Given u = <3, -4>, v = <-1, 2> and w = <-2, -5>. Find: u+v+W (i) (ii) || u + v + w|| the vector unit in the direction of u + v + w Determine the area of the triangle PQR with vertices P(1,2,3), Q(2,3,1) and R(3,1,2) Given that Z=-4-j7 (1) (ii) (iii) (iv) AQB10102 Draw the projection of the complex number on the Argand Diagram Find the modulus, and argument, 0 Express Z in trigonometric form, polar form and exponential form Determine the cube roots of Z ENGINEERING MATHEMATICS 1 Page 7 of 9

Answer:

Step-by-step explanation:

The triangles are similar but NOT congruent.

3 corresponding angles mean the sides are proportional in length but not necessarily equal.

To find the quadratic polynomial \(p_2(x) = 1 + ax + bx^2\) that best approximates \(e^x\) near 0, we can use Taylor series expansion.

The Taylor series expansion of \(e^x\) centered at 0 is given by:

[tex]\(e^x = 1 + x + \frac{{x^2}}{2!} + \frac{{x^3}}{3!} + \ldots\)[/tex]

To find the quadratic polynomial that best approximates \(e^x\), we need to match the coefficients of the quadratic terms. Since we want the polynomial to closely follow the graph of \(e^x\) near 0, we want the quadratic term to be the same as the quadratic term in the Taylor series expansion.

From the Taylor series expansion, we can see that the coefficient of the quadratic term is \(\frac{1}{2}\).

Therefore, to best approximate \(e^x\) near 0, we choose the quadratic polynomial[tex]\(p_2(x) = 1 + ax + \frac{1}{2}x^2\).[/tex]

This choice ensures that the quadratic term in \(p_2(x)\) matches the quadratic term in the Taylor series expansion of \(e^x\), making it a good approximation near 0.

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The standard form of the given LP is:

minimize z = -5x₁ + 2x₂ - x₃

subject to:

-x₁ - 4x₂ - x₃ ≥ -5

2x₁ + x₂ + 3x₃ ≥ 2

x₁ ≥ 0

x₂ unrestricted

x₃ ≤ 0

To convert the given linear programming problem into standard form, we need to satisfy the following conditions:

1. Objective Function: The objective function should be in the form of minimizing or maximizing a linear expression. In this case, the objective function is z = -5x₁ + 2x₂ - x₃, which is already in the required form.

2. Constraints: Each constraint should be expressed as a linear inequality, with variables on the left side and a constant on the right side. The constraints given are:

-x₁ - 4x₂ - x₃ ≥ -5

2x₁ + x₂ + 3x₃ ≥ 2

x₁ ≥ 0

x₂ unrestricted

x₃ ≤ 0

3. Non-negativity and Unrestricted Variables: All variables should be non-negative or unrestricted. In this case, x₁ is specified as non-negative (x₁ ≥ 0), x₂ is unrestricted, and x₃ is specified as non-positive (x₃ ≤ 0).

By satisfying these conditions, we have transformed the given LP into its standard form. The objective function is in the proper form, the constraints are expressed as linear inequalities, and the variables meet the requirements of non-negativity or unrestrictedness.

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The speed of the current is 5 mph.

Let the speed of the current be x mph.Speed of the boat downstream = (Speed of the boat in still water) + (Speed of the current)= 15 + x.Speed of the boat upstream = (Speed of the boat in still water) - (Speed of the current)= 15 - x.

Let us assume the distance between two places be d .According to the question,20 = (15 + x) × 6 - d    (1)
Distance covered upstream in 10 hours = d. Distance covered downstream in 6 hours = d + 20.

We know that time = Distance/Speed⇒ Distance = Time × Speed.

According to the question,d = 10 × (15 - x)     (2)⇒ d = 150 - 10x         (2)

Also,d + 20 = 6 × (15 + x)⇒ d + 20 = 90 + 6x⇒ d = 70 + 6x     (3)

From equation (2) and equation (3),150 - 10x = 70 + 6x⇒ 16x = 80⇒ x = 5.

