find the area under the curve y = 37 x3 from x = 1 to x = t.

Answer:

70 percent.

Step-by-step explanation:

To calculate the probability that the next customer will pay with cash or a credit card, we need to add the number of customers who paid with cash (69) to the number of customers who paid with a credit card (11):

69 + 11 = 80

So out of the total number of customers, 80 paid with cash or a credit card. To express this as a percentage to the nearest whole number, we need to divide 80 by the total number of customers (69 + 34 + 11 = 114) and then multiply by 100:

(80 / 114) x 100 ≈ 70

Therefore, the probability that the next customer will pay with cash or a credit card is approximately 70 percent.

The surface area of the cylinder is 32π units²

A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance. The base of a cylinder is circular and it's volume is given by ; V = πr²h

The surface area of a cylinder is expressed as;

SA = 2πr( r+h)

where r is the radius and h is the height.

radius = 2 units

height = 6 units

SA = 2×2 π( 2+6)

SA = 4π × 8

SA = 32π units²

Therefore the surface area of the cylinder in term of pi is 32π units².

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The indefinite integral of ln(e^(8x) - 5) dx is x - ln|e^(8x) - 5| + C.To find the indefinite integral, we can use the substitution method.

Let u = e^(8x) - 5, then du = 8e^(8x) dx. Rearranging, we have dx = du / (8e^(8x)). Substituting these into the integral, we get ∫(ln(u) / (8e^(8x))) du. Simplifying further, we have (1/8) ∫ln(u) du.

Using the integration formula for ln(u), we obtain (1/8)(u ln|u| - u) + C. Substituting back u = e^(8x) - 5, we get the final result of x - ln|e^(8x) - 5| + C, where C represents the constant of integration.

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B) "Some goat was eaten by a wolf" is a correct interpretation of the statement, because it means that there exists at least one goat that was eaten by a wolf.

The given statement can be translated as: "For all goats z, if z is eaten by a wolf, then there exists a wolf w such that w has eaten z."

A) "Every goat is eaten by a wolf" is not a correct interpretation of the statement. The correct interpretation is that if a goat is eaten by a wolf, then there exists at least one wolf that has eaten a goat.

B) "Some goat was eaten by a wolf" is a correct interpretation of the statement, because it means that there exists at least one goat that was eaten by a wolf.

C) "There is a wolf who has eaten every goat" is not a correct interpretation of the statement. The correct interpretation is that for each goat that is eaten, there exists at least one wolf that has eaten it.

D) "Every goat has eaten a wolf" is not a correct interpretation of the statement. The correct interpretation is that if a goat is eaten by a wolf, then there exists at least one wolf that has eaten a goat, but it does not imply that every goat has eaten a wolf.

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Answer:

The missing side length is 13 - 5 = 8 cm.

To find the instantaneous velocity at t = 3, we need to take the derivative of the position function with respect to time:

s'(t) = 10t + 3

Then, we can plug in t = 3 to find the instantaneous velocity:

s'(3) = 10(3) + 3 = 33

Therefore, the instantaneous velocity at t = 3 is 33.

So, to find the instantaneous velocity of the object at t = 3, we first need to find the derivative of the position function s(t) = 5t² + 3t + 2 with respect to time (t). This derivative represents the velocity function, v(t).

Step 1: Differentiate s(t) with respect to t
v(t) = ds/dt = d(5t² + 3t + 2)/dt = 10t + 3

Step 2: Evaluate v(t) at t = 3
v(3) = 10(3) + 3 = 30 + 3 = 33

So, the instantaneous velocity of the object moving in a straight line at t = 3 is 33 units per time unit.

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The probability that the lifetime of the computer exceeds 10 years is approximately 0.9512.

Let X denote the lifespan of the computer. Since X follows an exponential distribution, we know that its probability density function is given by: f(x) = λe^(-λx)

where λ is the rate parameter. We are given that P(X > 3) = 0.0027, which means: ∫3 to ∞ λe^(-λx) dx = 0.0027

Using integration by parts, we can solve for λ: -λe^(-λx) | from 3 to ∞ = 0.0027, Taking the limit as the upper bound approaches infinity, we get: 0 + λe^(-3λ) = 0.0027

Solving for λ, we get: λ = 0.0003

Now, we can find the probability that the lifetime exceeds 10 years: P(X > 10) = ∫10 to ∞ λe^(-λx) dx = e^(-3λ) ≈ 0.9512

Therefore, the probability that the lifetime of the computer exceeds 10 years is approximately 0.9512.

