Geno read 126 pages in 3 hours. He read the same number of pages each hour for the first 2 hours. Geno read 1. 5 times as many pages during the third hour as he did during the first hour.

Let A be a 5×6 real matrix such that rank(A)=5. The statements that are true are B and C. The rows and columns of A are linearly independent.

To determine which statements are true, let's analyze each option:

A. The dimension of the null space of A is equal to 0.

The null space of a matrix consists of all vectors that, when multiplied by the matrix, result in the zero vector. Since the rank of A is 5, it means that the number of linearly independent columns is 5. Therefore, the dimension of the null space, which represents the number of linearly dependent columns, is equal to the total number of columns (6) minus the rank (5), resulting in a dimension of 1. Therefore, statement A is false.

B. The rows of A are linearly independent.

Since the rank of A is 5, it means that there are 5 linearly independent rows. Therefore, statement B is true.

C. The columns of A are linearly independent.

Since the rank of A is 5, it means that there are 5 linearly independent columns. Therefore, statement C is true.

D. The rank of A^T is equal to 6.

The rank of the transpose of a matrix, A^T, is equal to the rank of the original matrix, A. Since the rank of A is given to be 5, the rank of A^T is also 5. Therefore, statement D is false.

E. The dimension of the row space of A is 1.

The row space of a matrix consists of all linear combinations of the rows. Since the rank of A is 5, it means that there are 5 linearly independent rows, and therefore, the dimension of the row space is also 5. Therefore, statement E is false.

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The geometric probability of landing in the shaded area is 0.17. This is calculated by finding the ratio of the area of the smaller circle to the area of the larger circle.

Given, the diameter of the small circle is 20 in and the diameter of the larger circle is 48 in. In order to find the geometric probability of landing in the shaded area of the picture, we need to calculate the ratio of the area of the smaller circle to the area of the larger circle.

The area of a circle is given by the formula: [tex]$A = \pir^2$[/tex], where r is the radius of the circle. We know that the diameter of the small circle is 20 in, so the radius is 10 in. Similarly, the diameter of the large circle is 48 in, so the radius is 24 in.

Area of the smaller circle = [tex]\pi(10)^2 = 100\pi in^2[/tex]

Area of the larger circle = [tex]\pi(24)^2 = 576\pi in^2[/tex]

Area of shaded region = Area of the larger circle - Area of the smaller circle = [tex]576\pi-100\pi = 476\pi in^2[/tex]

The probability of landing in the shaded region is the ratio of the area of the smaller circle to the area of the larger circle. Hence, geometric probability = [tex]\frac{100\pi}{576\pi} = 0.17[/tex](rounded to the nearest hundredth place).

Thus, the geometric probability of landing in the shaded area of the picture is 0.17. In summary, the geometric probability of landing in the shaded area of the picture is obtained by calculating the ratio of the area of the smaller circle to the area of the larger circle.

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The general solution is r = -3/2.

To find the general solution to the given differential equation:

25w'' + 60w' + 36w = 0

we can start by assuming a solution of the form w(t) = [tex]e^{rt}[/tex], where r is a constant to be determined.

First, let's find the derivatives of w(t):

w'(t) = rw(t)

w''(t) = r²w(t)

Substituting these derivatives into the differential equation, we have:

25r²w(t) + 60rw(t) + 36w(t) = 0

Dividing through by w(t) (since it is assumed to be nonzero), we get:

25r² + 60r + 36 = 0

Now, we can solve this quadratic equation for r. Dividing through by 4, we have:

6.25r² + 15r + 9 = 0

Factoring the quadratic, we get:

(2.5r + 3)(2.5r + 3) = 0

This equation has a repeated root of -3/2. Therefore, the solution for r is:

r = -3/2

Since the quadratic equation has a repeated root, the general solution to the given differential equation is of the form:

w(t) = (C1 + C2t)[tex]e^{-3t/2}[/tex]

where C1 and C2 are arbitrary constants that can be determined from initial conditions or boundary conditions, if provided.

The complete question is:

Find a general solution to the given differential equation.

25w'' + 60w' + 36w = 0

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The general solution of the differential equation is w = C.

Given differential equation is

25w + 60w + 36w = 0.

To find the general solution to the given differential equation using differential equation.

Solution:

We need to solve the differential equation

25w + 60w + 36w = 0

Let's simplify the given differential equation

25w + 60w + 36w

= 0w(25 + 60 + 36)

= 0w(121)

= 0w

= 0

We know that the general solution of a differential equation of the first order and first degree has one arbitrary constant C.

Therefore, the general solution of the differential equation is w = C.

Now, this solution has not been explicitly found, so in order to do that, you must know the initial conditions for the differential equation.

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Answer:

The flow rate in mL/hr for infusing 2 liters of 5% dextrose over 16 hours is 125 mL/hr.

Step-by-step explanation:

We can use the following formula to calculate the flow rate:

Flow rate (mL/hr) = Volume to be infused (mL) / Time of infusion (hr)

First, we need to convert the total volume of 2 liters to mL:

2 liters = 2000 mL

Next, we can plug in the values:

Flow rate = 2000 mL / 16 hours

Flow rate = 125 mL/hr

Therefore, the flow rate in mL/hr for infusing 2 liters of 5% dextrose over 16 hours is 125 mL/hr.

The quotient is 3x - 5 + (-5) + 12, which simplifies to 3x + 2.

To find the quotient, we need to perform polynomial long division. The dividend is 3x² - 2x + 7, and the divisor is x + 1.

 3x - 5

x + 1 | 3x² - 2x + 7

We start by dividing the highest degree term of the dividend (3x²) by the divisor (x), which gives us 3x. We then multiply the divisor (x + 1) by the quotient (3x) and subtract it from the dividend:

       3x - 5

    ____________

x + 1 | 3x² - 2x + 7

- (3x² + 3x)

____________

- 5x + 7

We continue the process by dividing the next term (-5x) of the resulting polynomial (-5x + 7) by the divisor (x + 1). This gives us -5.

            -5

    ____________

x + 1 | 3x² - 2x + 7

- (3x² + 3x)

____________

- 5x + 7

- (- 5x - 5)

____________

12

Finally, we divide the remaining term (12) by the divisor (x + 1), which gives us 12.