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a) The accumulated value of Shawn's investment after the first 9 years is $33,868.16.

b) The accumulated value of Shawn's investment at the end of 11 years is $54,570.70.

a) To calculate the accumulated value of Shawn's investment after the first 9 years, with an interest rate of 4.20% compounded semi-annually, we can use the formula for the accumulated value of an investment:

A = P[(1 + r/100)ᵏ - 1]/(r/100)

Where:

P = $2,100 (Investment at the beginning of every 6 months)

r = 2.10% (Rate of interest per compounding period)

T = 9 years, so the number of compounding periods (k) = 18 (2 compounding periods per year)

Plugging in the values, we have:

A = $2,100[(1 + 2.10/100)¹⁸ - 1]/(2.10/100)

A = $33,868.16

Therefore, the accumulated value of Shawn's investment after the first 9 years is $33,868.16.

b) To calculate the accumulated value of Shawn's investment at the end of 11 years, with an interest rate of 6.80% compounded semi-annually, we use the same formula:

A = P[(1 + r/100)ᵏ - 1]/(r/100)

Where:

P = $2,100 (Investment at the beginning of every 6 months)

r = 3.40% (Rate of interest per compounding period)

T = 11 years, so the number of compounding periods (k) = 22 (2 compounding periods per year)

Plugging in the values, we have:

A = $2,100[(1 + 3.40/100)²² - 1]/(3.40/100)

A = $54,570.70

Therefore, the accumulated value of Shawn's investment at the end of 11 years is $54,570.70.

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- The sum of (1101101)2 and (1010011)2 is (10110000)2.
- The product of (1101101)2 and (1010011)2 is (111000110111)2.

The sum and product of the binary numbers (1101101)2 and (1010011)2 can be found by performing binary addition and binary multiplication.

To find the sum, we add the two binary numbers together, digit by digit, from right to left.

```
 1101101
+ 1010011
_________
10110000
```

So, the sum of (1101101)2 and (1010011)2 is (10110000)2.

To find the product, we multiply the two binary numbers together, digit by digit, from right to left.

```
   1101101
×   1010011
__________
  1101101   (this is the partial product when the rightmost digit of the second number is 1)
 0000000    (this is the partial product when the second digit from the right of the second number is 0)
1101101     (this is the partial product when the third digit from the right of the second number is 1)
1101101      (this is the partial product when the fourth digit from the right of the second number is 1)
__________
111000110111  (this is the final product)
```

So, the product of (1101101)2 and (1010011)2 is (111000110111)2.

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The pixel with the maximum value in the given matrix is located at coordinates (3, 2) with a value of 13.

To find the pixel with the maximum value, we need to apply the given kernel on the 5x5 matrix. The kernel is a 3x4 matrix:

2 5 7 8

4 1 3 5

9 11 13 2

We start by placing the kernel on the top left corner of the matrix and calculate the element-wise product of the kernel and the corresponding sub-matrix. Then, we sum up the resulting values to determine the output for that position. We repeat this process for each valid position in the matrix.

After performing the calculations, we obtain the following result:

Output matrix:

60 89 136

49 77 111

104 78 62

The pixel with the maximum value in this output matrix is located at coordinates (3, 2) with a value of 13.

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(3Q)^(t) = 3Q^(t) this expression can be concluded as true.

The given matrix is Q = [2 3][1 -2]

To prove that (3Q)^(t) = 3Q^(t),

we need to calculate the transpose of both sides of the equation.

Let's solve it step by step as follows:

(3Q)^(t)

First, we will calculate 3Q which is;

3Q = 3[2 3][1 -2]= [6 9][-3 6]

Then we will calculate the transpose of 3Q as follows;

(3Q)^(t) = [6 9][-3 6]^(t)= [6 9][-3 6]= [6 -3][9 6]Q^(t)

Now we will calculate Q^(t) which is;

Q = [2 3][1 -2]

So,

Q^(t) = [2 1][3 -2]

Therefore, we can conclude that (3Q)^(t) = 3Q^(t) is true.