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The correct response is 15. 15 degrees of freedom must be in the denominator.

The degrees of freedom (df) in the denominator for the F-ratio is equal to the degrees of freedom for the MSW.
The formula to calculate the degrees of freedom for the MSW is:
dfW = n - k
Where n is the total number of observations (trucks) and k is the number of groups (manufacturers).
In this case, the company purchased 18 trucks in total, with 5 from manufacturer A, 4 from manufacturer B, and 5 from manufacturer C. Therefore:
n = 18
k = 3
Substituting these values in the formula, we get:
dfW = n - k = 18 - 3 = 15
Therefore, the answer is: 15.
The denominator degrees of freedom for the F-test in this ANOVA analysis can be calculated using the formula:
Denominator Degrees of Freedom = Total Number of Trucks - Number of Manufacturers.

To apply the F-test in ANOVA, we need to calculate the mean square between groups (MSB) and mean square within groups (MSW) and then calculate the F-ratio.

In this case, the company purchased a total of 18 trucks from 3 different manufacturers (A, B, and C). Using the formula:
Denominator Degrees of Freedom = 18 - 3 = 15
So, the correct answer is 15.

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By applying Pythagoras theorem we get length of Hypotenuse of triangle A is 11.66 m, Perpendicular of triangle B is 9.74 m,  Perpendicular of triangle C is 10.24 m, and  Base of triangle D is 6.32 m.

By applying Pythagoras theorem we can calculate the length of sides of the given right angle triangle as,

For triangle A :

Hypotenuse² = Base² + Perpendicular²

⇒ Hypotenuse² = (10)² + (6)²

⇒ Hypotenuse² =  136

⇒ Hypotenuse = 11.66 m (approximately)

For triangle B :

Hypotenuse² = Base² + Perpendicular²

⇒ Perpendicular² = Hypotenuse² -  Base²

⇒ Perpendicular² = 12² - 7²

⇒ Perpendicular² = 95

⇒ Perpendicular = 9.74 m (approximately)

For triangle C :

Hypotenuse² = Base² + Perpendicular²

⇒ Perpendicular² = Hypotenuse² -  Base²

⇒ Perpendicular² = 13² - 8²

⇒ Perpendicular² = 105

⇒ Perpendicular = 10.24 m (approximately)

For triangle D :

Hypotenuse² = Base² + Perpendicular²

⇒ Base² = Hypotenuse² -  Perpendicular²

⇒ Base² = 11² - 9²

⇒ Base² = 40

⇒ Base = 6.32 m (approximately)

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Anwer: a) (4,1)

            b)(2,-1)

Step-by-step explanation:

The answer is d. the mean difference of a relationship is not measured and described by a correlation.

Correlation is a statistical technique used to measure the degree of association between two variables. It is used to describe the direction, form, and strength of the relationship between the variables. The direction of a relationship can be positive or negative, depending on whether the variables move in the same or opposite direction. The form of a relationship can be linear or nonlinear, depending on whether the relationship is a straight line or a curve. The strength of a relationship can be weak or strong, depending on how closely the variables are related to each other. However, correlation does not measure the mean difference of a relationship, which is a measure of central tendency that describes the average difference between two groups or variables.

In summary, correlation measures the direction, form, and strength of a relationship between two variables. It does not measure the mean difference of a relationship, which is a measure of central tendency. Correlation is a useful tool in understanding the relationship between variables, but it should be used in conjunction with other statistical techniques to provide a comprehensive understanding of the data.

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The line integral of (9x+5y)ds over the line segment C is sqrt(13)(11/2), and the vector parametric equation for the line segment C is r(t) = <3-3t, 2t>.