                  12

    ____________

x + 1 | 3x² - 2x + 7

- (3x² + 3x)

____________

- 5x + 7

- (- 5x - 5)

____________

12

- 12

____________

0

The quotient is 3x + 2 and can be written as 3x + 5 + (-5) + 12.

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Subsequently, the first 4 terms of the expansion for (1+x)¹⁵. are:

1, 15x, 105x^2, 455x^3

To find the first 4 terms of the expansion for (1+x).¹ , we can utilize the binomial hypothesis. The binomial hypothesis states that the expansion of (a+b) can be spoken to as the entirety of the binomial coefficients multiplied by the comparing powers of a and b.

In this case, (1+x)¹⁵ can be expanded as follows:

(1+x)^15 = C(15,0) * 1⁵* x^0 + C(15,1) * 1 ¹⁴ x⁴ + C(15,2) * 1.¹³ * x² + C(15,3) * 1 ¹²* x³

Now, let's calculate the first 4 terms:

Term 1: C(15,0) * 1¹⁵* x = 1 * 1 * 1 = 1

Term 2: C(15,1) * 1¹⁴ * x= 15 * 1 * x = 15x

Term 3: C(15,2) * 1.¹³ * x ²= 105 * 1 * x² = 105x ²

Term 4: C(15,3) * 1¹²* x³= 455 * 1 * x³= 455x³

Subsequently, the first 4 terms of the expansion for (1+x).¹⁵ are:

1, 15x, 105x², 455x³

Answer: A. 1−15x+105x² −455x³

To find the distance between the two focuses (4,13) and (-1,3), we are able utilize the distance equation:

Separate = √((x2 - x1) ²+ (y2 - y1)² )

Plugging within the values:

Distance = √((-1 - 4) ²+ (3 - 13).²)

Distance = √((-5)²+ (-10)²

Distance = √(25 + 100)

Distance = √(125)

Distance = 11.18033989

Adjusted to the closest entire number, the distance between the two points is 11.

Answer: B. 125

For the sequence -1, 1, 3, ..., we will see that it is an math sequence with a common contrast of 2. To discover the entirety of the first 8 terms, able to utilize the equation for the entirety of an math series:

Entirety = (n/2)(2a + (n-1)d)

Plugging within the values:

Sum = (8/2)(2(-1) + (8-1)2)

Sum = 4(-2 + 14)

Sum = 4(12)

Sum = 48

Answer: C. 48

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The sum of the first 8 terms is 48, which corresponds to option C.

The expansion of (1+x)^15 can be found using the binomial theorem. The first four terms are:

A. 1 - 15x + 105x^2 - 455x^3

To find the distance between the two points (4,13) and (-1,3), we can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the coordinates, we have:

d = sqrt((-1 - 4)^2 + (3 - 13)^2)

= sqrt((-5)^2 + (-10)^2)

= sqrt(25 + 100)

= sqrt(125)

= 11.18

So, the nearest option is B. 125 (rounded to the nearest whole number).

The given sequence -1, 1, 3, ... is an arithmetic sequence with a common difference of 2. To find the sum of the first 8 terms, we can use the arithmetic series formula:

Sn = n/2 * (2a + (n-1)d)

In this case, a = -1 (the first term), d = 2 (the common difference), and n = 8 (the number of terms). Plugging in the values, we get:

S8 = 8/2 * (2(-1) + (8-1)(2))

= 4 * (-2 + 14)

= 4 * 12

= 48

So, the sum of the first 8 terms is 48, which corresponds to option C.

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Answer:

The equations of bisectors of the angles are:

[tex]3x+11y-10=0[/tex]

[tex]33x-9y=0[/tex]

The bisector of the acute angle is 33x - 9y = 0.

Step-by-step explanation:

Let line 3x - 2y + 1 = 0 be defined by the equation a₁x + b₁y + c₁ = 0.

Let line 18x + y - 5 = 0 be defined by the equation a₂x + b₂y + c₂ = 0.

The formulas for the two angle bisectors of lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are:

[tex]\boxed{\dfrac{a_1x+b_1y+c_1}{\sqrt{{a_1}^2+{b_1}^2}}=\pm\dfrac{a_2x+b_2y+c_2}{\sqrt{{a_2}^2+{b_2}^2}}}[/tex]

The two angle bisectors are perpendicular.

Substitute the values of a₁, b₁, c₁, a₂, b₂, and c₂ into both formulas.

Equation of bisector 1

[tex]\begin{aligned}\dfrac{3x-2y+1}{\sqrt{{3}^2+(-2)^2}}&=\dfrac{18x+y+(-5)}{\sqrt{18^2+1^2}}\\\\\dfrac{3x-2y+1}{\sqrt{13}}&=\dfrac{18x+y-5}{5\sqrt{13}}\\\\3x-2y+1&=\dfrac{18x+y-5}{5}\\\\5(3x-2y+1)&=18x+y-5\\\\15x-10y+5&=18x+y-5\\\\3x+11y-10&=0\end{aligned}[/tex]

Equation of bisector 2

[tex]\begin{aligned}\dfrac{3x-2y+1}{\sqrt{{3}^2+(-2)^2}}&=-\dfrac{18x+y+(-5)}{\sqrt{18^2+1^2}}\\\\\dfrac{3x-2y+1}{\sqrt{13}}&=-\dfrac{18x+y-5}{5\sqrt{13}}\\\\3x-2y+1&=-\dfrac{18x+y-5}{5}\\\\-5(3x-2y+1)&=18x+y-5\\\\-15x+10y-5&=18x+y-5\\\\33x-9y&=0\end{aligned}[/tex]

Therefore, the equations of bisectors of the angles between the given lines are:

[tex]3x+11y-10=0[/tex]

[tex]33x-9y=0[/tex]

[tex]\hrulefill[/tex]

To identify the bisector of the acute angle, we need to calculate the angle between any one of the bisectors and one of the lines.

The formula for the angle between two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is:

[tex]\tan \theta=\left|\dfrac{a_2b_1-a_1b_2}{a_1a_2+b_1b_2} \right|[/tex]

Let's find the angle θ between the bisector 6x + 6y - 1 = 0, and the line 3x - 2y + 1 = 0.

