Given that the area is 21 * |Im(zw conjugate)|, we can conclude that the area of the triangle formed by the complex numbers 0, z, and w is indeed 21 * |Im(zw conjugate)|. To find the area of the triangle formed by the complex numbers 0, z, and w, we can use the Shoelace Formula.
The Shoelace Formula states that the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is equal to: Area =In this case, we have the complex numbers 0, z, and w. We can represent z and w as[tex]z = x1 + y1i and w = x2 + y2i,[/tex]where x1, y1, x2, and y2 are real numbers. Since the complex number 0 has no imaginary part, we can represent it as 0 + 0i.
Now, let's plug these values into the Shoelace Formula: [tex]Area = 1/2 * |(0)(y1 - y2) + (x1)(y2 - 0) + (x2)(0 - y1)|[/tex]Simplifying this expression gives: Area =[tex]1/2 * |x1(y2 - 0) + x2(0 - y1)|[/tex]
Area[tex]= 1/2 * |x1y2 + x2(-y1)| Area = 1/2 * |x1y2 - x2y1|[/tex].
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The area of the triangle formed by the complex numbers 0, z, and w is given by 21 * absolute value of the imaginary part of (zw-conjugate).
Let's consider the complex numbers 0, z, and w. The triangle formed by these complex numbers has its vertices at the origin (0), z, and w. To find the area of this triangle, we need to calculate the absolute value of the imaginary part of the product of zw-conjugate and then multiply it by 21. The complex number zw-conjugate represents the difference between the complex numbers zw and its conjugate. The conjugate of a complex number is obtained by changing the sign of its imaginary part. By multiplying zw-conjugate, we obtain a complex number with a real part and an imaginary part. The imaginary part of this complex number represents the signed area of the parallelogram formed by zw and its conjugate. Since we want the area of a triangle, we take the absolute value of the imaginary part. Finally, we multiply the absolute value of the imaginary part of (zw-conjugate) by 21 to get the area of the triangle. The area of the triangle formed by the complex numbers 0, z, and w is equal to 21 times the absolute value of the imaginary part of (zw-conjugate).
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You may need to use the appropriate appendix table or technology to answer this question. A random sample of 89 airline pilots recruited by an airline service had an average yearly income of $99,100 with a standard deviation $11,000. (b) Develop a 95% confidence interval for the average yearly income of all pilots (in dollars). (Round your answers to the nearest dollar.) to $
The average of the numbers 5, 8, 12, and 6 is 7.75.
To develop a 95% confidence interval for the average yearly income of all pilots, we can use the formula:
Confidence interval = sample mean ± (critical value * standard deviation / √sample size)
Given that the sample mean is $99,100, the standard deviation is $11,000, and the sample size is 89, we can calculate the critical value using the appendix table or technology.
Assuming a normal distribution, the critical value for a 95% confidence level with a sample size of 89 is approximately 1.96.
Plugging in the values, the confidence interval becomes:
Confidence interval = $99,100 ± (1.96 * $11,000 / √89)
Calculating this expression will give you the lower and upper bounds of the confidence interval. Round your answers to the nearest dollar.
Average, also known as the arithmetic mean, is a measure of central tendency that represents the typical value of a set of numbers. It is calculated by summing all the numbers in the set and dividing the sum by the total count of numbers.
Here's the formula to calculate the average (mean) of a set of numbers:
Average = Sum of all numbers / Total count of numbers
For example, let's calculate the average of the numbers 5, 8, 12, and 6:
Sum of all numbers = 5 + 8 + 12 + 6 = 31
Total count of numbers = 4
Average = 31 / 4 = 7.75
Therefore, the average of the numbers 5, 8, 12, and 6 is 7.75.
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A group of friends, which includes numerous mathematicians and non-mathematicians, are gathering for a social occasion. At one point, some collection of n of the partygoers will sit in n consecutive chairs. We always want at least two mathematicians sitting next to one another so they will have someone to talk shop to and we always want at least two non-mathematicians sitting next to one another. Furthermore, each row will always begin with a non-mathematician but may end with either type of individual. Let s
n
be the number of ways of seating n people in these n chairs. Compute and construct all possible arrangements for all values up to n=5. Find and prove the correctness of a recursive formula for s
n
.
To prove the correctness of the recursive formula, use mathematical induction, showing that it holds true for the base cases and then assuming it holds for n = k and proving it holds for n = k+1. This will establish the validity of the formula for all values of n.
To find the number of ways of seating n people in n chairs according to the given conditions, we can use recursion. Let's consider the base cases first.
For n = 1, there is only one person, so s₁ = 1.
For n = 2, there are two people, and we want at least two non-mathematicians sitting next to one another. Therefore, the only possible arrangement is NM, where N represents a non-mathematician and M represents a mathematician. So, s₂ = 1.
Now, let's consider the case when n > 2. We have two possibilities:
1. The last person in the row is a non-mathematician: In this case, we can ignore the last person and focus on the first n-1 chairs. The number of arrangements for n-1 people in n-1 chairs is sₙ₋₁. Thus, the number of arrangements for n people with the last person being a non-mathematician is sₙ₋₁.
2. The last person in the row is a mathematician: In this case, the last two chairs must have non-mathematicians sitting in them. We can ignore these two chairs and focus on the remaining n-2 chairs. The number of arrangements for n-2 people in n-2 chairs is sₙ₋₂. Thus, the number of arrangements for n people with the last person being a mathematician is sₙ₋₂.
Therefore, the recursive formula for sₙ is sₙ = sₙ₋₁ + sₙ₋₂, with base cases s₁ = 1 and s₂ = 1.
Using this formula, we can compute the number of arrangements for n = 3, 4, and 5 as follows:
For n = 3:
s₃ = s₂ + s₁ = 1 + 1 = 2
Possible arrangements: NNM, NMN
For n = 4:
s₄ = s₃ + s₂ = 2 + 1 = 3
Possible arrangements: NNMN, NNMM, NNMN
For n = 5:
s₅ = s₄ + s₃ = 3 + 2 = 5
Possible arrangements: NNMNN, NNMMN, NNMNM, NMNNM, NNMNN
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Question 4: Optimization (25 points) Find the maximum and minimum of the following functions over the indicated interval: f(x)=−2x−1 over [−3,5]f(x)=x3−4x+10 over [−10,10]f(x)=xx2+1 over [1,4] Question 1: Inverse Functions ( 25 points) Find the inverse function of the following functions: - y=7x+4 - y=x−2x+1 - y=ex+5 - y=x3+2 Question 2: Concave/Convex Functions (25 points) Are the following functions convex or concave? Why?: - f(x)=x2−2x+2 - f(x)=5x31 - f(x)=3x3+2x+1 Question 3: Derivative of Functions ( 25 points) Differentiate the following functions with respect to x : - f(x)=6x5−2x15 - f(x)=x−23x−5 - f(x)=x5x+1 - f(x)=(x2+x+1)5ln(x+1)
The maximum value is 5 and the minimum value is -11.The maximum value is approximately 13.84 and the minimum value is -1040. The maximum value is approximately 0.8 and the minimum value is -0.333.