To find a vector parametric equation for the line segment C, we can use the two given points P and Q as the initial and terminal points of the vector, respectively. Let r(t) be the position vector of a point on the line segment C at time t, where t ranges from 0 to 1. Then, we have:

r(0) = P = <3, 0>

r(1) = Q = <0, 2>

The vector connecting P to Q is:

Q - P = <0, 2> - <3, 0> = <-3, 2>

So, a vector parametric equation for the line segment C is:

r(t) = <3, 0> + t<-3, 2> = <3-3t, 2t>

Now, we can use this vector parametric equation to compute the line integral:

∫C (9x+5y)ds = ∫[0,1] (9(3-3t) + 5(2t))|r'(t)| dt

where r'(t) is the derivative of r(t) with respect to t. We have:

r'(t) = <-3, 2>

|r'(t)| = sqrt(9 + 4) = sqrt(13)

Substituting these values, we get:

∫C (9x+5y)ds = ∫[0,1] (27-27t+10t) sqrt(13) dt

= sqrt(13) ∫[0,1] (37t-27) dt

= sqrt(13) [(37/2)t^2 - 27t] from 0 to 1

= sqrt(13) (37/2 - 27/1)

= sqrt(13) (11/2)

Therefore, the line integral of (9x+5y)ds over the line segment C is sqrt(13)(11/2), and the vector parametric equation for the line segment C is r(t) = <3-3t, 2t>.

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The particular solution is: y_p = (-1/2)e^(2x). So the general solution to the differential equation is: y = C_1e^(2x) + C_2xe^(2x) - (1/2)e^(2x)

To find a particular solution of y'' - 4y' + 4y = e^(2x), we can use the method of undetermined coefficients. Since the right-hand side is e^(2x), we assume a particular solution of the form y_p = Ae^(2x), where A is a constant to be determined.

Taking the first and second derivatives of y_p, we get:

y'_p = 2Ae^(2x)
y''_p = 4Ae^(2x)

Substituting these expressions into the differential equation, we get:

4Ae^(2x) - 4(2Ae^(2x)) + 4(Ae^(2x)) = e^(2x)

Simplifying and solving for A, we get:

-2Ae^(2x) = e^(2x)

A = -1/2

Therefore, the particular solution is:

y_p = (-1/2)e^(2x)

So the general solution to the differential equation is:

y = C_1e^(2x) + C_2xe^(2x) - (1/2)e^(2x)

where C_1 and C_2 are constants determined by any initial or boundary conditions.

To find a particular solution of the given differential equation, y'' - 4y' + 4y = e^(2x), you can use the method of undetermined coefficients. First, identify the form of the particular solution, which in this case is y_p = Ae^(2x), where A is a constant to be determined. Differentiate y_p twice and plug the results into the given equation to find the value of A. Then, the particular solution will be y_p = Ae^(2x) with the determined value of A.

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The slope-intercept form of the equations are y = -2x/3 + 3 and y = 3x/2 + 4 and the line are perpendicular to each other.

We know that,

The meaning of slope intercept form is the equation of a straight line in the form y = mx + b where m is the slope of the line and b is its y-intercept.

Given that two equations, we need to find their slope intercept form, and determine if the lines are parallel, perpendicular, or neither (intersecting).

The given equations are;

A) 2x + 3y = 9

B) 2y - 3x = 8

The general equation of a line, in a slope intercept form, is given by,

y = mx + c, where m is the slope of the line and c is the y-intercept,

A) 2x + 3y = 9

3y = 9-2x

y = -2x/3 + 3....(i)

B) 2y - 3x = 8

2y = 3x+8

y = 3x/2 + 4.....(ii)

Here, the slope are -2/3 and 3/2, we can say that both the slopes are negative reciprocal of each other,

We know that slopes of two perpendicular lines are negative reciprocal of each other,

Therefore, the given two line are perpendicular to each other.

Hence, the slope-intercept form of the equations are y = -2x/3 + 3 and y = 3x/2 + 4 and the line are perpendicular to each other.

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The ticket price that maximizes revenue from ticket sales is $6,400.50.

Given that, a magician performs in a hall that has a seating capacity of 1,000 spectators.

Let P be the ticket price and A be the average attendance.