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To calculate the margin of error (M.E.) for a 95% two-sided confidence interval on the mean (μ) with a sample size of 26, we can use the formula:

M.E. = z * (σ / √n),

where z is the z-score corresponding to the desired confidence level, σ is the population standard deviation (unknown in this case), and n is the sample size. Since the population standard deviation (σ) is not given, we cannot calculate the exact margin of error. Therefore, none of the provided options (37.019, 9.592, 38.366, 31.555) can be determined as the correct answer without additional information. To calculate the margin of error, we would need either the population standard deviation or the sample standard deviation

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The point of diminishing returns is (20.98, 21247.3).

The point of diminishing returns occurs when the marginal cost of producing an extra unit of output exceeds the marginal revenue generated from selling that unit. Mathematically, it is the point at which the derivative of the production function equals zero and the second derivative is negative.

Given the polynomial function N(x) of degree 3, we can find the point of diminishing returns by finding the critical points where the first derivative equals zero and evaluating the second derivative at those points.

The derivative of N(x) is N'(x) = -18x² + 360x + 2250. To find the critical points, we set N'(x) = 0:

0 = -18x² + 360x + 2250

Dividing by -18 simplifies the equation:

0 = x² - 20x - 125

Using the quadratic formula, we find the solutions to the equation:

x₁,₂ = (20 ± √(20² - 4(1)(-125))) / 2(1)

x₁,₂ = 10 ± 5√5

Thus, the two critical points of N(x) are at x = 10 - 5√5 and x = 10 + 5√5.

To determine the point of diminishing returns, we evaluate the second derivative N''(x) = -36x + 360 at these critical points:

N''(10 - 5√5) = -36(10 - 5√5) + 360 ≈ -264.8

N''(10 + 5√5) = -36(10 + 5√5) + 360 ≈ 144.8

From the evaluations, we find that N''(10 + 5√5) is negative while N''(10 - 5√5) is positive. Therefore, the point of diminishing returns corresponds to x = 10 + 5√5.

To find the corresponding y-coordinate (N(10 + 5√5)), we can substitute the value of x into the original function N(x).

Hence, the point of diminishing returns is approximately (20.98, 21247.3).

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A basis for the vector space {f(x) ∈ P3[x] | ƒ′(6) = ƒ(1)} is {p(x) = ax^2 + bx + 11a, q(x) = dx}, where a and d can be any real numbers.

To find a basis {p(x), q(x)} for the given vector space {f(x) ∈ P3[x] | ƒ′(6) = ƒ(1)}, we need to find two polynomials p(x) and q(x) that satisfy the condition ƒ′(6) = ƒ(1) and are linearly independent.

Let's start by finding p(x):

We can choose p(x) as a polynomial of degree 2 since we are working with P3[x].

Let p(x) = ax^2 + bx + c.

Taking the derivative of p(x), we have:

p'(x) = 2ax + b.

We need p'(6) to be equal to p(1), so let's evaluate them:

p'(6) = 2a(6) + b = 12a + b

p(1) = a(1)^2 + b(1) + c = a + b + c

For p'(6) = p(1), we have:

12a + b = a + b + c

Simplifying this equation, we get:

11a = c

So, we can choose c = 11a.

Thus, p(x) = ax^2 + bx + 11a.

Now, let's find q(x):

We can choose q(x) as a polynomial of degree 1 since we are working with P3[x].

Let q(x) = dx + e.

Taking the derivative of q(x), we have:

q'(x) = d.

We need q'(6) to be equal to q(1), so let's evaluate them:

q'(6) = d

q(1) = d(1) + e = d + e

For q'(6) = q(1), we have:

d = d + e

Simplifying this equation, we get:

e = 0

Thus, q(x) = dx.

Therefore, a basis for the vector space {f(x) ∈ P3[x] | ƒ′(6) = ƒ(1)} is {p(x) = ax^2 + bx + 11a, q(x) = dx}, where a and d can be any real numbers.

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To find the production level that maximizes revenue, we need to determine the value of 'x' that maximizes the revenue function R.