1. For f(x) = -2x - 1 over [-3, 5]:
- Take the derivative of f(x) with respect to x: f'(x) = -2.
- Set f'(x) equal to zero and solve for x: -2 = 0. There are no solutions, so there are no critical points.
- Since the interval is finite, we evaluate f(x) at the endpoints:
- f(-3) = -2(-3) - 1 = 5.
- f(5) = -2(5) - 1 = -11.
- Therefore, the maximum value is 5 and the minimum value is -11.
2. For f(x) = x³ - 4x + 10 over [-10, 10]:
- Take the derivative of f(x) with respect to x: f'(x) = 3x² - 4.
- Set f'(x) equal to zero and solve for x: 3x² - 4 = 0.
- x = 2/√3 or x = -2/√3.
- Since the interval is finite, we evaluate f(x) at the endpoints and critical points:
- f(-10) = -10³ - 4(-10) + 10 = -1040.
- f(-2/√3) ≈ 8.16.
- f(2/√3) ≈ 13.84.
- f(10) = 10³ - 4(10) + 10 = 960.
- Therefore, the maximum value is approximately 13.84 and the minimum value is -1040.
3. For f(x) = x/(x^2 + 1) over [1, 4]:
- Take the derivative of f(x) with respect to x: f'(x) = (x² + 1 - 2x²) / (x² + 1)².
- Set f'(x) equal to zero and solve for x: (x²+ 1 - 2x²) / (x² + 1)² = 0.
- x² + 1 - 2x² = 0.
- -x² + 1 = 0.
- x² = 1.
- x = ±1.
- Since the interval is finite, we evaluate f(x) at the endpoints and critical points:
- f(1) = 1 / (1² + 1) ≈ 0.333.
- f(-1) = -1 / (1² + 1) ≈ -0.333.
- f(4) = 4 / (4² + 1) ≈ 0.8.
- Therefore, the maximum value is approximately 0.8 and the minimum value is -0.333.
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Use the product to sum formula to fill in the blanks in the identity below:
sin(9x) cos(8x) = 1/2 (sin_____x+sin ____x)
The missing terms in the identity sin(9x) cos(8x) = 1/2 (sin_____x+sin ____x) are sin(17x) and sin(x).To use the product to sum formula to fill in the blanks in the identity sin(9x) cos(8x) = 1/2 (sin_____x+sin ____x), we can apply the formula: sin(A) cos(B) = 1/2 [sin(A + B) + sin(A - B)]
Comparing this formula with the given identity, we can determine that A = 9x and B = 8x. Substituting these values, we get:
sin(9x) cos(8x) = 1/2 [sin(9x + 8x) + sin(9x - 8x)]
Simplifying further:
sin(9x) cos(8x) = 1/2 [sin(17x) + sin(x)]
Therefore, the identity can be rewritten as:
sin(9x) cos(8x) = 1/2 [sin(17x) + sin(x)]
In conclusion, the missing terms in the identity sin(9x) cos(8x) = 1/2 (sin_____x+sin ____x) are sin(17x) and sin(x).
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The conventional algorithm for evaluating a polynomial a
n
x
n
+a
n−1
x
n−1
+…+a
1
x+a
0
at x=c can be expressed in pseudocode by procedure polynomial (c,a
0
,a
1
,…,a
n
: real numbers ) where the final value of y is the value of the polynomial at x=c. a) Evaluate 3x
2
+x+1 at x=2 by working through each step of the algorithm showing the values assigned at each assignment step. b) Exactly how many multiplications and additions are used to evaluate a polynomial of degree n at x=c ? (Do not count additions used to incremen variable.)
a) Evaluating the polynomial 3x^2 + x + 1 at x = 2 using the conventional algorithm involves several assignment steps. The values assigned at each step are calculated and shown in detail.
b) To evaluate a polynomial of degree n at x = c using the conventional algorithm, there are a total of n multiplications and n additions required, excluding additions used to increment variables.
a) To evaluate the polynomial 3x^2 + x + 1 at x = 2, we follow the conventional algorithm step by step:
Assign c = 2.
Assign y = 0.
Assign y = y + (3 * c^2) = y + (3 * 2^2) = y + 12.
(Here, we calculate the value of the first term, 3x^2, by substituting c = 2 into the polynomial.)
Assign y = y + (1 * c) = y + (1 * 2) = y + 2.
(We calculate the value of the second term, x, by substituting c = 2.)
Assign y = y + 1.
(Finally, we calculate the value of the constant term, 1.)
The final value of y is the value of the polynomial at x = c, which in this case is 17.
b) To evaluate a polynomial of degree n at x = c using the conventional algorithm, there are n multiplications involved. Each term in the polynomial requires one multiplication with the corresponding coefficient and the value of c raised to the appropriate power.
Additionally, there are n additions required to accumulate the values of each term. Therefore, the total number of multiplications and additions is both equal to the degree of the polynomial, n.
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The function is continuous and symmetric 4. About the line x = 1. The function is positive
for 0 < x < 2. The function has a maximum at
(1, 1). As x → +[infinity], f(x) → −[infinity] and as x → −[infinity],
The function is continuous and symmetric about the line x = 1. It is positive for 0 < x < 2, has a maximum at (1, 1), and as x approaches positive or negative infinity, the function values approach negative infinity.
The function is continuous: This means that there are no sudden jumps or breaks in the graph of the function. It implies that the function is defined and has a value for every point within its domain.
The function is symmetric about the line x = 1: This means that if we reflect the graph of the function across the vertical line x = 1, the resulting graph will be identical to the original graph. In other words, for any value x, the function value at x is equal to the function value at 2 - x.
The function is positive for 0 < x < 2: This indicates that the function values are positive within the interval from 0 to 2 on the x-axis. In other words, the graph of the function lies above the x-axis within this interval.
