximately bell-shaped and symmetric, we can use the normal model to approximate the sampling distribution of the sample mean.

Regarding the Success/Failure Condition, it is a condition that applies when we are using the normal model to approximate the sampling distribution of a proportion. The condition states that if np and n(1-p) are both at least 10, where n is the sample size and p is the probability of success, then we can use the normal model to approximate the sampling distribution of the proportion.

However, in this case, it is not clear whether the data is related to proportions or not. Therefore, the statement that "the Success/Failure Condition is satisfied for sample sizes of at least 75" may not be applicable in this context.

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What is the median of the wave-height distribution? (Round your answer to three decimal places.)For0 < p < 1,give a general expression for the 100pth percentile of the wave-height distribution (p) using the given values of and .(p) =as a model for 1-hour significant wave height (m) at a certain site.

Answers

The 95th percentile of the wave-height distribution would be -0.052 meters.

The median of a distribution is the value that divides the data into two equal halves. To find the median of the wave-height distribution, we need to arrange the wave heights in order from lowest to highest and find the middle value. If there is an odd number of values, then the median is the middle value.

If there is an even number of values, then the median is the average of the two middle values. Since we do not have any data or values given for the wave-height distribution, we cannot determine the median.

The 100pth percentile of the wave-height distribution is the value below which 100p% of the data falls. In other words, if we rank all the wave heights from lowest to highest, the 100pth percentile is the height at which 100p% of the data lies below. A general expression for the 100pth percentile of the wave-height distribution (p) can be given as:

(p) = (1 - p) x + p y

Where x is the wave height corresponding to the (n-1)p-th rank, and y is the wave height corresponding to the np-th rank, where n is the number of observations in the distribution.

Using the model (p) = m as a 1-hour significant wave height (m) at a certain site, we can calculate the 100pth percentile for any given value of p. For example, if p = 0.95, then the 95th percentile of the wave-height distribution would be:

(0.95) = (1 - 0.95) x + 0.95 y

Simplifying this expression, we get:

y = (0.95 - 0.05x)/0.95

Substituting the value of (p) = m, we get:

m = (0.95 - 0.05x)/0.95

Solving for x, we get:

x = (0.95 - 0.95m)/0.05

Therefore, the value of the wave height corresponding to the 5th percentile of the distribution would be:

(0.05) = (1 - 0.05) x + 0.05 y

Simplifying this expression, we get:

x = (0.05y - 0.05)/(0.95)

Substituting the value of (p) = m, we get:

m = (0.05y - 0.05)/(0.95)

Solving for y, we get:

y = (0.95m + 0.05)/(0.05)

Therefore, the value of the wave height corresponding to the 95th percentile of the distribution would be:

(0.95) = (1 - 0.95) x + 0.95 y

Substituting the values of x and y, we get:

(0.95) = (1 - 0.95) [(0.95 - 0.95m)/0.05] + 0.95 [(0.95m + 0.05)/(0.05)]

Simplifying this expression, we get:

m = 0.95y - 0.05x

Substituting the values of x and y, we get:

m = 0.95 [(0.95m + 0.05)/(0.05)] - 0.05 [(0.95 - 0.95m)/0.05]

Simplifying this expression, we get:

m = 19m + 1 - 19

Solving for m, we get:

m = -0.052

Therefore, the 95th percentile of the wave-height distribution would be -0.052 meters.

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(-2,-5)(-3,-6) slope

Answers

Answer: Slope = 1

Step-by-step explanation:

Slope = Y2 - Y1   -6 -- 5      -6 + 5    = -1    =   1

              X2 - X1    -3 --2       -3 + 2       -1

Colin & Brian share a lottery win of £2100 in the ratio 4 : 1.
Colin then shares his part between himself, his wife & their son in the ratio 2 : 5 : 3.
How much more does his wife get over their son?

Answers

Answer:

£336 more

Step-by-step explanation:

2100/5 = 420 One fifth of the total lottery.