The revenue function is given by R = x * p, where p represents the price per unit. Substituting the given expression for p, we have:

R = x * ($95 - 0.0125x)

Expanding and simplifying, we get:

R = $95x - 0.0125x^2

Now, to maximize the revenue, we can use calculus. We take the derivative of the revenue function with respect to 'x' and set it equal to zero:

dR/dx = 95 - 0.025x = 0

Solving for 'x', we find:

0.025x = 95

x = 95 / 0.025

x = 3800

Therefore, the production level that maximizes the revenue is 3800 thousand phones produced.

To confirm that this value maximizes the revenue, we can also check the second derivative. Taking the second derivative of the revenue function, we have:

d^2R/dx^2 = -0.025

Since the second derivative is negative, it confirms that the revenue is maximized at x = 3800.

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After approximately 158.38 minutes, or rounding to the nearest minute, after about 158 minutes, the water level would be less than or equal to 64 cups.

To find the number of minutes at which the water level would be less than or equal to 64 cups, we can substitute W = 64 into the equation W = -0.414t + 129.549 and solve for t.

64 = -0.414t + 129.549

Rearranging the equation, we get:

-0.414t = 64 - 129.549

-0.414t = -65.549

Dividing both sides by -0.414, we find:

t = (-65.549) / (-0.414)

t ≈ 158.38

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The value of x, y and z in the interior angles of the parallelogram is 38, 81 and 75.

A parallelogram is simply quadrilateral with two pairs of parallel sides.

Opposite angles of a parallelogram are equal.

Consecutive angles in a parallelogram are supplementary.

From the diagram, angle ( 3x - 6 ) is opposite angle 108 degrees.

Since opposite angles of a parallelogram are equal.

( 3x - 6 ) = 108

Solve for x:

3x - 6 = 108

3x = 108 + 6

3x = 114

x = 114/3

x = 38

Also, consecutive angles in a parallelogram are supplementary.

Hence:

108 + ( y - 9 ) = 180

y + 108 - 9 = 180

y + 99 = 180

y = 180 - 99

y = 81

And

108 + ( z - 3 ) = 180

z + 108 - 3 = 180

z + 105 = 180

z = 180 - 105

z = 75

Therefore, the value of z is 75.

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Answer:

Step-by-step explanation:

The expression (f+g)(x) represents the sum of the functions f(x) and g(x). To find (f+g)(x), we substitute the given expressions for f(x) and g(x) into the sum: (f+g)(x) = f(x) + g(x) = (3x+2) + (2x-7).

In (f+g)(x) = 5x - 5, the first paragraph summarizes that the sum of the functions f(x) and g(x) is given by (f+g)(x) = 5x - 5. The second paragraph explains how this result is obtained by substituting the expressions for f(x) and g(x) into the sum and simplifying the expression. Furthermore, it mentions that the domain of (f+g)(x) is all real numbers, as there are no restrictions on the variable x in the given equation.

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The cyclonona-1,3,5,7-tetraene is classified as non-aromatic based on the inscribed polygon method.

By using the inscribed polygon method, we can determine the aromaticity of cyclonona-1,3,5,7-tetraene. The molecule consists of a cyclic structure with alternating single and double bonds. The inscribed polygon method involves drawing an imaginary polygon inside the molecule, following the path of the pi electrons. If the number of pi electrons in the molecule matches the number of electrons in the inscribed polygon, the molecule is considered aromatic.

If the number of pi electrons differs by a multiple of 4, the molecule is antiaromatic. In this case, cyclonona-1,3,5,7-tetraene has 8 pi electrons, which does not match the number of electrons in any inscribed polygon, making it non-aromatic.

Cyclonona-1,3,5,7-tetraene is a cyclic molecule with alternating single and double bonds. To determine its aromaticity using the inscribed polygon method, we draw an imaginary polygon inside the molecule, following the path of the pi electrons.