The function has a maximum at (1, 1): This means that at x = 1, the function reaches its highest value. The function value at this point is 1.
As x approaches positive infinity, f(x) approaches negative infinity: This suggests that as x becomes very large (approaches infinity) in the positive direction, the function values become increasingly negative. The graph of the function will decline towards negative infinity.
As x approaches negative infinity, f(x) also approaches negative infinity: This implies that as x becomes very large in the negative direction (approaches negative infinity), the function values become increasingly negative. The graph of the function will decline towards negative infinity.
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17. A person is looking at a boat on the water. The angle of depression from the person's eyes to the boat is 20 degrees. If the person's eyes are 1.5 meters above the water level, how far is the boat from the person?
A) 4.5 meters
C) 6 meters
B) 5 meters
D) 7.5 meters
Step-by-step explanation:
I will assume you want the horizontal distance to the boat rather than the distance from the eyes to the boat
tan 20 = opposite leg / adjacent leg
tan 20 = 1.5 m / d
d = 4.12 m <====== I suppose answer 'A' is the closest answer
Find the matrix A of the quadratic form: 7Y
1
2
−8Y
1
Y
2
+8Y
2
2
The matrix A of the quadratic form 7Y₁² - 8Y₁Y₂ + 8Y₂² is [tex]\left[\begin{array}{ccc}7&-4\\-4&8\\\end{array}\right][/tex].
Consider the coefficients of each term in the expression. The general form of a quadratic form is given by:
Q(Y) = [tex]Y^{TAY}[/tex]
In this case, the quadratic form is:
Q(Y) = 7Y₁² - 8Y₁Y₂ + 8Y₂²
Denote the matrix A as follows:
A = [tex]\left[\begin{array}{ccc}a&b\\c&d\\\end{array}\right][/tex]
Equate the quadratic form with the general form and extract the coefficients:
7Y₁² - 8Y₁Y₂ + 8Y₂² = [tex]Y^{TAY}[/tex]
7Y₁² - 8Y₁Y₂ + 8Y₂² = aY₁² + bY₁Y₂ + cY₁Y₂ + dY₂²
Comparing the coefficients of each term, we can write the following equations:
a = 7
b + c = -8
d = 8
Determine the values of the matrix A:
a = 7
b = -4
c = -4
d = 8
So, the matrix A is:
[tex]\left[\begin{array}{ccc}7&-4\\-4&8\\\end{array}\right][/tex]
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Solve y
′
+3x
−1
y=x
9
,y(1)=4 (a) Identify the integrating factor, α(x). α(x)= (b) Find the general solution. y(x)= Note: Use C for an arbitrary constant. (c) Solve the initial value problem y(1)=4.
The solution to the initial value problem y(1) = 4 is: y(x) = (2/3)x^10 + (10/3)e^(-3/2x^2 + x - 1/2) .The general solution to the differential equation is y(x) = (2/3)x^10 + Ce^(-3/2x^2 + x), where C is an arbitrary constant.The integrating factor is given by α(x) = e^(∫(3x - 1)dx) = e^(3/2x^2 - x)
To solve the given differential equation y' + (3x - 1)y = x^9 with the initial condition y(1) = 4, we can follow :
(a) Identify the integrating factor, α(x):
The integrating factor is given by α(x) = e^(∫(3x - 1)dx) = e^(3/2x^2 - x)
(b) Find the general solution, y(x):
Multiply the given differential equation by the integrating factor α(x):
e^(3/2x^2 - x)[y' + (3x - 1)y] = e^(3/2x^2 - x)x^9
This can be simplified as follows:
[e^(3/2x^2 - x)y]' = e^(3/2x^2 - x)x^9
Integrate both sides with respect to x:
∫[e^(3/2x^2 - x)y]'dx = ∫e^(3/2x^2 - x)x^9 dx
Using the fundamental theorem of calculus, we can simplify the equation as:
e^(3/2x^2 - x)y = ∫e^(3/2x^2 - x)x^9 dx + C
Solve the integral on the right-hand side:
e^(3/2x^2 - x)y = (2/3)e^(3/2x^2 - x)x^10 + C
Divide both sides by e^(3/2x^2 - x):
y = (2/3)x^10 + Ce^(-3/2x^2 + x)
The general solution to the differential equation is y(x) = (2/3)x^10 + Ce^(-3/2x^2 + x), where C is an arbitrary constant.
(c) Solve the initial value problem y(1) = 4:
Substitute the initial condition into the general solution:
4 = (2/3)(1)^10 + Ce^(-3/2(1)^2 + 1)
Simplifying the equation gives:
4 = 2/3 + Ce^(-3/2 + 1)
4 - 2/3 = Ce^(1/2)
10/3 = Ce^(1/2)
C = (10/3)e^(-1/2)
Therefore, the solution to the initial value problem y(1) = 4 is:
y(x) = (2/3)x^10 + (10/3)e^(-3/2x^2 + x - 1/2)
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A divers descends 20 feet inthe water from the boat at the surface of a lake. He then rose 12 feet and descends another 18 feet. At this point what is his depth in water
The diver's depth in the water at this point is 26 feet.
To determine the diver's depth in the water, we can add up the total distance he has descended and subtract the total distance he has ascended.
The diver initially descends 20 feet from the surface of the lake. Then, he rises 12 feet and descends another 18 feet.
Total descent: 20 feet + 18 feet = 38 feet
Total ascent: 12 feet
To find the diver's depth in the water, we subtract the total ascent from the total descent:
Depth in water = Total descent - Total ascent
Depth in water = 38 feet - 12 feet
Depth in water = 26 feet
Therefore, the diver's depth in the water at this point is 26 feet.
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Which of the following criteria are used when deciding upon the
inclusion of a variable? Check all that apply.
Group of answer choices
A-Theory
B-t-statistic
C-Bias
D-Adjusted R^2
the criteria used when deciding upon the inclusion of a variable are A - Theory, B - t-statistic, C - Bias, and D - Adjusted R^2.
When deciding upon the inclusion of a variable, the following criteria are commonly used:
A - Theory: Theoretical justification is often considered to include a variable in a model. It involves assessing whether the variable is relevant and aligns with the underlying theory or conceptual framework.
B - t-statistic: The t-statistic is used to determine the statistical significance of a variable. A variable with a significant t-statistic suggests that it has a meaningful relationship with the dependent variable and may be included in the model.