Colin has four shares so 4X420 = 1680

He then splits it into 10 (2+3+5) parts. 1680/10=168

Wife: 168X5=840

Son: 168X3=504

Wife-Son 840-504=336

Answer:

336£

Step-by-step explanation:

To get the amount of money colin gets

(2100*4)/5=1680

Lets do a table:

2 is himself5 is his wife3 is his son

2. 5. 3. Sum:10

(2*1680)/10. (5*1680)/10. (3*1680)/10

=336. =840. =504

Then the wife gets 840-504=336£

Given a Figure whose points are A(9, 1), B(2, 3), C(-2, 4).
What would be the image of that figure if it goes under a translation (x, y) -> (x - 5, y + 8), what is the
coordinate of B'?
A. B'(-14, 11)
B. B'(3, 11)
C. B'(14, 11)
D. B'(-3, 11)

Given a Figure whose points are A(9, 1), B(2, 3), C(-2, 4).What would be the image of that figure if

Answers

Answer:

B' (-3,11)

Step-by-step explanation:

B(2,3)  The rule is saying subtract 5 from x and 8 add to y

(2 -5, 3 + 8)

(-3,11)

solve for x for both of these plssss i need help

solve for x for both of these plssss i need help

Answers

Answer:

top one is -7

bottom one is -3

Step-by-step explanation:

A rancher has 3000 feet of fencing with which to construct adjacent, equally sized rectangular pens as shown in the figure above. What dimensions should these pens have to maximize the enclosed area?

Answers

The required dimensions of a rectangular rancher with perimeter for fencing, 3000 feet are equal the 500 feet and 375 feet at maximum area of 3,75,000 ft².

We have a rancher which to construct adjacent, equally sized rectangular pens with the fencing of 3000 feet. Suppose that the rancher need to be fenced in the way shown in the attached figure. Then, the perimeter is 4x + 3y = 3000

=> x \( =\frac{3000 - 3y}{4} \).

The area of rectangle is represented by A=L × W --(1) , where L and W are dimensions of rectangles. The total area will be A = 2× x × y --(2)

=> A = 2× (\(\frac{3000 - 3y}{4}\))× y

= ( \( 1500 - \frac{3y}{2} \))y

= \( 1500y - \frac{3y²}{2} \)

Now, differentiate above area function, with respect to y, and equate to 0, for determining the critical points on the graph, \(\frac{dA}{dy} \) = A'(y) =\( 1500 - \frac{6y}{2} \) = 0

=> y = \(\frac{1500}{3} = 500 \)

Also, x \( =\frac{3000 - 3× 500}{4} \).

\( =\frac{1500}{4} = 375 \).

Maximum area = 375 × 500 × 2 = 375000 ft². Hence, the dimensions that will give the maximum area are 500 feet and 375 feet.

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Complete question : attached figure complete the question.

A rancher has 3000 feet of fencing with which to construct adjacent, equally sized rectangular pens as

Plot the frequency response and the impulse response of the LTI system having the output y = 2te-ul for the input x) = -(t).

Answers

Given an LTI system that has an output y = 2te^-ul for the input x(t) = -(t). Here, e^-ul is the decay constant and is a real positive constant. The impulse response of the system can be found by considering the impulse input x(t) = δ(t).

The output of the system with an impulse input is given by y(t) = h(t), where h(t) is the impulse response of the system.We have,x(t) = -(t)..........(1)Applying derivative on both sides of the above equation, we get,y(t) = 2te^-ul, Applying derivative on both sides of the above equation, we get,

Plot of frequency response and Impulse response of the system Hence, the plot of frequency response and the impulse response of the LTI system is shown in the figure below:, the frequency response is given by H(jω) = (2/(-jω + ul)^2) where ω is the angular frequency. The impulse response of the system is given by h(t) = (2/ul^2)t.e^-ul.

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not every linearly independent set in set of real numbers r superscript ℝn is an orthogonal set. T/F?

Answers

The given statement is true.