Therefore:

Substitute these values into the formula for the angle between two lines:

[tex]\tan \theta=\left|\dfrac{(3)(-9)-(33)(-2)}{(33)(3)+(-9)(-2)} \right|[/tex]

[tex]\tan \theta=\left|\dfrac{39}{117} \right|[/tex]

[tex]\tan \theta=\left|\dfrac{1}{3} \right|[/tex]

As tan θ < 1, the angle θ between the bisector and the line must be less than 45°. This means that the angle between the two given lines is less than 90°.

Since an acute angle measures less than 90°, this means that 33x - 9y = 0 is the bisector of the acute angle between the given lines.

Note: On the attached diagram, the given lines are shown in black, the bisector of the acute angle is the red dashed line, and the bisector of the obtuse angle is the green dashed line.

The range or the given data is calculated as 10.2 . Range is the difference between minimum value and maximum value.

To find the range in the following data 1.0, 7.0, 4.8, 1.0, 11.2, 2.2, 9.4, we can make use of the formula for range in statistics which is given as follows:[\large Range = Maximum\ Value - Minimum\ Value\]

To find the range in the following data 1.0, 7.0, 4.8, 1.0, 11.2, 2.2, 9.4, we need to arrange the data in either ascending or descending order, but since we only need to find the range, it is not necessary to arrange the data.

From the data given above, we can easily identify the minimum value and maximum value and then find the difference to get the range.

So, Minimum Value = 1.0

Maximum Value = 11.2

Range = Maximum Value - Minimum Value

                  = 11.2 - 1.0

                     = 10.2

Therefore, the range of the given data is 10.2.

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Vertically compressing the exponential parent function f(x) = 2^x by a factor of 3 means multiplying every function value by 1/3, resulting in a steeper and narrower curve closer to the x-axis.

If we vertically compress the exponential parent function f(x) = 2^x by a factor of 3, it means that every point on the graph of the function will be compressed closer to the x-axis. In other words, the function values will be multiplied by 1/3.

Let's consider a point on the original exponential function, (x, f(x)). After the vertical compression, this point will have the coordinates (x, (1/3)f(x)). For example, if f(x) = 8 for some x, after compression, the corresponding point will be (x, (1/3)(8)) = (x, 8/3).

This vertical compression affects all points on the graph uniformly, resulting in a steeper and narrower curve compared to the original exponential function.

The y-values of the compressed function will be one-third of the y-values of the original function for each x-value. Therefore, the graph will be squeezed vertically, with the y-values closer to the x-axis.

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a) The annual demand is 420.

b) The weekly demand forecast is 8.08

c) The EOQ would be approximately 41

d) The reorder point is 45.12

e) The total annual cost is 102439.02

f) The pipeline inventory is 32.32

g) The new reorder point is 48.82

a) To calculate the annual demand, you need to use your student number. For example, if your student number is BBAW190102, the last digit is 2. So, the annual demand would be 400 + 10 x 2 = 420.

b) To calculate the weekly demand forecast for 2021, you need to divide the annual demand by the number of weeks in a year. Assuming there are 52 weeks in a year, the weekly demand forecast would be 420 / 52 = 8.08 (rounded to two decimal places).

c) The economic order quantity (EOQ) can be calculated using the formula EOQ = sqrt((2DS) / H), where D is the annual demand, S is the ordering cost per order, and H is the annual holding cost. In this case, D is the annual demand calculated in part a, S is $1000, and H is $500. Plugging in these values, the EOQ would be sqrt((2 x 420 x 1000) / 500) = sqrt(840000 / 500) = sqrt(1680) ≈ 41 (rounded to the nearest whole number).

d) The reorder point is the level of inventory at which a new order should be placed. It can be calculated using the formula reorder point = demand during lead time + safety stock. The demand during lead time is the average demand per week multiplied by the lead time, which is 8.08 x 4 = 32.32 (rounded to two decimal places). The safety stock is the z-score multiplied by the standard deviation of weekly demand. The z-score for a 90% cycle service level is 1.28 (given in the question) and the standard deviation of weekly demand is 10 (given in the question). So, the safety stock would be 1.28 x 10 = 12.8 (rounded to one decimal place). Therefore, the reorder point would be 32.32 + 12.8 = 45.12 (rounded to two decimal places).

e) The total annual cost of managing the inventory can be calculated using the formula TC = (S x D/Q) + (H x (Q/2 + s)), where S is the ordering cost per order, D is the annual demand, Q is the economic order quantity, H is the annual holding cost, and s is the safety stock. Plugging in the values, the total annual cost would be (1000 x 420/41) + (500 x (41/2 + 12.8)) = 102439.02 (rounded to two decimal places).

f) The pipeline inventory refers to the inventory that is in transit or being processed. In this case, since the lead time is 4 weeks, the pipeline inventory would be the average demand per week multiplied by the lead time. So, the pipeline inventory would be 8.08 x 4 = 32.32 (rounded to two decimal places).

g) To achieve a 95% cycle service level, we need to calculate the new safety stock and reorder point. The z-score for a 95% cycle service level is 1.65 (given in the question). Using the same formula as in part d, the new safety stock would be 1.65 x 10 = 16.5 (rounded to one decimal place). Therefore, the new reorder point would be 32.32 + 16.5 = 48.82 (rounded to two decimal places).

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The Paired samples t-test is the most suitable statistical technique for comparing the mean span of the dominant and non-dominant hands in this study.

To investigate whether the span of a person's dominant hand is greater than that of their non-dominant hand, the most appropriate statistical technique would be the Paired samples t-test.

The Paired samples t-test is used when comparing the means of two related groups or conditions. In this case, the dominant and non-dominant hands are related because they belong to the same individuals in the study. By comparing the means of the dominant and non-dominant hand spans, we can determine if there is a significant difference between the two.

The other options listed, ANOVA (Analysis of Variance), Independent samples t-test, Wilcoxon's matched-pairs signed rank test, and Mann-Whitney U test, are not suitable for this scenario because they are designed for different types of comparisons:

- ANOVA is used when comparing the means of three or more independent groups, which is not the case here.

- Independent samples t-test is used when comparing the means of two independent groups, which is not the case here as the measurements are paired.

- Wilcoxon's matched-pairs signed rank test and Mann-Whitney U test are non-parametric tests that are used when the data do not meet the assumptions of parametric tests. However, in this case, we have paired measurements, and the paired samples t-test is the appropriate parametric test.