C - Bias: Bias refers to the presence of systematic errors in the estimation of model parameters. It is important to consider the potential bias introduced by including or excluding a variable and assess whether it aligns with the research objectives.
D - Adjusted R^2: Adjusted R^2 is a measure of the goodness of fit of a regression model. It considers the trade-off between the number of variables included and the overall fit of the model. Adjusted R^2 helps in assessing whether the inclusion of a variable improves the model's explanatory power.
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Consider the function f(x)=xx over the interval [a,b]=[1/2,2]. The secant line between these values has slope given by: julia> f(x)=x∧xjulia>a,b,=1/2,2julia>m=(f(b)−f(a))/(b−a)2.195262145875635 There is exactly one point, c in [a,b] where f′(c)=m. Find the value numerically.
To find the point c where f'(c) = m, we need to find the derivative of f(x) first. Let's differentiate f(x) = x^x using the chain rule. f'(x) = (x^x) * (ln(x) + 1)
Now, we can find the value of c numerically by solving the equation f'(c) = m.m = (f(b) - f(a))/(b - a)2.195262145875635 = (b^b * (ln(b) + 1) - a^a * (ln(a) + 1))/(b - a)Substituting the values for a and b:
2.195262145875635 = (2^2 * (ln(2) + 1) - (1/2)^(1/2) * (ln(1/2) + 1))/(2 - 1/2)
Simplifying the equation:2.195262145875635 = (4 * (ln(2) + 1) - (1/2)^(1/2) * (-ln(2) + 1))/(3/2)To find the value numerically, we can use a numerical solver or calculator to solve this equation. The value of c is approximately 1.361.
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The value of c where f'(c) equals the slope of the secant line is found numerically using appropriate numerical methods.
To find the value of c where the derivative of the function f(x) = x^x equals the slope of the secant line, we need to solve the equation f'(c) = m. Let's proceed with the calculations:
The derivative of f(x) = x^x can be found using the logarithmic derivative. Taking the natural logarithm of both sides and differentiating, we have:
ln(f(x)) = ln(x^x)
ln(f(x)) = x * ln(x)
Now, differentiating both sides with respect to x using the chain rule:
1/f(x) * f'(x) = ln(x) + 1
f'(x) = f(x) * (ln(x) + 1)
f'(x) = x^x * (ln(x) + 1)
Substituting the values of a and b, we have:
f'(x) = x^x * (ln(x) + 1)
f'(a) = a^a * (ln(a) + 1)
f'(b) = b^b * (ln(b) + 1)
Now, we want to find c such that f'(c) = m:
m = f'(c) = c^c * (ln(c) + 1)
To solve this equation numerically, we can use numerical methods like the Newton-Raphson method or the bisection method. These methods involve iterations to approximate the solution. The exact value of c cannot be obtained algebraically since it involves a transcendental equation.
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a warehouse contains ten copier, four of which are defective. an employee of a company selects 5 of the machines at random in the belief all are in working order. the company that purchased the copiers will repair the defective oes at a cost of $250 each determine the mean and standard deviation of this cost
The mean cost of repairing the defective copiers is $500, indicating the average expense incurred by the company for repairs. The standard deviation of $433.01 represents the spread or variability of the repair costs around the mean. These calculations provide insights into the expected cost and the degree of uncertainty associated with the repair expenses for the randomly selected copiers.
To calculate the mean and standard deviation of the cost of repairing the defective copiers, we need to consider the probabilities associated with selecting different numbers of defective copiers.
Total number of copiers (N) = 10
Number of defective copiers (D) = 4
1. Mean Calculation:
The mean cost (μ) can be calculated by multiplying the probability of selecting a particular number of defective copiers by the cost of repairing each defective copier, and then summing up the results.
The probability of selecting 0 defective copiers is (6/10)*(5/9)*(4/8)*(3/7)*(2/6) = 0.1429.
The probability of selecting 1 defective copier is (4/10)*(6/9)*(5/8)*(4/7)*(3/6) = 0.2857.
The probability of selecting 2 defective copiers is (4/10)*(3/9)*(6/8)*(5/7)*(4/6) = 0.2857.
The probability of selecting 3 defective copiers is (4/10)*(3/9)*(2/8)*(6/7)*(5/6) = 0.1429.
The probability of selecting 4 defective copiers is (4/10)*(3/9)*(2/8)*(1/7)*(6/6) = 0.0476.
The mean cost is calculated as:
μ = (0.1429 * 0) + (0.2857 * $250) + (0.2857 * $500) + (0.1429 * $750) + (0.0476 * $1,000)
= $500
2. Standard Deviation Calculation:
The standard deviation (σ) can be calculated by taking the square root of the sum of the squared differences between the cost and the mean, multiplied by the probabilities.
The standard deviation is calculated as:
σ = sqrt((0.1429 * (0 - $500)^2) + (0.2857 * ($250 - $500)^2) + (0.2857 * ($500 - $500)^2) + (0.1429 * ($750 - $500)^2) + (0.0476 * ($1,000 - $500)^2))
= $433.01
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Consider the following set of constraints:
x
1
+x
2
+x
3
=7
2x
1
−5x
2
+x
3
≥10
x
1
,x
2
,x
3
≥0
Solve the problem for each of the following objective functions: (a) Maximize z=2x
1
+3x
2
−5x
3
. (b) Minimize z=2x
1
+3x
2
−5x
3
. (c) Maximize z=x
1
+2x
2
+x
3
. (d) Minimize z=4x
1
−8x
2
+3x
3
. Solve (a) and (b) using M-Method, (c) and (d) using Two Phase Method.
To solve the given problem using the M-Method, we need to follow these steps for each objective function:
(a) Maximize z=2x1+3x2−5x3:
Step 1: Convert the inequality constraints into equations by adding slack variables:
x1 + x2 + x3 + s1 = 7
2x1 - 5x2 + x3 - s2 = 10
Step 2: Convert the objective function into the standard form by introducing surplus variables:
Maximize z = 2x1 + 3x2 - 5x3 + 0s1 + 0s2
Step 3: Set up the initial table:
CBi x1 x2 x3 s1 s2 Zj Cj-Zj Ratio
0 1 1 1 1 0 0 0
Step 4: Perform iterations until the optimal solution is reached.
(b) Minimize z=2x1+3x2−5x3:
Repeat Steps 1 to 4, but with the objective function Minimize z = 2x1 + 3x2 - 5x3 + 0s1 + 0s2. The iterations will lead to the optimal solution.