We can set up the following equation to solve the problem:

47P = 640A

Since a $1 decrease in the ticket price results in an increase in attendance of 20, we can use the following equation to solve the problem:

P - 1 = 20(A - 640)

Solving for P and replacing A with its original equation, we get:

P = 1 + 20(47P - 640)

Simplifying the equation:

P = 1 + 940P - 12800

Solving for P:

2P = 12,801

P = $6,400.50

Therefore, the ticket price that maximizes revenue from ticket sales is $6,400.50.

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To solve the problem, let's go step by step.

1) Draw the K-map and find all prime implicants:

The given function f has 4 variables (d, c, b, a). So, we need to draw a 4-variable Karnaugh map (K-map). The K-map will have 2^4 = 16 cells.

The K-map for f is as follows:

```

        cd

ab     00 01 11 10

00 |   1   2   8   7

01 |   3   4   12 15

11 |   X   5   X   14

10 |   X   9   X   10

```

Now, let's find all the prime implicants:

- A: Group (1, 2, 3, 4) with d = 0.

- B: Group (2, 3, 7, 8) with b = 0.

- C: Group (3, 4, 12, 15) with a = 0.

- D: Group (2, 3, 4, 5) with c = 1.

- E: Group (7, 8, 14, 15) with b = 1.

- F: Group (4, 5, 9, 10) with a = 1.

- G: Group (8, 9, 12, 15) with c = 0.

- H: Group (12, 14, 15, 10) with d = 1.

2) Minimize f:

To minimize f, we need to simplify it by combining the prime implicants.

The minimized form of f can be expressed as the sum of prime implicants A, B, D, and E:

f = A + B + D + E

This can be further simplified, if desired, using Boolean algebra techniques.

Note: Please double-check the given function f and the tabulation method steps to ensure accuracy in the K-map and prime implicant identification.

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The Mean Value Theorem states that for a function f(x) that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), there exists a value "c" in the open interval (a,b) such that:f'(c) = (f(b) - f(a))/(b-a).

So, if we have the function or equation, we can find the values of "c" using the above formula. However, since that information is not provided, we cannot answer that part of the question. Additionally, "pean" is not a known mathematical term.
To determine whether the Mean Value Theorem (MVT) can be applied to a function on the closed interval [a, b], we need to check two conditions:

1. The function is continuous on the closed interval [a, b].
2. The function is differentiable on the open interval (a, b).

Since the given function is f(x) = 11, it is a constant function. Constant functions are always continuous and differentiable on their domain, which means both conditions are satisfied.

Therefore, the Mean Value Theorem can be applied to f(x) = 11 on the closed interval [a, b].

According to the MVT, there exists at least one value 'c' in the open interval (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a)

So, the Mean Value Theorem can be applied, and there are infinitely many values of 'c' in the open interval (a, b) that satisfy the theorem, since f'(c) = 0 for all 'c'.

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The iterated integral ∫∫(sino + siny) dy dx equals zero.

The double integral ∫∫R (24*2) dA, where R = {(x,y): 0≤y≤5, 1/2≤x≤2}, equals 98.

We have ∫∫(sino + siny) dy dx, where the limits of integration are not given. Assuming the limits of y to be a and b, and limits of x to be c and d, we can evaluate the integral as follows:

∫c^d ∫a^b (sino + siny) dy dx

= ∫c^d [-cos(y)]_a^b dx (using integration formula of sin)

= ∫c^d [cos(a) - cos(b)] dx

= [sin(c)(cos(a) - cos(b)) - sin(d)(cos(a) - cos(b))] (using integration formula of cos)

= 0 (since sin(0) = sin(2π) = 0, and cos(a) - cos(b) is a constant)

Therefore, the iterated integral ∫∫(sino + siny) dy dx equals zero.

We have to find the double integral ∫∫R (242) dA, where R = {(x,y): 0≤y≤5, 1/2≤x≤2}. We can evaluate the integral as follows:

∫1/2^2 ∫0^5 (242) dy dx

= 48∫1/2^2 (5) dx

= 48*(5/2) (using integration formula of constants)

= 120

Therefore, the double integral ∫∫R (24*2) dA, where R = {(x,y): 0≤y≤5, 1/2≤x≤2}, equals 120.

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The volume of the solid obtained by rotating the region bounded by the curves about the axis y = -3 is (49π + 2)/72 cubic units.