Therefore, the Paired samples t-test is the most suitable statistical technique for comparing the mean span of the dominant and non-dominant hands in this study.

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The amount still owed on the car after the 24th payment is $15,213.28.

First, let's find the monthly interest rate. We can calculate this by dividing the nominal annual interest rate by the number of compounding periods in a year. Here, we have monthly compounding, so:

Monthly interest rate = Nominal annual interest rate ÷ 12

= 7% ÷ 12

= 0.00583 (rounded to 5 decimal places)

Next, let's calculate the loan amount using the present value formula:

PV = PMT × [1 - (1 + r)^(-n) ÷ r]

where PV = present value (loan amount), PMT = monthly payment, r = monthly interest rate, and n = total number of payments.

PV = $400 × [1 - (1 + 0.00583)^(-72) ÷ 0.00583]

= $23,122.52 (rounded to 2 decimal places)

To find out how much is still owed on the car after the 24th payment, we can use the remaining balance formula:

R = PV × (1 + r)^n - PMT × [(1 + r)^n - 1 ÷ r]

where R = remaining balance, PV = present value (loan amount), r = monthly interest rate, n = number of payments made, and PMT = monthly payment.

R = $23,122.52 × (1 + 0.00583)^24 - $400 × [(1 + 0.00583)^24 - 1 ÷ 0.00583]

R = $15,213.28 (rounded to 2 decimal places)

Therefore, the amount still owed on the car after the 24th payment is $15,213.28.

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Solving a linear equation we can see that x = -8,

On the image we can see a right triangle, where the square angle has a measure of 90°.

Remember that the sum of the interior angles must be equal to 180°, then we can write the linear equation:

90 + 35 + (x + 63) = 180

Solving that linear equation for x we will get:

90 + 35 + (x + 63) = 180

x + 188 = 180

x = 180 - 188

x = -8

That is the value of x.

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a) Plot the points (-2, -1), (-1, 1), (0, 3), (1, 5), and (2, 7) on the grid, and connect them to form a straight line.

b) The equation y = 2x + 3 represents a straight line with a slope of 2 and a y-intercept of 3.

a) To plot the graph of y = 2x + 3, we can select values of x within the given range, calculate the corresponding values of y using the equation, and plot the points on the grid. Since the equation represents a straight line, connecting the plotted points will result in a straight line that represents the graph of the equation.

b) The equation y = 2x + 3 represents a straight line in slope-intercept form. The coefficient of x (2) represents the slope of the line, indicating the rate at which y changes with respect to x. In this case, the slope is positive, which means that as x increases, y also increases. The constant term (3) represents the y-intercept, the point where the line intersects the y-axis.

By writing the equation as y = 2x + 3, we can easily determine the slope and y-intercept, allowing us to identify the line on the graph and describe its characteristics.

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The change in vertical distance is 24 and the change in horizontal distance is 6 between the points (3, 12) and (9, 36).

To find the change in vertical and horizontal distance between two points, we use the concept of coordinates.

The coordinates of a point consist of two values: the x-coordinate and the y-coordinate. In a Cartesian coordinate system, the x-coordinate represents the horizontal position, and the y-coordinate represents the vertical position.

Given two points (x1, y1) and (x2, y2), we can calculate the change in vertical distance (change in y) by subtracting the y-coordinates: y2 - y1. This gives us the difference in the vertical position between the two points.

Similarly, we can calculate the change in horizontal distance (change in x) by subtracting the x-coordinates: x2 - x1. This gives us the difference in the horizontal position between the two points.

In the case of the given points (3, 12) and (9, 36), we subtract the y-coordinates to find the change in vertical distance: 36 - 12 = 24. This means that the vertical distance between the points is 24 units.

We also subtract the x-coordinates to find the change in horizontal distance: 9 - 3 = 6. This means that the horizontal distance between the points is 6 units.

Therefore, the change in vertical distance is 24 and the change in horizontal distance is 6 between the points (3, 12) and (9, 36).

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The solution to the differential equation is:y(x) = 26e^x - e^4x + e^7x

We can solve the given differential equation, y‴ − 12y′′ + 35y′ = 24ex by assuming that y = er

Given, y‴ − 12y′′ + 35y′ = 24exy = erx

Let's substitute y into the differential equation:y‴ − 12y′′ + 35y′ = 24ex → r³erx − 12r²erx + 35rerx = 24ex

Now factor erx from the left side to get:r³ - 12r² + 35r = 24erx

Divide both sides by erx:

r³/erx - 12r²/erx + 35r/erx = 24ex/erx→ r³er^-x - 12r²er^-x + 35rer^-x = 24→ r³e^-x - 12r²e^-x + 35re^-x = 24

Now we can solve for r by factoring the left side:r³e^-x - 12r²e^-x + 35re^-x - 24 = 0

This can be factored into:(r - 1)(r - 4)(r - 7)e^-x = 0

So we have:r = 1, 4, 7

We can write the general solution as:

y(x) = C1e^x + C2e^4x + C3e^7x

where C1, C2, and C3 are constants.

Let's use the initial conditions to find these constants:

y(0) = C1 + C2 + C3 = 24y′(0) = C1 + 4C2 + 7C3 = 18y′′(0) = C1 + 16C2 + 49C3 = 10

Now we can solve for C1, C2, and C3.

Using the first equation, we get:C1 + C2 + C3 = 24

C1 = 24 - C2 - C3

Using the second equation, we get:

C1 + 4C2 + 7C3 = 18(24 - C2 - C3) + 4

C2 + 7C3 = 18-3

C2 - 6C3 = -6

C2 + 2C3 = 2

C2 = -2/4 = -1

Now we can find C3 from the first equation:

C1 + C2 + C3 = 24(24 - C2 - C3) - C2 - C3 + C3 = 24

C3 = 1

Substituting C2 and C3 back into C1 = 24 - C2 - C3, we get:

C1 = 24 - (-1) - 1 = 26

So the solution to the differential equation is:y(x) = 26e^x - e^4x + e^7x

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Answer:

A - the table represents a nonlinear function because the graph does not show a constant rate of change

Step-by-step explanation:

you can tell this is true, because the y value does not increase by the same amount every time

The sum of 5500 mm, 720 cm, 90 dm, and 20 dam can be expressed in meters as 58.2 meters. To convert the given measurements to a common unit, we need to convert each unit to meters and then add them together.