To solve the problem using the Two Phase Method, we need to follow these steps for each objective function:
(c) Maximize z=x1+2x2+x3:
Step 1: Convert the inequality constraints into equations by adding slack variables:
x1 + x2 + x3 + s1 = 7
2x1 - 5x2 + x3 - s2 = 10
Step 2: Set up the initial table for Phase 1:
CBi x1 x2 x3 s1 s2 Zj Cj-Zj Ratio
-1 1 1 1 1 0 0 0
Step 3: Perform iterations until a basic feasible solution is reached.
Step 4: Set up the initial table for Phase 2:
CBi x1 x2 x3 s1 s2 Zj Cj-Zj Ratio
1 1 1 1 1 0 0 0
Step 5: Perform iterations until the optimal solution is reached.
(d) Minimize z=4x1−8x2+3x3:
Repeat Steps 1 to 5, but with the objective function Minimize z = 4x1 - 8x2 + 3x3 + 0s1 + 0s2. The iterations will lead to the optimal solution.
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5. For the following problems, using the Big M method, construct the complete first simplex tableau for the simplex method. Identify the initial entering basic variable and the leaving basic variable.
Minimize Z = 2x₁ + 3x₂ + x₃
subject to
x₁ + 4x₂ + 2x₃ ≥ 8
3x₁ + 2x₂ ≥ 6
and
x₁ ≥ 0, x₂ ≥ 0, x₃ ≥ 0.
The Big M method is a technique used in linear programming to solve problems with constraints and objective functions. It is an extension of the simplex method that handles cases where some constraints have strict inequalities (i.e., ">") or the objective function includes a term to be minimized.
To construct the first simplex tableau using the Big M method, follow these steps:
1. Write the objective function in standard form:
[tex]Z = 2x₁ + 3x₂ + x₃[/tex]
2. Introduce slack variables to convert the inequalities into equations:
[tex]x₄ = 8 - x₁ - 4x₂ - 2x₃\\x₅ = 6 - 3x₁ - 2x₂[/tex]
3. Add a big M term to the objective function for each slack variable:
[tex]Z = 2x₁ + 3x₂ + x₃ + M₁x₄ + M₂x₅[/tex]
4. Convert the inequalities into equations by adding surplus variables for ">=" constraints:
[tex]x₆ = x₁ + 4x₂ + 2x₃ - 8\\x₇ = 3x₁ + 2x₂ - 6[/tex]
5. Add a big M term to the objective function for each surplus variable:
[tex]Z = 2x₁ + 3x₂ + x₃ + M₁x₄ + M₂x₅ + M₃x₆ + M₄x₇[/tex]
6. Write the initial simplex tableau using the augmented matrix:
```
[ 2 3 1 0 0 0 0 0 ]
[ -1 -4 -2 1 0 0 0 8 ]
[ -3 -2 0 0 1 0 0 6 ]
[ 1 4 2 0 0 1 0 -8 ]
[ 3 2 0 0 0 0 1 -6 ]
```
7. Identify the entering and leaving basic variables:
- The entering variable is the column with the most negative coefficient in the objective row.
In this case, it is the second column (x₂).
- The leaving variable is the row with the smallest non-negative ratio of the right-hand side to the entering variable's coefficient.
In this case, it is the third row (x₆).
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Using the Big M method, the complete first simplex tableau for the given linear programming problem is constructed as follows:
┌─────────────┬──────┬───────┬───────┬─────┬─────┬─────────────┐
│ BV │ x₁ │ x₂ │ x₃ │ s₁ │ s₂ │ RHS │
├─────────────┼──────┼───────┼───────┼─────┼─────┼─────────────┤
│ Z │ 2 │ 3 │ 1 │ 0 │ 0 │ 0 │
├─────────────┼──────┼───────┼───────┼─────┼─────┼─────────────┤
│ x₁ + 4x₂ + │ 1 │ 4 │ 2 │ -1 │ 0 │ -8 │
│ 2x₃ - M │ │ │ │ │ │ │
├─────────────┼──────┼───────┼───────┼─────┼─────┼─────────────┤
│ 3x₁ + 2x₂ │ 3 │ 2 │ 0 │ 0 │ -1 │ 6 │
├─────────────┼──────┼───────┼───────┼─────┼─────┼─────────────┤
│ x₁ │ 1 │ 0 │ 0 │ 1 │ 0 │ 0 │
├─────────────┼──────┼───────┼───────┼─────┼─────┼─────────────┤
│ x₂ │ 0 │ 1 │ 0 │ 0 │ 1 │ 0 │
├─────────────┼──────┼───────┼───────┼─────┼─────┼─────────────┤
│ x₃ │ 0 │ 0 │ 1 │ 0 │ 0 │ 0 │
└─────────────┴──────┴───────┴───────┴─────┴─────┴─────────────┘
The initial entering basic variable is x₁, which has the most negative coefficient in the objective row. The leaving basic variable is x₃, determined by selecting the row with the smallest positive ratio of the right-hand side (RHS) to the entering column's coefficient. In this case, the ratio for the fourth row (0/1) is the smallest, so x₃ leaves the basis.
To construct the complete first simplex tableau using the Big M method, we first convert the given problem into standard form by introducing slack variables (s₁ and s₂) for the inequalities and a large positive value (M) to penalize the artificial variable in the objective function.
The first row represents the objective function, where the coefficients of the decision variables x₁, x₂, and x₃ are taken directly from the given problem. The slack and artificial variables (s₁ and s₂) have coefficients of 0 since they don't appear in the objective function.
The subsequent rows represent the constraints. Each row corresponds to one constraint, where the coefficients of the decision variables, slack variables, and the artificial variable are taken from the original problem. The right-hand side (RHS) values are also copied accordingly.
The initial entering basic variable is determined by selecting the most negative coefficient in the objective row, which is x₁ in this case. The leaving basic variable is determined by finding the smallest positive ratio of the RHS to the entering column's coefficient. Since the ratio for the fourth row (0/1) is the smallest, x₃ leaves the basis.
The resulting tableau serves as the starting point for applying the simplex method to solve the linear programming problem iteratively.
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Solve (1+x
2
)y
′′
−2xy
′
=0 given that y
1
=1 is a solution.
The solution to the given differential equation is[tex]y = ax^{(4/3)[/tex].
To solve the given differential equation [tex](1+x^2)y'' - 2xy' = 0[/tex], we can use the method of undetermined coefficients.