Consider linearly independent vectors of \(R_{n}\)

Where n = 2

\(V = \[\begin{array}{ccc}4&\\2\end{array}\right]\)  

\(U = \[\begin{array}{ccc}5&\\6\end{array}\right]\)

Now product of these vectors

UV = 4x5 + 2x6

     = 32

Hence these vectors are not orthogonal.

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Based on Exercise 9, prove that τ and σ are multiplicative. That is, prove that if m and n are relatively prime, then τ(mn)=τ(m)τ(n) and σ(mn)= σ(m)σ(n).

Answers

To prove that τ and σ are multiplicative, we need to show that if m and n are relatively prime, then τ(mn) = τ(m)τ(n) and σ(mn) = σ(m)σ(n).

To prove τ(mn) = τ(m)τ(n), we can start by considering the prime factorization of m and n. Let's say m = p₁^a₁ * p₂^a₂ * ... * pₖ^aₖ and n = q₁^b₁ * q₂^b₂ * ... * qₙ^bₙ, where p₁, p₂, ..., pₖ and q₁, q₂, ..., qₙ are distinct prime numbers. Since m and n are relatively prime, none of their prime factors overlap.

Now, the divisors of mn are the numbers of the form p₁^x₁ * p₂^x₂ * ... * pₖ^xₖ * q₁^y₁ * q₂^y₂ * ... * qₙ^yₙ, where 0 ≤ xᵢ ≤ aᵢ and 0 ≤ yⱼ ≤ bⱼ. Thus, the number of divisors of mn, τ(mn), can be calculated by multiplying the number of divisors of m, τ(m), with the number of divisors of n, τ(n). Hence, τ(mn) = τ(m)τ(n).

To prove σ(mn) = σ(m)σ(n), we can again use the prime factorization of m and n. Similar to the previous proof, the sum of divisors of mn, σ(mn), can be calculated by multiplying the sum of divisors of m, σ(m), with the sum of divisors of n, σ(n). Hence, σ(mn) = σ(m)σ(n).

Therefore, we have proved that τ and σ are multiplicative when m and n are relatively prime.

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We have shown that both τ and σ are multiplicative functions, satisfying the properties τ(mn) = τ(m)τ(n) and σ(mn) = σ(m)σ(n) for relatively prime numbers m and n.

To prove that τ (the number of positive divisors) and σ (the sum of positive divisors) are multiplicative functions, we need to show that for any two relatively prime numbers, m and n, the following properties hold:

1. τ(mn) = τ(m)τ(n)

2. σ(mn) = σ(m)σ(n)

Let's start with property 1:

1. τ(mn) = τ(m)τ(n)

If m and n are relatively prime, it means that they do not share any prime factors. Therefore, the divisors of mn are formed by taking a divisor of m and a divisor of n. Each divisor of mn can be uniquely expressed as the product of a divisor of m and a divisor of n.

Since the number of divisors of mn is equal to the product of the number of divisors of m and the number of divisors of n, we can conclude that τ(mn) = τ(m)τ(n).

Now let's move on to property 2:

2. σ(mn) = σ(m)σ(n)

Similar to property 1, we can express the sum of divisors of mn as the sum of products, where each product is formed by taking a divisor of m and a divisor of n. Again, since m and n are relatively prime, their divisors are distinct.

Using the distributive property, we can expand the sum of products as the sum of two separate terms: one involving the divisors of m and another involving the divisors of n. The sum of divisors of mn is therefore equal to the product of the sum of divisors of m and the sum of divisors of n.

Hence, we can conclude that σ(mn) = σ(m)σ(n).

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8(3x-4)=15x+40
24x-32=15x+40
?
9x=72
x=8

What’s the next step after 24x-32=15x+40

Answers

24x-15x=40+32. I hope this is the answer. Thank you.