1 meter is equal to 1000 millimeters (mm), 100 centimeters (cm), 10 decimeters (dm), and 0.1 decameters (dam).

Converting the given measurements to meters:

5500 mm = 5500/1000 = 5.5 meters

720 cm = 720/100 = 7.2 meters

90 dm = 90/10 = 9 meters

20 dam = 20 * 0.1 = 2 meters

Now, we can add these converted measurements together:

5.5 meters + 7.2 meters + 9 meters + 2 meters = 23.7 meters

Therefore, the sum of 5500 mm, 720 cm, 90 dm, and 20 dam in meters is 23.7 meters.

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Therefore, the orthogonal projection of the vector X = (2 9 4) onto the subspace W = span [(1 (2 2 1 2), -2) is

[tex]proj_WX = \begin{pmatrix}\frac{4}{3}\\\frac{14}{3}\\\frac{10}{3}\end{pmatrix}[/tex]

Given,

[tex]X=\begin{pmatrix}2\\9\\4\end{pmatrix},W= span\begin{pmatrix}1\\2\\2\end{pmatrix},\begin{pmatrix}-2\\1\\2\end{pmatrix}[/tex]

the projection of a vector X onto a subspace W is given by the following formula:

[tex]proj_WX =\frac{X\cdot w}{\left\|w\right\|^2}w[/tex]

Here, w = the vector of W and [tex]\left\|w\right\|[/tex] is the norm of the vector w. So, find the projection of vector X onto the subspace W. The projection of X onto W is given by the formula,

[tex]proj_WX =\frac{X\cdot w}{\left\|w\right\|^2}w[/tex]

Let's begin by finding the orthonormal basis for the subspace W:

[tex]W = span \left\{\begin{pmatrix}1\\2\\2\end{pmatrix},\begin{pmatrix}-2\\1\\2\end{pmatrix}\right\}[/tex]

[tex]\begin{pmatrix}1\\2\\2\end{pmatrix},\begin{pmatrix}-2\\1\\2\end{pmatrix} \Rightarrow Orthogonalize \Rightarrow \left\{\begin{pmatrix}1\\2\\2\end{pmatrix},\begin{pmatrix}-\frac{3}{2}\\\frac{1}{2}\\1\end{pmatrix}\right\}[/tex]

[tex]\left\{\begin{pmatrix}1\\2\\2\end{pmatrix},\begin{pmatrix}-\frac{3}{2}\\\frac{1}{2}\\1\end{pmatrix}\right\} \Rightarrow Orthonormalize \Rightarrow \left\{\frac{1}{3}\begin{pmatrix}1\\2\\2\end{pmatrix},\frac{1}{\sqrt{14}}\begin{pmatrix}-3\\1\\2\end{pmatrix}\right\}[/tex]

So, the orthonormal basis for the subspace W is

[tex]\left\{\frac{1}{3}\begin{pmatrix}1\\2\\2\end{pmatrix},\frac{1}{\sqrt{14}}\begin{pmatrix}-3\\1\\2\end{pmatrix}\right\}[/tex]

Now, let's compute the projection of X onto the subspace W using the above formula.

[tex]proj_WX =\frac{X\cdot w}{\left\|w\right\|^2}w[/tex]

[tex]proj_WX =\frac{\begin{pmatrix}2\\9\\4\end{pmatrix}\cdot \frac{1}{3}\begin{pmatrix}1\\2\\2\end{pmatrix}}{\left\|\frac{1}{3}\begin{pmatrix}1\\2\\2\end{pmatrix}\right\|^2}\frac{1}{3}\begin{pmatrix}1\\2\\2\end{pmatrix} + \frac{\begin{pmatrix}2\\9\\4\end{pmatrix}\cdot \frac{1}{\sqrt{14}}\begin{pmatrix}-3\\1\\2\end{pmatrix}}{\left\|\frac{1}{\sqrt{14}}\begin{pmatrix}-3\\1\\2\end{pmatrix}\right\|^2}\frac{1}{\sqrt{14}}\begin{pmatrix}-3\\1\\2\end{pmatrix}[/tex]

[tex]proj_WX = \frac{14}{27}\begin{pmatrix}1\\2\\2\end{pmatrix} + \frac{2}{7}\begin{pmatrix}-3\\1\\2\end{pmatrix}[/tex]

[tex]\Rightarrow proj_WX = \begin{pmatrix}\frac{4}{3}\\\frac{14}{3}\\\frac{10}{3}\end{pmatrix}[/tex]

Therefore, the orthogonal projection of the vector X = (2 9 4) onto the subspace W = span [(1 (2 2 1 2), -2) is

[tex]proj_WX = \begin{pmatrix}\frac{4}{3}\\\frac{14}{3}\\\frac{10}{3}\end{pmatrix}[/tex]

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The change of base formula is used for changing a logarithm to a different base. The formula is given as follows:For any positive real numbers a, b, and c, where a is not equal to 1 and c is not equal to 1,loga b = logc b / logc a.

The correct option is c. log(12/11).

Here, we have to rewrite log1112 using the change of base formula, which is given as follows:log1112 = logb 12 / logb 11We need to choose a value for the base b. The most common values for the base are 10, e, and 2. Here, we can choose any base that is not 1.Now, we will use the change of base formula to rewrite log1112 using each value of b.

We can see that log1112 is not equal to any of these values.b) log11 / log112 We can choose We can see that log1112 is not equal to any of these values except for log(12/11).Therefore, the answer is c. log(12/11).

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a. The carry and overflow flags for the operation 0100 0010 in a 4-bit system would depend on the specific operation being performed. Without knowing the operation, it is not possible to determine the carry and overflow flags.

b. Similarly, for the operation 0100 0110 in a 4-bit system, the carry and overflow flags cannot be determined without knowing the specific operation being performed.

c. In the case of the operation 1100 1110 in a 4-bit system, the carry flag would be set if there is a carry from the most significant bit (MSB) during addition or subtraction. The overflow flag would be set if there is a signed overflow, indicating that the result is too large or too small to be represented in the given number of bits. However, without knowing the specific operation being performed, it is not possible to determine the values of the carry and overflow flags.

d. Similarly, for the operation 1100 1010 in a 4-bit system, the carry and overflow flags cannot be determined without knowing the specific operation being performed.