To find the volume of the solid obtained by rotating the region bounded by the curves y = 0, y = cos(6x), and x = π/12, x = 0 about the axis y = -3, we can use the method of cylindrical shells.

To use the cylindrical shells method, we need to integrate the volume of each cylindrical shell. The volume of a cylindrical shell is given by:

V = 2πrhΔx

where r is the distance from the axis of rotation to the shell, h is the height of the shell, and Δx is the width of the shell.

In this case, the axis of rotation is y = -3, so the distance from the axis to a point (x, y) on the curve y = cos(6x) is r = y + 3. The height of the shell is h = x - 0 = x, and the width of the shell is Δx = π/12 - 0 = π/12.

Thus, the volume of each cylindrical shell is:

V = 2π(x)(cos(6x) + 3)(π/12)

To find the total volume, we need to integrate this expression from x = 0 to x = π/12:

V = ∫0^(π/12) 2π(x)(cos(6x) + 3)(π/12) dx

This integral can be evaluated using integration by parts or a table of integrals. The result is:

V = π/24 + (1/36)sin(6π/12) + 3π/4

Simplifying this expression, we get:

V = π/24 + (1/36) + 3π/4

V = (49π + 2)/72

Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = 0, y = cos(6x), and x = π/12, x = 0 about the axis y = -3 is (49π + 2)/72 cubic units.

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The given statement "In the 1st law of thermodynamics for a CV, the W cv term includes all forms of power (rate of work) done on or by the CV EXCEPT flow work" is True because the first law of thermodynamics for a control volume (CV) states that the net change in energy within the CV is equal to the net energy transfer into or out of the CV, plus the net rate of work done on or by the CV.

The term W cv in this equation represents the net rate of work done on or by the CV, but it excludes flow work, which is the work done by or against the pressure forces as a fluid flows into or out of the CV.

However, it does not include flow work. Flow work represents the energy required to push the fluid into or out of the control volume. This energy is already accounted for separately in the enthalpy term within the 1st law of thermodynamics for a CV. Thus, the Wcv term does not include flow work. Therefore, W cv includes all forms of power (rate of work) done on or by the CV except flow work.

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The total amount donated to the funds is A = 72 million

Given data ,

Let's denote the total amount of the charity's donated funds by x. We can set up the following equation to represent the given information:

0.41x = 29.6 million

To solve for x, we can divide both sides by 0.41:

x = 29.6 million / 0.41

On simplifying the equation , we get

x = 72.19512195 million

Rounding this to the nearest million dollars, we get:

x ≈ 72 million

Hence , the total amount of the charity's donated funds is approximately $72 million

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The appropriate p-value for a t-value of 2.73 with 5 degrees of freedom is approximately 0.05. This indicates that there is a 5% chance of observing a t-value as extreme as 2.73 or more extreme, assuming the null hypothesis is true.

In statistics, the p-value measures the strength of evidence against the null hypothesis. The null hypothesis states that there is no significant difference or effect in the population being studied. The p-value is calculated by determining the probability of obtaining a test statistic (in this case, the t-value) as extreme as or more extreme than the observed value, assuming the null hypothesis is true.

To determine the appropriate p-value for a t-value, we typically consult a t-distribution table or use statistical software. In this case, with 5 degrees of freedom and a t-value of 2.73, we look up the critical value or use software to find the corresponding p-value. The p-value associated with a t-value of 2.73 and 5 degrees of freedom is approximately 0.05.

The p-value of 0.05 indicates that there is a 5% chance of obtaining a t-value as extreme as 2.73 or more extreme, assuming the null hypothesis is true. Generally, a p-value of 0.05 or lower is considered statistically significant, implying that the observed result is unlikely to have occurred by chance alone. If the p-value is below a predetermined significance level (often denoted as α, commonly set at 0.05), we reject the null hypothesis in favor of an alternative hypothesis. If the p-value is above the significance level, we fail to reject the null hypothesis.

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The angle measure of the minor arc JL in the given circle is equal to 82°.

The portion of a boundary of a circle is known as an arc is what it means to be an arc of a circle. A chord of the circle is a straight line that connects an arc's two end points.

By connecting any two points on the circle that have been marked, there are two arcs created. The longer arc of the two, is known as the major arc, and the shorter one is known as the minor arc. The arc here is referred to as a semicircular arc if its length precisely equals the half of the circle's diameter.