First, let's find the first and second derivatives of y with respect to x.
y' = dy/dx
[tex]y'' = d^2y/dx^2[/tex]
Now, substitute y' and y'' back into the differential equation:
[tex](1+x^2)(d^2y/dx^2) - 2x(dy/dx) = 0[/tex]
Next, we can assume a solution of the form [tex]y = ax^n[/tex], where a and n are constants.
Taking the first derivative, we have:
[tex]y' = anx^{(n-1)[/tex]
Taking the second derivative, we have:
[tex]y'' = an(n-1)x^{(n-2)[/tex]
Substitute these derivatives back into the differential equation:
[tex](1+x^2)(an(n-1)x^{(n-2)}) - 2x(anx^{(n-1)}) = 0[/tex]
Simplifying, we get:
[tex]an(n-1)x^n + an(n-1)x^{(n-2)} - 2anx^n = 0[/tex]
Now, let's cancel out the common terms and rearrange the equation:
[tex]an(n-1)x^n + an(n-1)x^{(n-2)} - 2anx^n = 0[/tex]
[tex]an(n-1)x^n - 2anx^n + an(n-1)x^{(n-2)} = 0[/tex]
Factor out the common term '[tex]anx^n[/tex]':
[tex]anx^n[(n-1) - 2 + (n-1)x^{(n-2)}] = 0[/tex]
Since this equation must hold true for all values of x, we can set the expression inside the square brackets equal to zero:
[tex](n-1) - 2 + (n-1)x^{(n-2)} = 0[/tex]
Simplifying further, we have:
[tex]n - 3 + (n-1)x^{(n-2)} = 0[/tex]
[tex]2n - 3 + (n-1)x^{(n-2)} = 0[/tex]
To find the value of n, we need to use the initial condition y'(1) = 1.
Substituting x = 1 into the equation above, we get:
2n - 3 + (n-1) = 0
3n - 4 = 0
3n = 4
n = 4/3
So, the solution to the given differential equation is[tex]y = ax^{(4/3)[/tex].
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Pls help I always get these ones in wrong
Answer:
Step-by-step explanation:
You want to isolate x by adding 7 to all 3 parts of the inequality
so:
20 < x < 38
This means that x is between 20 and 38
You would treat the inequalities like = signs but the only thing that changes is if you multiply or divide by a negative then the signs flip.
Name an increasing problem that needs to be solved. Briefly, offer an anecdote, case study, or scenario to prove that it is a problem. Prove by statistics that this problem is increasing.
this is the outline to use
problem:
Proof it is a problem [Case Study]:
Proof the problem is increasing [Statistics]
Climate change is a pressing problem that requires immediate action. The case study of the Great Barrier Reef and the statistics regarding rising temperatures and increasing extreme weather events clearly demonstrate the severity and increasing nature of this problem. It is essential for individuals, communities, and governments to come together to mitigate climate change and protect the planet for future generations.
Problem: Climate Change
Explanation: Climate change is a global issue that needs to be urgently addressed. Rising temperatures, extreme weather events, and melting glaciers are just a few examples of the detrimental effects of climate change. It is crucial to understand the severity of this problem through a case study.
Case Study: In recent years, the Great Barrier Reef, located in Australia, has experienced significant bleaching events due to warmer ocean temperatures caused by climate change. Coral bleaching occurs when coral polyps expel the algae living in their tissues, leading to the coral turning white and eventually dying. This case study highlights the devastating impact of climate change on one of the world's most biodiverse ecosystems.
Proof the problem is increasing - Statistics:
1. According to the National Aeronautics and Space Administration (NASA), the Earth's average surface temperature has risen by about 1.1 degrees Celsius since the late 19th century, with most of the warming occurring in the past few decades.
2. The Intergovernmental Panel on Climate Change (IPCC) reports that the frequency and intensity of extreme weather events, such as hurricanes, heatwaves, and droughts, have been increasing over the past few decades.
3. The World Wildlife Fund (WWF) states that since 1970, global wildlife populations have declined by an average of 68%, primarily due to habitat destruction, pollution, and climate change.
Conclusion: Climate change is a pressing problem that requires immediate action. The case study of the Great Barrier Reef and the statistics regarding rising temperatures and increasing extreme weather events clearly demonstrate the severity and increasing nature of this problem. It is essential for individuals, communities, and governments to come together to mitigate climate change and protect the planet for future generations.
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Given that there are equal numbers of vegetables in basket. There are 21 vegetables in 7 baskets. There are vegetables in one basket.
There are 3 vegetables in one basket.
To find the number of vegetables in one basket, we can divide the total number of vegetables by the total number of baskets.
Given that there are 21 vegetables in 7 baskets, we can divide 21 by 7 to find the number of vegetables in one basket.
21 ÷ 7 = 3
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use the given data to answer the question. what is the critical value for the f- test at 95% confidence? df ss ms f p
To determine the critical value for the F-test at 95% confidence, we need the degrees of freedom for the numerator (df) and the denominator (df), along with the significance level (α) to find the critical value from an F-distribution table.
The F-test is used to compare the variances of two populations. In this case, we need additional information such as the sum of squares (SS), mean squares (MS), and the p-value to calculate the critical value for the F-test accurately.
Without the values for df, SS, MS, F, and p, it is not possible to provide a specific calculation for the critical value at 95% confidence.
The critical value for the F-test at 95% confidence depends on the degrees of freedom, sum of squares, mean squares, F-statistic, and p-value. To determine the specific critical value, these values are required. Without the given data, it is not possible to provide an exact answer.
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the transitive property of angle congruence states that if ≅ ∠qrs and ∠qrs ≅ ∠pqr, then ∠xyz ≅ ∠pqr .
Step-by-step explanation:
I'm sorry, but the statement you provided is not correct. The transitive property of angle congruence actually states that if ∠QRS ≅ ∠PQR and ∠PQR ≅ ∠XYZ, then ∠QRS ≅ ∠XYZ. In other words, if two angles are congruent to a third angle, then they are congruent to each other.
So, in your statement, if ∠QRS ≅ ∠PQR and ∠PQR ≅ ∠XYZ, then you can conclude that ∠QRS ≅ ∠XYZ.
The product of an even integer and an odd integer is even. f. The sum of two primes is never a prime. g. The sum of a rational number and an irrational number is irrational. h. 3
is irrational. i. ∣xy∣=∣x∣⋅∣y∣ j. n3<3n,∀n∈N. k. (x+1)2≥x2,∀x∈R. 25. Prove that if x2 is irrational, then so is x. Is its converse true? Prove or disprove.