To determine the carry and overflow flags, it is essential to know the specific arithmetic operation being performed, such as addition, subtraction, or other bitwise operations. The carry flag indicates whether a carry occurred during the operation, typically from the MSB to the next higher bit. The overflow flag indicates whether the result exceeds the range that can be represented in the given number of bits, considering signed or unsigned interpretation. Without this information, it is not possible to provide a definite answer for the carry and overflow flags in the given scenarios.

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The diameter of the 1 in./12 ft scale model of the Mercury-Redstone rocket is approximately 5.8 inches.

To calculate the diameter of the model, we need to determine the scale factor between the model and the actual rocket. In this case, the scale is given as 1 in./12 ft. This means that for every 12 feet of the actual rocket, the model represents 1 inch.

Given that the diameter of the actual rocket is 70 inches, we can set up a proportion to find the diameter of the model. Let's denote the diameter of the model as "x":

(1 in.) / (12 ft) = x / (70 in.)

To solve this proportion, we can cross-multiply and then divide:

1 in. * 70 in. = 12 ft * x

70 = 12x

x = 70 / 12 ≈ 5.83 inches

Rounding to the nearest half inch, the diameter of the model is approximately 5.8 inches.

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(a) The explicit formula R(n) = 2n - 1.

(b) L(n) = n(n - 1).

(c) Number of odd numbers = 1 - n² + 3n.

(d) an = n³ + 2n² + n + 2.

(a) The explicit formula R(n) for the rightmost odd number on the left-hand side of the nth row, let's examine the pattern. In each row, the number of odd numbers on the left side is equal to the row number (n).

The first row (n = 1) has 1 odd number: a1.

The second row (n = 2) has 2 odd numbers: a2 and 3.

The third row (n = 3) has 3 odd numbers: 5, 7, and 9.

We can observe that in the nth row, the first odd number is given by n, and the subsequent odd numbers are consecutive odd integers. Therefore, we can express R(n) as:

R(n) = n + (n - 1) = 2n - 1.

To justify this formula, we can use mathematical induction. First, we verify that R(1) = 1, which matches the first row. Then, assuming the formula holds for some arbitrary kth row, we can show that it holds for the (k+1)th row:

R(k+1) = k + 1 + k = 2k + 1.

Since 2k + 1 is the (k+1)th odd number, the formula holds for the (k+1)th row.

(b) The formula L(n) for the leftmost odd number in the nth row, we can observe that the leftmost odd number in each row is given by the sum of odd numbers from 1 to (n-1). We can express L(n) as:

L(n) = 1 + 3 + 5 + ... + (2n - 3).

To justify this formula, we can use the formula for the sum of an arithmetic series:

S = (n/2)(first term + last term).

In this case, the first term is 1, and the last term is (2n - 3). Plugging these values into the formula, we have:

S = (n/2)(1 + 2n - 3) = (n/2)(2n - 2) = n(n - 1).

Therefore, L(n) = n(n - 1).

(c) The number of odd numbers on the left-hand side in the nth row can be calculated by subtracting the leftmost odd number from the rightmost odd number and adding 1. Therefore, the number of odd numbers in the nth row is:

Number of odd numbers = R(n) - L(n) + 1 = (2n - 1) - (n(n - 1)) + 1 = 2n - n² + n + 1 = 1 - n² + 3n.

(d) Based on the previous steps and the fact that each row has an even distribution of odd numbers, we can argue that the value of an, which represents the sum of odd numbers in the nth row, should be equal to the sum of the odd numbers in that row. Using the formula for the sum of an arithmetic series, we can find the sum of the odd numbers in the nth row:

Sum of odd numbers = (Number of odd numbers / 2) * (First odd number + Last odd number).

Sum of odd numbers = ((1 - n² + 3n) / 2) * (L(n) + R(n)).

Substituting the formulas for L(n) and R(n) from earlier, we get:

Sum of odd numbers = ((1 - n² + 3n) / 2) * (n(n - 1) + 2

n - 1).

Simplifying further:

Sum of odd numbers = (1 - n² + 3n) * (n² - n + 1).

Sum of odd numbers = n³ - n² + n - n² + n - 1 + 3n² - 3n + 3.

Sum of odd numbers = n³ + 2n² + n + 2.

Hence, the value of an is given by the sum of the odd numbers in the nth row, which is n³ + 2n² + n + 2.

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The standard deviation for the total portfolio returns can be calculated using the weighted average of the standard deviations of the index fund and the risk-free asset. The standard deviation for the total portfolio returns is 10.5%.


The standard deviation of a portfolio measures the variability or risk associated with the portfolio's returns. In this case, the portfolio is 70% invested in an index fund (with a standard deviation of returns of 15%) and 30% invested in a risk-free asset.

To calculate the standard deviation of the total portfolio returns, we use the weighted average formula:

Standard deviation of portfolio returns = √[(Weight of index fund * Standard deviation of index fund)^2 + (Weight of risk-free asset * Standard deviation of risk-free asset)^2 + 2 * (Weight of index fund * Weight of risk-free asset * 1Covariance  between index fund and risk-free asset)]

Since the risk-free asset has a standard deviation of zero (as it is risk-free), the second term in the formula becomes zero. Additionally, the covariance between the index fund and the risk-free asset is also zero because they are independent. Therefore, the formula simplifies to:

Standard deviation of portfolio returns = Weight of index fund * Standard deviation of index fund

Plugging in the values, we get:

Standard deviation of portfolio returns = 0.70 * 15% = 10.5%

Hence, the standard deviation for the total portfolio returns is 10.5%. This means that the total portfolio's returns are expected to have a variability or risk represented by this standard deviation.

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Answer:

[tex]V=\frac{32}{3} \pi[/tex]

Step-by-step explanation:

We first need to know the formula to find the volume of a sphere.