Both the length and angle of an arc can be determined when measuring an arc. To find the minor arc's measure of the given circle, which is the angle of the arc JL's measure, and that is what is asked in the question and is required to be found here.

When the arc's end points are connected to the circle's center, an angle is created at that location and we can use that to measure the arc's angle.

Thus, we get ∠JKL = 82°

Here, the measure of arc JL = 82°

Therefore, the measure of the minor arc JL equals 82°.

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Note that the full question is:

(check the attached image)

Note that the coordinates of B after it has been translated will be B'(4, 1) .

A translation is a geometric transformation in Euclidean geometry that moves every point in a figure, shape, or space by the same distance in the same direction. A translation may alternatively be understood as the addition of a constant vector to each point or as altering the coordinate system's origin.

The translation formula or vertical translation equation is g(x) = f(x+k) + C.

The four basic translations or transformations in geometry are:

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4 miles by 3.14 is 12.6 miles.

To round 4 miles by 3.14 to the nearest tenth, we need to look at the digit in the hundredth place, which is 4. Since 4 is less than 5, we round down and leave the tenths place as 1. Therefore, the rounded answer is 12.6 miles.

It's important to understand the concept of rounding, as it is commonly used in mathematical calculations and in everyday life. Rounding helps us simplify numbers and make them easier to work with. However, it's important to keep in mind that rounding can lead to inaccuracies if not done correctly.

In addition, it's important to have all necessary information before making a decision. In the case of the given problem, we needed to know the value of pi (3.14) in order to calculate the answer. Similarly, in other situations, we may need to gather more data or conduct statistical tests before making a decision. This is where statistical tests come into play. They allow us to analyze data and make informed decisions based on the results. Therefore, it's important to have a solid understanding of mathematical concepts and statistical tests to make accurate decisions.

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a + b = 3 + 1/2 = 7/2. To find a + b, we need to use the properties of moment-generating functions to relate them to the mean and variance of X.

Specifically, we will use the fact that the nth moment of X is given by the nth derivative of the moment generating function evaluated at t=0.
First, we find the first two derivatives of Mx(t):
Mx'(t) = a*e^at / (1-bt^2)^2
Mx''(t) = (a^2 + 2abt^2 + b) * e^at / (1-bt^2)^3
Next, we evaluate these derivatives at t=0 to get the first two moments of X:
E(X) = Mx'(0) = a
E(X^2) = Mx''(0) + [Mx'(0)]^2 = a^2 + 1/b
Using the given information that E(X) = 3 and Var(X) = 2, we can set up a system of equations to solve for a and b:
a = 3
a^2 + 1/b = E(X^2) = Var(X) + [E(X)]^2 = 2 + 3^2 = 11
Substituting a=3 into the second equation, we get:
9 + 1/b = 11
1/b = 2
b = 1/2
Therefore, a + b = 3 + 1/2 = 7/2.

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The solution of the equations 7³ˣ = 1/343, 9 (3)ˣ = 1/3, and (2/3)ˣ⁺¹ = (3/2)²ˣ will be -1, -3, and -1/3, respectively.

Given that:

Equations, 7³ˣ = 1/343, 9 (3)ˣ = 1/3, and (2/3)ˣ⁺¹ = (3/2)²ˣ

Simplify the equation 7³ˣ = 1/343, then

7³ˣ = 1/343

3x log 7 = log (1/343)

3x = -3

x = -1

Simplify the equation 9 (3)ˣ = 1/3, then

9 (3)ˣ = 1/3

x log 3 = log (1/27)

x = -3

Simplify the equation 9 (3)ˣ = 1/3, then

(2/3)ˣ⁺¹ = (3/2)²ˣ

(x + 1) log (2/3) = 2x log (3/2)

x + 1 = - 2x

3x = - 1

x = - 1/3

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The rate of change of revenue per unit is 0.0071 when the cost per unit is changing by $12 and the revenue is $4500.

To find the rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is $4500, we can follow these steps:

1. Use the given formula: c = R^2 / 112,000 + 5807
2. Plug in the given values: cost per unit (c) and revenue per unit (R).
3. Differentiate both sides of the equation with respect to x.
4. Solve for dR/dx when the cost per unit is changing by $12 and the revenue is $4500.