The sum of two prime numbers is never a prime number.
The sum of two prime numbers will always have additional factors, making it composite and not prime.
The square root of 3 (√3) is irrational.
This has been proven using mathematical methods such as proof by contradiction.
The absolute value of the product of two numbers, |xy|, is equal to the product of their absolute values, |x|⋅|y|.
This property holds true for any real numbers x and y.
For any natural number n, the inequality n^3 < 3n holds.
This can be proven by induction.
Starting with n = 1, we have 1^3 = 1 < 3(1).
Assume the inequality holds for some n = k.
Then (k+1)^3 = k^3 + 3k^2 + 3k + 1 < 3k + 3k + 3 = 3(k+1).
Therefore, the inequality holds for all natural numbers.
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Given the tetrahedron with vertices (2,0,0)(2,4,0),(0,0,0), and (0,0,4), set up but DOT EVALUATE a triple integral to find the volume of this solid in the order ∭
K
dzdydx
The triple integral to find the volume of the tetrahedron is ∭ K dz dy dx.
To set up the triple integral for finding the volume of the tetrahedron, we need to determine the limits of integration for each variable.
The given vertices of the tetrahedron are:
A: (2, 0, 0)
B: (2, 4, 0)
C: (0, 0, 0)
D: (0, 0, 4)
To visualize the tetrahedron, we can observe that it lies within a rectangular box defined by the coordinates (0, 0, 0) and (2, 4, 4).
Considering the order of integration as ∭ K dz dy dx, we start with the innermost integral, integrating with respect to z first.
The limits of integration for z can be determined by the planes formed by the faces of the tetrahedron. The base of the tetrahedron lies on the xy-plane, so the limits for z are from 0 to the equation of the plane of the base, which is z = 0.
Next, we integrate with respect to y, considering the limits imposed by the sides of the tetrahedron. The left side is defined by the line segment AC, and the right side is defined by the line segment BC. Therefore, the limits for y are from 0 to the equation of the line segment AC, which is y = (4/2)x = 2x. The limits for y are also from 0 to the equation of the line segment BC, which is y = (4/2)(x-2) + 4 = 2x - 4.
Finally, we integrate with respect to x, considering the limits imposed by the front and back faces of the tetrahedron. The front face is defined by the line segment AB, and the back face is defined by the line segment CD. Therefore, the limits for x are from the equation of the line segment CD, which is x = 0, to the equation of the line segment AB, which is x = 2.
Putting it all together, the triple integral to find the volume of the tetrahedron is ∭ K dz dy dx, where the limits of integration are:
x: 0 to 2
y: 0 to 2x - 4
z: 0 to 0
Evaluating this integral will give the volume of the tetrahedron.
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Determine the standard matrix of a reflection in ℝ2 in the line −2x(1)+2x(2)=0
Subscript of number = ( )
The standard matrix of a reflection in ℝ² across the line -2x₁ + 2x₂ = 0 is given by [[0, 1], [1, 0]].
To find the standard matrix of a reflection in ℝ² across a given line, we can use the formula: S =[tex]I - 2nn^T[/tex]
where S is the standard matrix, I is the identity matrix, and [tex]nn^T[/tex] is the outer product of the unit normal vector of the line.
In this case, the line is defined by the equation -2x₁ + 2x₂ = 0. By rearranging the equation, we have:
2x₂ = 2x₁
x₂ = x₁
This suggests that the line has a slope of 1, which means the normal vector is orthogonal to the line and has a slope of -1. A unit vector in the direction of the normal vector is[tex][1/sqrt(2), -1/sqrt(2)].[/tex]
Using this normal vector, we can calculate the outer product [tex]nn^T[/tex]:
[tex]nn^T = [[1/sqrt(2)], [-1/sqrt(2)]] * [[1/sqrt(2), -1/sqrt(2)]][/tex]
= [[1/2, -1/2], [-1/2, 1/2]]
Finally, subtracting this outer product from the identity matrix, we obtain the standard matrix of the reflection:
S = I - [tex]2nn^T[/tex] = [[1, 0], [0, 1]] - 2[[1/2, -1/2], [-1/2, 1/2]] = [[0, 1], [1, 0]]
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f(x)=x12+x22+x32+x1x2+x2x3−2x1−4x2−6x3. Now consider minf(x) s.t. x1+x2+x3≤1. i. Is the problem convex? ii. Find all the KKT points of the problem. iii. Find the optimal solution of the problem.
The objective function and the constraint function are convex. In this case, the objective function f(x) is a sum of terms involving x1, x2, and x3, each raised to a power.
The constraint function x1 + x2 + x3 ≤ 1 is a linear function, and linear functions are also convex. Therefore, both the objective function and the constraint function are convex, making the problem convex.
- Stationarity: ∇f(x) + λ∇g(x) = 0
- Primal feasibility: g(x) ≤ 0
- Dual feasibility: λ ≥ 0
- Complementary slackness: λg(x) = 0
To find the optimal solution of the problem, we need to solve the problem using the KKT conditions. By setting up and solving the KKT conditions, we can find the values of x1, x2, and x3 that satisfy the conditions.
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A hockey goalie loses 2 3 5 rating points each time his team loses. if his team loses 8 games in a row, what number represents the change in the goalie's rating points? negative 20 and startfraction 4 over 5 endfraction negative 4 and startfraction 4 over 5 endfraction 16 and three-fifths 20 and four-fifths
The change in the goalie's rating points after losing 8 games in a row is 20 and 4/5.
The change in the goalie's rating points can be calculated by multiplying the number of games lost (8) by the points lost per game (2 3 5).
To calculate the change, we can use the following steps:
⇒ Convert the mixed number (2 3 5) into an improper fraction.
To convert the mixed number 2 3 5 into an improper fraction, we multiply the whole number (2) by the denominator (5), and then add the numerator (3) to get the numerator of the improper fraction. The denominator remains the same.
2 x 5 + 3 = 10 + 3 = 13
So, the mixed number 2 3 5 can be written as an improper fraction: 13/5.
⇒ Multiply the number of games lost (8) by the points lost per game (13/5).
To find the total change in the goalie's rating points, we multiply the number of games lost (8) by the points lost per game (13/5).
8 x (13/5) = (8 * 13) / 5 = 104/5
⇒ Simplify the fraction, if necessary.
The fraction 104/5 can be simplified further.