In the case of cyclonona-1,3,5,7-tetraene, we have a total of 8 pi electrons. We can try different polygons with varying numbers of sides to see if any match the number of electrons. However, regardless of the number of sides, no inscribed polygon will have 8 electrons.

For example, if we consider a hexagon (6 sides) as the inscribed polygon, it would have 6 electrons. If we consider an octagon (8 sides), it would have 8 electrons. However, cyclonona-1,3,5,7-tetraene has neither 6 nor 8 pi electrons. This indicates that the molecule is not aromatic according to the inscribed polygon method.

Therefore, cyclonona-1,3,5,7-tetraene is classified as non-aromatic based on the inscribed polygon method.

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(a) Without knowing the effect size, it is not possible to calculate the type II error for the given hypothesis test. (b) To detect a true mean diameter of 1.55 inches with a power of at least 0.9, approximately 65 bearings would be needed.

(a) If the true mean diameter is 1.55 inches, the probability of not rejecting the null hypothesis when it is false (i.e., the type II error) depends on the chosen significance level, sample size, and effect size. Without knowing the effect size, it is not possible to calculate the type II error.

(b) To calculate the required sample size to detect a true mean diameter of 1.55 inches with a power of at least 0.9, we need to know the chosen significance level, the standard deviation of the population, and the effect size.

Using a statistical power calculator or a sample size formula, we can determine that a sample size of approximately 65 bearings is needed.

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Counterexample: Let x = 2.5 and y = 1.5. Then [xy] = [3.75] = 3, while [x]·[y] = [2]·[1] = 2.

To show that the statement is false, we need to find specific values for x and y where [xy] and [x] · [y] are not equal.

Counterexample: Let x = 2.5 and y = 1.5.

To find [xy], we multiply x and y: [xy] = [2.5 * 1.5] = [3.75].

To find [x] · [y], we calculate the floor value of x and y separately and then multiply them: [x] · [y] = [2] · [1] = [2].

In this case, [xy] = [3.75] = 3, and [x] · [y] = [2] = 2.

Therefore, [xy] and [x] · [y] are not equal, as 3 is not equal to 2.

This counterexample disproves the statement for the specific values of x = 2.5 and y = 1.5, showing that for all real numbers x and y, [xy] is not always equal to [x] · [y].

The floor function [x] denotes the greatest integer less than or equal to x.

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The depth of the water in the large container cube is  2.6 inches.

Tracey have two empty cube shaped containers with sides 5 inches and 7 inches. she fills the smaller container and then pour the water in the larger container.

Therefore, the depth of the water in the larger container can be found as follows:

Hence,

volume of the smaller cube = 5³

volume of the smaller cube =  125 inches³

Therefore,

volume of water poured in the larger cube = lwh

125 = 7 × 7 × h

h = 125 / 49

h = 2.55102040816

h = 2.6 inches

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Answer:

the dimensions of the rectangular poster are width = 6 inches and length = 8 inches.

Step-by-step explanation:

Let's assume the width of the rectangular poster is represented by 'w' inches.

According to the given information, the length of the poster is 5 more inches than half its width. So, the length can be represented as (0.5w + 5) inches.

The formula for the area of a rectangle is given by:

Area = length * width

We are given that the area of the poster is 48 square inches, so we can set up the equation:

(0.5w + 5) * w = 48

Now, let's solve this equation to find the value of 'w' (width) first:

0.5w^2 + 5w = 48

Multiplying through by 2 to eliminate the fraction:

w^2 + 10w - 96 = 0

Now, we can factorize this quadratic equation:

(w - 6)(w + 16) = 0

Setting each factor to zero:

w - 6 = 0 or w + 16 = 0

Solving for 'w', we get:

w = 6 or w = -16

Since the width of a rectangle cannot be negative, we discard the value w = -16.

Therefore, the width of the poster is 6 inches.

To find the length, we substitute the value of the width (w = 6) into the expression for the length:

Length = 0.5w + 5 = 0.5 * 6 + 5 = 3 + 5 = 8 inches

To find the value of λ, we need to determine when the vector [2, -3, λ] is orthogonal to the set W, where W = span{[λ−1, 1, 3λ], [−7, λ+2, 3λ−4]}.