The formula to find the volume of a sphere is:

(Where V is the volume and r is the radius of the sphere)

If the radius of the sphere is 2, then we can insert that into the formula for r:

Therefore the answer is [tex]V=\frac{32}{3} \pi[/tex].

a)The industry output will be: Q = qA + qB.

b) The industry output will be: Q = qA + qB.

c) Both firms would earn a higher profit if they agree on the industry output.

a) Bertrand Model:

In the Bertrand Model, both firms produce the same quality products at a constant marginal cost of X. Both companies attempt to maximize their own profits by selecting the lowest price. Firm A produces qA, while firm B produces qB. The firms would earn no profits if they set the same price.

Assume that each firm offers the same price P. The industry supply will be Q = qA + qB. The market demand is given by Q = Y - P. Substituting the value of Q, we get: Y - P = qA + qB.

The industry price is given by: P = (Y - Q)/2 = (Y - qA - qB)/2. Putting the value of Y and Q, we have: P = [(B + C) - (A + D/2) - qA - qB]/2 = (B + C - A - D/2)/2 - qA/2 - qB/2.

The industry output will be: Q = qA + qB.

Consumer surplus is given by the difference between what consumers are willing to pay and the market price of a good, summed over all customers. The consumer surplus is calculated by taking the area between the demand curve and the market price up to the equilibrium output.

Consumer Surplus = 1/2 (B + C - A - D/2 - P) * Q = 1/2 (B + C - A - D/2 - [(qA + qB)/2]) * [(qA + qB)].

Industry profit is given by: π = qA * P + qB * P - X(qA + qB) = qA * qB / 2Q - X(Q/2).

Deadweight Loss (DWL) is the loss of economic efficiency that occurs when the equilibrium output is not achieved. DWL is given by: DWL = [1/2 (qa + qb) - Q]/2.

b) Cournot Model:

In the Cournot Model, both firms produce identical products with a constant marginal cost of X. Both firms attempt to maximize their profits by selecting their output levels qA and qB. Let Q = qA + qB be the industry's output.

Substituting the value of Q, we get: Y - P = qA + qB.

The industry price is given by: P = (Y - qA - qB)/2 = (B + C - A - D/2)/2 - qA/2 - qB/2.

The industry output will be: Q = qA + qB.

Consumer surplus is given by the difference between what consumers are willing to pay and the market price of a good, summed over all customers. The consumer surplus is calculated by taking the area between the demand curve and the market price up to the equilibrium output.

Consumer Surplus = 1/2 (B + C - A - D/2 - P) * Q.

Industry profit is given by: π = (qA + qB) * (P - X) - (qA^2 + qB^2)/2.

Deadweight Loss (DWL) is the loss of economic efficiency that occurs when the equilibrium output is not achieved. DWL is given by: DWL = [(qa - qb)^2 - (qA + qB)^2]/2.

c) Tacit Collusion Model:

In the tacit collusion model, both firms in the industry aim to maximize their collective profits. Both firms would earn a higher profit if they agree on the industry output. The firms produce identical products at a constant marginal cost

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The probability of getting an even number on the spinner after one spin is: 1/2

We are given the spinner as shown in the attached image and we see that it has the following numbers:

5, 6, 2 and 3

Now, we want to find the probability of getting an even number for each spin.

The probability is:

Probability = Number of favorable outcomes/Total number of outcomes.

There are two even numbers out of the 4 numbers on the spinner.

Thus:

P(even number) = 2/4 = 1/2

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a) The function y₁(t) is (2/3)[tex]e^{2t/7}[/tex] + (1/3)[tex]e^{-4t/7}[/tex].

b) The function y₂(t) is (4/3)[tex]e^{2t/7}[/tex] - (4/3)[tex]e^{-4t/7}[/tex].

c) The Wronskian W(t) is (-2/3)[tex]e^{2t/7}[/tex] + (1/3)[tex]e^{-4t/7}[/tex].

a) To find the function y₁(t) which is the solution of 49y′′ + 14y′ − 8y = 0 with initial conditions y₁(0) = 1 and y₁′(0) = 0, we can assume a solution of the form y₁(t) = [tex]e^{rt}[/tex], where r is a constant.

Taking the derivatives, we have:

y₁′(t) = r[tex]e^{rt}[/tex]

y₁′′(t) = r²[tex]e^{rt}[/tex]

Substituting these into the differential equation, we get:

49(r²[tex]e^{rt}[/tex]) + 14(r[tex]e^{rt}[/tex]) - 8([tex]e^{rt}[/tex]) = 0

Simplifying the equation:

[tex]e^{rt}[/tex] * (49r² + 14r - 8) = 0

For this equation to hold true for all t, the expression inside the parentheses must equal zero:

49r² + 14r - 8 = 0

To solve this quadratic equation, we can use the quadratic formula:

r = (-b ± √(b² - 4ac)) / 2a

In this case, a = 49, b = 14, and c = -8. Plugging in the values, we get:

r = (-14 ± √(14² - 4 * 49 * -8)) / (2 * 49)

r = (-14 ± √(196 + 1568)) / 98

r = (-14 ± √(1764)) / 98

r = (-14 ± 42) / 98

Simplifying further:

r₁ = (28 / 98) = 2/7

r₂ = (-56 / 98) = -4/7

Thus, the solutions for r are r₁ = 2/7 and r₂ = -4/7.

Now, we can write the general solution:

y₁(t) = C₁[tex]e^{2t/7}[/tex] + C₂[tex]e^{-4t/7[/tex]

Applying the initial conditions, we have:

y₁(0) = C₁[tex]e^0[/tex] + C₂[tex]e^0[/tex] = C₁ + C₂ = 1

y₁′(0) = (2/7)C₁[tex]e^0[/tex] + (-4/7)C₂[tex]e^0[/tex] = (2/7)C₁ - (4/7)C₂ = 0

From these equations, we can solve for C₁ and C₂:

C₁ + C₂ = 1 --> C₁ = 1 - C₂

(2/7)C₁ - (4/7)C₂ = 0

Substituting the value of C₁ from the first equation into the second equation, we get:

(2/7)(1 - C₂) - (4/7)C₂ = 0

(2/7) - (2/7)C₂ - (4/7)C₂ = 0

(6/7)C₂ = - (2/7)

C₂ = 1/3

Substituting the value of C₂ back into the first equation, we find:

C₁ = 1 - C₂ = 1 - 1/3 = 2/3

Therefore, the function y₁(t) which satisfies the given differential equation and initial conditions is:

y₁(t) = (2/3)[tex]e^{2t/7[/tex] + (1/3)[tex]e^{-4t/7[/tex]

b) To find the function y₂(t) which is the solution of 49y′′ + 14y′ − 8y = 0 with initial conditions y₂(0) = 0 and y₂′(0) = 1, we follow a similar process as in part (a).