Step 1:
c = R^2 / 112,000 + 5807

Step 2:
Given: c is changing by $12 (dc/dx = 12) and R = $4500
Plug in R = $4500 into the equation:
c = (4500^2) / 112,000 + 5807

Step 3:
Differentiate both sides of the equation with respect to x:

dc/dx = (d/dx) [R^2 / 112,000 + 5807]

Using the chain rule, we get:
dc/dx = (2R * dR/dx) / 112,000

Step 4:
Solve for dR/dx when dc/dx = 12 and R = $4500:

12 = (2 * 4500 * dR/dx) / 112,000
12 * 112,000 / (2 * 4500) = dR/dx
dR/dx = 56/15
dR/dx = (4500/x)^2/56,000 + 5807

dR/dC = (dR/dx) / (dC/dx) = ((4500/x)^2/56,000 + 5807) / ((2R/112,000) * dR/dx) = ((4500/x)^2/56,000 + 5807) / ((2(4500/x))/112,000 * (4500/x)^2/56,000 + 5807)^2/56,000

Plugging in the values, we get:

dR/dC = ((4500/x)^2/56,000 + 5807) / ((2(4500/x))/112,000 * (4500/x)^2/56,000 + 5807)^2/56,000

dR/dC = 0.0071

The rate of change of revenue per unit when the cost per unit is changing by $12 and the revenue is $4500 is approximately 56/15 or 3.73 dollars per unit.

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A) The particular solution for the first equation is:

y = 1/2

B) The particular solution to the second equation is:

y = 1/7 (x + 1)⁴ + 12/7 (x + 1)⁻³

a) dy/dx + 2xy = x

First, we need to find the integrating factor:

μ(x) = e∫2x dx = eˣ²

Multiplying both sides by the integrating factor, we get:

eˣ² dy/dx + 2xeˣ²y = xeˣ²

Using the product rule, we can simplify the left-hand side as follows:

d/dx (eˣ² y) = xeˣ²

Integrating both sides with respect to x, we obtain:

eˣ²) y = ∫xeˣ² dx = 1/2 eˣ² + C

Thus, the general solution is:

y = 1/2 + Ce⁻ˣ²

To find the particular solution, we can use the initial condition y(0) = 1/2:

1/2 = 1/2 + Ce⁻₀²

C = 0

Therefore, the particular solution is:

y = 1/2

b) (x + 1) dx - 3y = (x + 1)⁴, x = 0, y = -1/2; x = 1, y = 16

First, we need to rearrange the equation in the standard form:

dy/dx + 3y/(x + 1) = (x + 1)³

Next, we need to find the integrating factor:

μ(x) = e∫3/(x + 1) dx = (x + 1)³

Multiplying both sides by the integrating factor, we get:

(x + 1)³ dy/dx + 3(x + 1)² y = (x + 1)⁶

Using the product rule, we can simplify the left-hand side as follows:

d/dx [(x + 1)³ y] = (x + 1)⁶

Integrating both sides with respect to x, we obtain:

(x + 1)³ y = 1/7 (x + 1)⁷ + C

Thus, the general solution is:

y = 1/7 (x + 1)⁴ + C/(x + 1)³

To find the particular solution, we can use the initial conditions:

y(0) = -1/2

y(1) = 16

Substituting these values, we get a system of equations:

C = -1/7

1/7 (2⁴) - 1/7 = 16

C = 12/7

Therefore, the particular solution is:

y = 1/7 (x + 1)⁴ + 12/7 (x + 1)⁻³

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Answer:

Let the length of the rectangle be 2x and the width be 3x, since the ratio of length to width is 2:3. The area of the rectangle is given by:

length x width = (2x) x (3x) = 6x^2

We know that the area of the rectangle is 150in^2, so we can set up the equation:

6x^2 = 150

Simplifying this equation, we get:

x^2 = 25

Taking the square root of both sides, we get:

x = 5

Therefore, the width of the rectangle is 3x = 15, and the length of the rectangle is 2x = 10.

Answer: The length of the rectangle is 10 inches.