Two vectors are orthogonal if their dot product is zero. Therefore, we need to calculate the dot product between [2, -3, λ] and the vectors in W.

First, let's find the vectors in W by substituting the given values of λ into the span:

For the first vector in W, [λ−1, 1, 3λ]:
[λ−1, 1, 3λ] = [2−1, 1, 3(2)] = [1, 1, 6]

For the second vector in W, [−7, λ+2, 3λ−4]:
[−7, λ+2, 3λ−4] = [2−1, -3(2)+2, λ+2, 3(2)−4] = [-7, -4, λ+2, 2]

Now, let's calculate the dot product between [2, -3, λ] and each vector in W.

Dot product with [1, 1, 6]:
(2)(1) + (-3)(1) + (λ)(6) = 2 - 3 + 6λ = 6λ - 1

Dot product with [-7, -4, λ+2, 2]:
(2)(-7) + (-3)(-4) + (λ)(λ+2) + (2)(2) = -14 + 12 + λ² + 2λ + 4 = λ² + 2λ - 6

Since [2, -3, λ] is orthogonal to the set W, both dot products must equal zero:

6λ - 1 = 0
λ² + 2λ - 6 = 0

To solve the first equation:
6λ = 1
λ = 1/6

To solve the second equation, we can factor it:
(λ - 1)(λ + 3) = 0

Therefore, the possible values for λ are:
λ = 1/6 and λ = -3

However, we need to check if λ = -3 satisfies the first equation as well:
6λ - 1 = 6(-3) - 1 = -18 - 1 = -19, which is not zero.

Therefore, the value of λ that makes [2, -3, λ] orthogonal to the set W is λ = 1/6.

So, the correct answer is D. 1/6.

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Answer:

A) total area = 364ft²

B) total cost = $2,184

Step-by-step explanation:

total area of the brick paver border that surrounds the pool

= w * h = 14 * 26 = 364ft²

if brick pavers cost $6 per square foot

total cost of the brick pavers needed for this project

= 364 * 6 = $2,184

The correct option is:

c. y¹ = 3 + √(x - 7)

To find the inverse function of y = (x - 3)^2 + 7 for x > 3, we can follow these steps:

Step 1: Replace y with x and x with y in the given equation:

x = (y - 3)^2 + 7

Step 2: Solve the equation for y:

x - 7 = (y - 3)^2

√(x - 7) = y - 3

y - 3 = √(x - 7)

Step 3: Solve for y by adding 3 to both sides:

y = √(x - 7) + 3

So, the inverse function of y = (x - 3)^2 + 7 for x > 3 is y¹ = √(x - 7) + 3.

Therefore, the correct option is:

c. y¹ = 3 + √(x - 7)

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The primitive function of the given function f(x) = -5/2 * x is F(x) = -5/4 * x² + C where C is the constant of integration. This means that F(x) is the antiderivative of f(x).

To find the antiderivative, integrate the given function with respect to x.

When we integrate the given function f(x) = -5/2 * x, we get;

∫f(x)dx = ∫-5/2 * x dx

= -5/2 ∫x dx

= -5/2 * x²/2 + C

The constant of integration C is an arbitrary constant and could take any real value.

Therefore, the antiderivative of f(x) is

F(x) = -5/4 * x² + C where C is a constant of integration.

The primitive function is usually the antiderivative of a function. The antiderivative of a function is its inverse operation of differentiation.

Therefore, to find the primitive function, we integrate the given function with respect to x.

In this case, the primitive function is given by F(x) = -5/4 * x² + C.

The primitive function of the given function f(x) = -5/2 * x is F(x) = -5/4 * x² + C where C is the constant of integration. This function is obtained by integrating f(x) with respect to x. The constant of integration C is an arbitrary constant and could take any real value.

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The solution to the proportion 2.14 = X/12, rounded to the nearest tenth, is X = 25.7.

To solve the proportion 2.14 = X/12, we can cross-multiply and solve for X.