Work out the volume of each of this prisms:

Work out the volume of each of this prisms:

Answers

The volumes of right prisms are listed below:

Case A: 150 cm³

Case B: 525 m³

Case C: 429 m³

How to determine the volume of a right prism

The volume of right prism (V), in cubic length units, is always equal to the product of the base area (A), in square length units, and height (h), in length units:

V = A · h

There are three right prisms, whose areas are combinations of right triangles and rectangles. The area formulas are shown below:

Rectangle

A = w · l

Right triangle

A = 0.5 · w · l

Where:

w - Width, in length units.l - Length, in length units.

The complete procedure is listed below:

Determine the base area.Calculate the volume of prism.

Case A

A = 0.5 · (4 cm) · (5 cm) + (3 cm) · (5 cm)

A = 25 cm²

V = (25 cm²) · (6 cm)

V = 150 cm³

Case B

A = (5 m) · (8 m) + (5 m) · (7 m)

A = 75 m²

V = (75 m²) · (7 m)

V = 525 m³

Case C

A = (8 m) · (4 m) + (3 m) · (9 m) + 0.5 · (5 m) · (5 m)

A = 71.5 m²

V = (71.5 m²) · (6 m)

V = 429 m³

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Use the given transformation to evaluate the integral. double integral 9xy dA R , where R is the region in the first quadrant bounded by the lines y = 2 3 x and y = 3x and the hyperbolas xy = 2 3 and xy = 3; x = u/v, y = v

Answers

It looks like the boundaries of \(R\) are the lines \(y=\dfrac23x \) and \(y=3x\), as well as the hyperbolas \(xy=\frac23\) and \(xy=3\). Naturally, the domain of integration is the set

\(R = \left\{(x,y) ~:~ \dfrac{2x}3 \le y \le 3x \text{ and } \dfrac23 \le xy \le 3 \right\}\)

By substituting \(x=\frac uv\) and \(y=v\), so \(xy=u\), we have

\(\dfrac23 \le xy \le 3 \implies \dfrac23 \le u \le 3\)

and

\(\dfrac{2x}3 \le y \le 3x \implies \dfrac{2u}{3v} \le v \le \dfrac{3u}v \implies \dfrac{2u}3 \le v^2 \le 3u \implies \sqrt{\dfrac{2u}3} \le v \le \sqrt{3u}\)

so that

\(R = \left\{(u,v) ~:~ \dfrac23 \le u \le 3 \text{ and } \sqrt{\dfrac{2u}3 \le v \le \sqrt{3u}\right\}\)

Compute the Jacobian for this transformation and its determinant.

\(J = \begin{bmatrix}x_u & x_v \\ y_u & y_v\end{bmatrix} = \begin{bmatrix}\dfrac1v & -\dfrac u{v^2} \\\\ 0 & 1 \end{bmatrix} \implies \det(J) = \dfrac1v\)

Then the area element under this change of variables is

\(dA = dx\,dy = \dfrac{du\,dv}v\)

and the integral transforms to

\(\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \int_{\sqrt{2u/3}}^{\sqrt{3u}} \frac{dv\,du}v\)

Now compute it.

\(\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \ln|v|\bigg|_{v=\sqrt{2u/3}}^{v=\sqrt{3u}} \,du \\\\ ~~~~~~~~ = \int_{2/3}^3 \ln\left(\sqrt{3u}\right) - \ln\left(\sqrt{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln(3u) - \ln\left(\frac{2u}3\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln\left(\frac{3u}{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \int_{2/3}^3 du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \left(3-\frac23\right) = \boxed{\frac76 \ln\left(\frac92\right)}\)

A random variable follows a binomial distribution with a probability
of success equal to 0.66. For a sample size of n = 6, find the
values below.
a. the probability of exactly 4 successes
b. the probability of 5 or more successes
c. the probability of exactly 6 successes
d. the expected value of the random variable

Answers

a. The probability of exactly 4 successes is approximately 0.2967.

b. The probability of 5 or more successes is approximately 0.5332.

c. The probability of exactly 6 successes is approximately 0.1399.

d. The expected value of the random variable is 3.96

To solve these problems, we'll use the binomial probability formula:

P(X = k) = C(n, k)× \(p^{k}\)× \((1-p)^{(n-k)}\)

where:

P(X = k) is the probability of getting exactly k successes,

n is the sample size,

p is the probability of success,

C(n, k) is the number of combinations of n items taken k at a time.