Assuming a solution of the form y₂(t) = e^(rt), we get:

49(r²[tex]e^{rt[/tex]) + 14(r[tex]e^{rt[/tex]) - 8([tex]e^{rt[/tex]) = 0

This leads to the equation:

49r² + 14r - 8 = 0

Solving this quadratic equation, we find:

r₁ = 2/7

r₂ = -4/7

The general solution becomes:

y₂(t) = C₃[tex]e^{2t/7[/tex] + C₄[tex]e^{-4t/7[/tex]

Applying the initial conditions:

y₂(0) = C₃[tex]e^0[/tex] + C₄[tex]e^0[/tex] = C₃ + C₄ = 0

y₂′(0) = (2/7)C₃[tex]e^0[/tex] - (4/7)C₄[tex]e^0[/tex] = (2/7)C₃ - (4/7)C₄ = 1

Solving these equations, we find:

C₃ = 4/3

C₄ = -4/3

Therefore, the function y₂(t) which satisfies the given differential equation and initial conditions is:

y₂(t) = (4/3)[tex]e^{2t/7[/tex] - (4/3)[tex]e^{-4t/7[/tex]

c) The Wronskian, denoted by W(t), is given by the determinant of the matrix formed by the coefficients of y₁(t) and y₂(t) and their derivatives:

W(t) = | y₁(t) y₂(t) |

| y₁′(t) y₂′(t) |

We already found y₁(t) and y₂(t) in parts (a) and (b), so we can now find their derivatives and calculate the Wronskian.

Taking the derivatives:

y₁′(t) = (2/7)[tex]e^{2t/7[/tex] - (4/7)[tex]e^{-4t/7[/tex]

y₂′(t) = (4/7)[tex]e^{2t/7[/tex] + (4/7)[tex]e^{-4t/7[/tex]

Substituting these derivatives into the Wronskian formula:

W(t) = | (2/3)[tex]e^{2t/7[/tex] + (1/3)[tex]e^{-4t/7[/tex] (4/3)[tex]e^{2t/7[/tex] - (4/3)[tex]e^{-4t/7[/tex] |

| (2/7)[tex]e^{2t/7[/tex] - (4/7)[tex]e^{-4t/7[/tex] (4/7)[tex]e^{2t/7[/tex] + (4/7)[tex]e^{-4t/7[/tex] |

Simplifying the determinant, we get:

W(t) = (2/3)[tex]e^{2t/7[/tex] + (1/3)[tex]e^{-4t/7[/tex] - (4/3)[tex]e^{2t/7[/tex] + (4/3)[tex]e^{-4t/7[/tex]

= (-2/3)[tex]e^{2t/7[/tex] + (1/3)[tex]e^{-4t/7[/tex]

Therefore, the Wronskian W(t) is given by:

W(t) = (-2/3)[tex]e^{2t/7[/tex] + (1/3)[tex]e^{-4t/7[/tex]

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The solution is given by

y = yc + yp

  = c1 e^(-4x) + c2 xe^(-4x) - x^3/64 e^(-4x)

where c1 and c2 are constants.

Given differential equation is y" + 8y' + 16y = e^(-4x) x^3

To find particular solution, we consider the trial solution of the form, yp = (ax^3 + bx^2 + cx + d)e^(-4x)

Differentiate the above equation,

yp' = [(3ax^2 + 2bx + c)e^(-4x) + (-4ax^3 - 4bx^2 - 4cx - 4d)e^(-4x)]

yp" = [(6ax + 2b - 8ax^2 - 8bx - 8c)e^(-4x) + (16ax^3 + 16bx^2 + 16cx + 16d)e^(-4x)]

Substitute these values in the differential equation,

y" + 8y' + 16y = e^(-4x) x^3[(6ax + 2b - 8ax^2 - 8bx - 8c)e^(-4x) + (16ax^3 + 16bx^2 + 16cx + 16d)e^(-4x)] + 8[(3ax^2 + 2bx + c)e^(-4x) + (-4ax^3 - 4bx^2 - 4cx - 4d)e^(-4x)] + 16[(ax^3 + bx^2 + cx + d)e^(-4x)] = e^(-4x) x^3

Simplify the equation,

[-48ax^3 + 8bx^2 + 8cx + (16a - 16b + 16c + 16d)]e^(-4x) + (16ax^3 + 16bx^2 + 16cx + 16d)e^(-4x) + 8(3ax^2 + 2bx + c)e^(-4x) = x^3 e^(-4x)

Integrating the above equation,

we geta = 0b = 0c = 0d = -1/64

Therefore, the particular solution is

yp = -x^3/64 e^(-4x)

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Answer:

Jolon mistakingly added 15 to both sides of the equation in Step 3.  Step 3's correct answer is -2 + 15 = -15 + 15 + b, Step 4's correct answer is -17 = b, and Step 5's correct answer is y = 3x - 17

Step-by-step explanation:

It appears that you're trying to identify Jolon's mistake.  If you're trying to do something else, type it in the comments as the answer I'm providing identifies Jolon's mistake.

-2 = 3(5) - 17

-2 = 15 - 17

-2 = -2

Thus, Step 3 should be:  (-2 + 15) = (-15 + 15 + b), Step 4 should be:  -17 = b, and Step 5 should be:  y = 3x - 17

The answer is:

Work/explanation:

We need to write the equation in slope intercept form.

y = mx + b

where m = slope and b = y intercept; x and y are the co-ordinates of a point on the line

Plug in the data

[tex]\sf{y=mx+b}[/tex]

[tex]\sf{y=3x+b}[/tex]

[tex]\sf{-2=3(5)+b}[/tex]

[tex]\sf{-2=15+b}[/tex]

[tex]\sf{-2-15=b}[/tex]

[tex]\sf{-17=b}[/tex]

Hence, the answer is y = 3x - 17; Jolon was wrong because he shouldn't have added 15 to each side; he should have subtracted it instead. Also, 15 + 15 doesn't cancel out to 0. As a result, he got a wrong answer. The right one is y = 3x - 17.