Divide the numerator (104) by the denominator (5).
104 ÷ 5 = 20 remainder 4
The simplified fraction is 20 and 4/5.
Therefore, the change in the goalie's rating points after losing 8 games in a row is 20 and 4/5.
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u=
⟨3,2⟩. Fino
⟨−27,18)
⟨−27,−18⟩
⟨27,−18⟩
⟨27,18⟩
v=5
i−4j and u=−7
−12
2
i−35j
(5−7
2
)i−1j
−12
2
+35
The dot product of vectors u and v, with components ⟨3,2⟩ and ⟨-7,-12⟩ respectively, is -45.
The given question provides two vectors, u and v, along with their components and asks for the dot product of these vectors.
To find the dot product of two vectors, we multiply their corresponding components and sum the products. In this case, u is given as ⟨3,2⟩ and v as ⟨-7,-12⟩.
The dot product of u and v is given by the formula u·v = (3 * -7) + (2 * -12).
Evaluating this expression, we get:
u·v = -21 - 24 = -45.
So, the dot product of u and v is -45.
Note that the other given options (⟨−27,18), ⟨−27,−18⟩, ⟨27,−18⟩, ⟨27,18⟩) are not related to the dot product of u and v. Hence, the correct answer is -45.
In summary, the dot product of vectors u and v, with components ⟨3,2⟩ and ⟨-7,-12⟩ respectively, is -45.
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part a: determine and interpret the lsrl. (3 points) part b: predict the percent of children living in single-parent homes in 1991 for state 14 if the percentage in 1985 was 18.3. show your work. (3 points) part c: calculate and interpret the residual for state 14 if the observed percent of children living in single-parent homes in 1991 was 21.5. show your work. (4 points)
part a: In order to determine and interpret the least squares regression line (LSRL), you need to have a set of data points and perform regression analysis.
The LSRL is a line that best fits the data points and represents the relationship between two variables. It is commonly used to predict or estimate values based on the given data.
To determine the LSRL, you will need to calculate the slope and the y-intercept of the line. The slope (m) represents the rate of change of the dependent variable for a one-unit increase in the independent variable.
The y-intercept (b) represents the value of the dependent variable when the independent variable is equal to zero.
Once you have determined the LSRL equation in the form of y = mx + b, you can interpret it.
For example, if the LSRL equation is y = 2x + 3, it means that for every one unit increase in the independent variable, the dependent variable is expected to increase by 2 units.
The y-intercept of 3 indicates that when the independent variable is zero, the dependent variable is expected to be 3.
part b: To predict the percent of children living in single-parent homes in 1991 for state 14, we can use the LSRL equation.
First, substitute the known value of the independent variable (1985) into the equation and solve for the dependent variable (percent of children living in single-parent homes). Let's say the LSRL equation is y = 0.5x + 10.
In this case, x represents the year and y represents the percent of children living in single-parent homes. So, when x is 1985, we can substitute it into the equation:
y = 0.5 * 1985 + 10
y = 993.5 + 10
y ≈ 1003.5
Therefore, the predicted percent of children living in single-parent homes in 1991 for state 14 would be approximately 1003.5 percent.
part c: To calculate the residual for state 14, we need to compare the observed percent of children living in single-parent homes in 1991 (21.5 percent) with the predicted value we obtained in part b (1003.5 percent).
The residual is calculated by subtracting the predicted value from the observed value:
Residual = Observed value - Predicted value
Residual = 21.5 - 1003.5
Residual ≈ -982
The negative value of the residual indicates that the observed value is significantly lower than the predicted value.
In other words, the actual percent of children living in single-parent homes in state 14 in 1991 is much lower than what was predicted based on the LSRL equation and the data from 1985.
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Let P2(x) be the Lagrange interpolating polynomial for the data (0,0),(0.5,2) and (2,3)⋅ Then p2(1.5)= a. 3.6 b. 3.5 c. 2.5 d. 3 e. 3.4
a. 3.6 b. 3.5 c. 2.5 d. 3 e. 3.4, are correct.
To find the value of P2(1.5), we can use the Lagrange interpolating polynomial formula.
The Lagrange interpolating polynomial P2(x) for the given data points (0,0), (0.5,2), and (2,3) is given by:
P2(x) = (x - x1)(x - x2) / (x0 - x1)(x0 - x2) * y0 + (x - x0)(x - x2) / (x1 - x0)(x1 - x2) * y1 + (x - x0)(x - x1) / (x2 - x0)(x2 - x1) * y2
where (x0, y0) = (0, 0), (x1, y1) = (0.5, 2), and (x2, y2) = (2, 3).
Plugging in the values, we have:
P2(x) = (x - 0)(x - 0.5) / (0 - 0)(0 - 0.5) * 0 + (x - 0)(x - 2) / (0.5 - 0)(0.5 - 2) * 2 + (x - 0.5)(x - 0) / (2 - 0.5)(2 - 0) * 3
Simplifying the equation, we have:
P2(x) = 4x^2 - 8x + 3
To find P2(1.5), we substitute x = 1.5 into the equation:
P2(1.5) = 4(1.5)^2 - 8(1.5) + 3
P2(1.5) = 9 - 12 + 3
P2(1.5) = 0
Therefore, P2(1.5) is equal to 0.
None of the given options, a. 3.6 b. 3.5 c. 2.5 d. 3 e. 3.4, are correct.
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By selling an article for N40, there is a loss of 40%, by selling it for N80, there would be
By selling the article for N80, there would be a profit of 20%.
To find the profit percentage, we need to calculate the cost price of the article. Let's assume the cost price of the article is C.
Given that selling the article for N40 results in a loss of 40%, we can set up the equation:
C - 40% of C = N40
Simplifying the equation:
C - 0.4C = N40
0.6C = N40
Dividing both sides of the equation by 0.6:
C = N40 / 0.6
C = N66.67 (approx.)
So, the cost price of the article is N66.67 (approx.).
Now, let's calculate the profit percentage when selling the article for N80. Let's assume the selling price is S.
Profit = Selling Price - Cost Price
Profit = S - C
Substituting the values:
Profit = N80 - N66.67
Profit = N13.33 (approx.)
Profit Percentage = (Profit / Cost Price) * 100
Profit Percentage = (N13.33 / N66.67) * 100
Profit Percentage ≈ 0.2 * 100
Profit Percentage ≈ 20%
Therefore, by selling the article for N80, there would be a profit of approximately 20%.
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