Now let's solve each part of the problem:

a. The probability of exactly 4 successes:

P(X = 4) = C(6, 4) × (0.66)⁴ × (1 - 0.66)⁽⁶⁻⁴⁾

C(6, 4) = 6! / (4! × (6 - 4)!) = 6! / (4! × 2!) = (6 × 5) / (2 × 1) = 15

P(X = 4) = 15 × (0.66)⁴ × (0.34)² ≈ 0.2967 (rounded to four decimal places)

b. The probability of 5 or more successes:

P(X ≥ 5) = P(X = 5) + P(X = 6)

P(X = 5) = C(6, 5) × (0.66)⁵ × (1 - 0.66)⁽⁶⁻⁵⁾ = 6 × (0.66)⁵ × (0.34)¹ ≈ 0.3933

P(X = 6) = C(6, 6) × (0.66)⁶ × (1 - 0.66)⁽⁶⁻⁶⁾ = 1 × (0.66)⁶× (0.34)⁰ = 0.1399

P(X ≥ 5) = P(X = 5) + P(X = 6) = 0.3933 + 0.1399 ≈ 0.5332 (rounded to four decimal places)

c. The probability of exactly 6 successes:

P(X = 6) = C(6, 6) × (0.66)⁶ × (1 - 0.66)⁽⁶⁻⁶⁾ = 1 × (0.66)⁶ × (0.34)⁰= 0.1399

d. The expected value of the random variable:

The expected value (mean) of a binomial distribution is given by:

E(X) = n × p

E(X) = 6 × 0.66 = 3.96

Therefore:

a. The probability of exactly 4 successes is approximately 0.2967.

b. The probability of 5 or more successes is approximately 0.5332.

c. The probability of exactly 6 successes is approximately 0.1399.

d. The expected value of the random variable is 3.96

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An experiment has 12 possible outcomes, all equally likely. An event can occur in 6 ways. Find the probability that the event occurs.
The probability that the event occurs is (Type an integer or a decimal. Do not round.)

This is on mymathlab please help

Answers

Answer:

P(event occuring) = 0.5

Steps:

P(event occuring) = 6/12

P(event occuring) = 1/2

P(event occuring) = 0.5

Probabilities are used to determine the chance of an event.

The probability that the event occurs is 0.5

The given parameters are:

\(\mathbf{Favorable\ Outcomes = 12}\)

\(\mathbf{Outcomes = 6}\)

So, the probability that the event occurs is:

\(\mathbf{Pr = \frac{Outcomes}{Favorable\ Outcomes}}\)

This gives

\(\mathbf{Pr = \frac{6}{12}}\)

Simplify

\(\mathbf{Pr = \frac{1}{2}}\)

Divide

\(\mathbf{Pr =0.5}\)

Hence, the probability that the event occurs is 0.5

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A right circular cone with a radius of 3 cm has a slant height of 5 cm. A right cylinder with a radius of 4 cm has a height of 6 cm. What is the number of full cones of water needed to completely fill the cylinder with water?.

Answers

About 3 full cones of water are needed to be added to the cylinder to completely fill it.

The radius of the right circular cone is 3cm and the slant height is 5 cm.

The height of the cone can be found as,

H = √(5)²- (3)²

H = 4cm

The volume of the cone is given by,

V = πR²H/3

R is the radius and H is the height of the cone,

Putting values,

v = π x 5 x 5 x 4/3

v = 104.71 cm³

Now, the height h of cylinder is given to be 6 cm and the radius r of the cylinder is 4cm.

The volume of the cylinder is given as,

V = πr²h

V = π x 4 x 4 x 6

V = 301.59

Let us assume that we have to add n full cones of water to completely fill the cylinder, so, we can write,

nv = V

Putting values,

n104.71 = 301.59

n = 2.85

So, we have to add roughly three full cones of water to completely fill the cylinder.

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NEED HELP IMMEDIATELY!

Simplify 10√2y + 5√2y + 3√2y.
A. 18√6y
B. 18√2y
C. 12√2y
D. 18√6y^3

(ANSWER IS NOT A)

Answers

18 root

2

Step-by-step explanation:

In this question imagine there is no y

It will be 10 root2 +5 root2 +3 root 2 it will be 18 root 2

10 + 5 + 3 = 18 therefore it is B

A researcher interviews 6 widows about their marriages and notices how many cats are wandering around. Is there a significant relationship between the number of times an old widow was married and the number of cats the old lady owns? ( You don't need to do the math to calculate it - the Pearson r is given).
Times Married: 1 1 2 2 3 3
Cats Owned: 3 2 4 5 5 6
Pearson r = +.91
Write up the conclusion for this study in APA format and be sure to include the r2.

Answers

There is a significant relationship between the number of cats she owns and the number of times an old widow was married (r = +0.91, p < 0.05, r² = 0.82).

Given, the Pearson correlation coefficient of +0.91,

There appears to be a strong +ve correlation between the number of cats she owns and the number of times an old widow was married.

It suggests that the more times a widow was married,the more cats she tends to own.

Approximately 82% of the variance in the number of cats owned can be explained by the number of times a widow was married is indicated by the coefficient of determination (r²).

Hence, we can say that there is a significant relationship between the number of cats she owns and the number of times an old widow was married (r = +0.91, p < 0.05, r² = 0.82).

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Part G
You’ve seen two methods for finding the area of ΔABC—using coordinate algebra (by hand) and using geometry software. How are the two methods similar? How are they different? Why might coordinate algebra be important in making and using geometry software?

Answers

The methods are similar as they both apply the fundamentals of triangle geometry in solving the area of a triangle using the formula for area of triangle 1/2×base of triangle×height of triangle.

How to convey the information

To find the area of a triangle using coordinate algebra, you would need to measure the triangle's coordinates (x,y) manually on the graph and then calculate the y coordinates based on the x coordinates. If you used a geometry program, it would automatically obtain these coordinates (depending on the graph you created in the program) and compute the triangle's area using the formula.

For the last question, it is easier to customize and get the best performance out of the program for your geometry calculations if you have a firm understanding of coordinate.

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Part GYouve seen two methods for finding the area of ABCusing coordinate algebra (by hand) and using

A cylinder with a base diameter of x units has a volume of πx3 cubic units. A cylinder with a base diameter of x units has a volume of pi x cubed cubic units. Which statements about the cylinder are true? Select two options. The radius of the cylinder is 2x units. The area of the cylinder’s base is One-fourthπx2 square units. The area of the cylinder’s base is One-halfπx2 square units. The height of the cylinder is 2x units. The height of the cylinder is 4x units.

Answers

Answer:

The height of the cylinder is 4 x units.

The area of the cylinder’s base is One-fourthπx2 square units

Step-by-step explanation:

Formula for volume of the cylinder:

V = r² π h

Volume of the cylinder=

πx^3

Diameter=x

Radius (r)=diameter/2

=x/2

V = r² π h

πx^3=(x/2)^2πh

πx^3=(x^2/4)πh

Divide both sides by π

x^3=(x^2/4)h

Make h the subject of the formula

h=x^3÷x^2/4

=x^3×4 / x^2

=4x^3 / x^2

=4*x*x*x / x*x

h=4x

Area of the base:

B = r² π

Recall, r=x/2

B=(x/2)^2 * π

=(x^2/4)π

=πx^2/4

=1/4(πx^2)

The area of the cylinder’s base is One-fourthπx2 square units.

Answer:

B & E

Step-by-step explanation:

Edge 2020

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