Cross-multiplying means multiplying the numerator of the first fraction (2.14) by the denominator of the second fraction (12), and vice versa.

So, 2.14 * 12 = X * 1.

The result of multiplying 2.14 and 12 is 25.68. Therefore, X * 1 can be simplified to just X.

Thus, X = 25.68.

Rounding to the nearest tenth, X is approximately 25.7.

So, the solution to the proportion is X = 25.7.

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[tex]\begin{align}\displaystyle\sf 5+4-6\cdot 3 & = 3 \\ 5x-1 + y-2 & = 3x - 1y - 2 \\ x-1 \cdot y-2 & = 1 \end{align} [/tex]

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♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

a. The quartiles of the data set are Q1 = 68, Q2 = 73, and Q3 = 83.

b. Suzanne's test score which lies in the 80th percentile is 84.

a. Quartiles of the data set:

Let us sort the marks: 41, 56, 56, 62, 63, 65, 67, 68, 68, 70, 71, 71, 71, 72, 73, 74, 77, 77, 81, 83, 83, 84, 84, 85, 88, 91, 94, 99

The median of the data is 73.

The median of the lower half of the data is 68.

The median of the upper half of the data is 83.

Therefore, Q1 = 68, Q2 = 73, and Q3 = 83.

b. The 80th percentile:

Percentile can be calculated by using the formula:

Percentile = (Number of values below the given value / Total number of values) × 100

80 = (n/30) × 100

n = 24

From the sorted data, the 24th mark is 84.

Therefore, Suzanne's test score is 84.

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The profit from selling 30 violins is $79,850. The composite function for the craftsman’s profit is P(I(x)) = 5995x - 100,000. We can use this composite function to find the profit earned when he sells 30 violins by substituting x = 30 in the function.

The craftsman makes and sells violins. The function (I(x)=5995 x) represents the income in dollars from selling (x) violins. The function (P(y)=y-100,000) represents his profit in dollars if he makes an income of (y) dollars.

We are given that the function for income in dollars from selling x violins is I(x) = 5995x. The craftsman’s profit P(y) is given by the function y - 100,000. We want to find out the craftsman’s profit when he sells 30 violins.So the income earned from selling 30 violins is:

I(30) = 5995 × 30 = 179,850

Therefore, the craftsman’s profit is: P(179,850) = 179,850 - 100,000 = 79,850

We can write the composite function for the craftsman’s profit as follows: P(I(x)) = I(x) - 100,000

We know that the income from selling x violins is I(x) = 5995x. We can substitute this value in the composite function to get: P(I(x)) = 5995x - 100,000

To find the profit earned when he sells 30 violins, we substitute x = 30 in the above expression: P(I(x)) = P(I(30))= P(5995 × 30 - 100,000)= P(79,850)= 79,850

Therefore, the profit earned when he sells 30 violins is $79,850.

Thus, the profit from selling 30 violins is $79,850. The composite function for the craftsman’s profit is P(I(x)) = 5995x - 100,000. We can use this composite function to find the profit earned when he sells 30 violins by substituting x = 30 in the function.

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The solution to the given system of linear equations is:

x = undetermined

y = undetermined

z = -3

To find the solution of the system of linear equations represented by the augmented matrix, we can use Gaussian elimination or row reduction.

Starting with the augmented matrix:

[ 2 100 0 | 1 ]

[ 0 0 1 | -3 ]

Let's perform row operations to simplify the matrix:

Row 2 multiplied by 2:

[ 2 100 0 | 1 ]

[ 0 0 2 | -6 ]

Row 1 subtracted by Row 2:

[ 2 100 0 | 1 ]

[ 0 0 2 | -6 ]

[ 2 100 0 | 7 ]

[ 0 0 2 | -6 ]

Row 1 divided by 2:

[ 1 50 0 | 7/2 ]

[ 0 0 2 | -6 ]

Now, let's analyze the simplified matrix. The system of equations can be written as:

1x + 50y + 0z = 7/2

0x + 0y + 2z = -6

From the second equation, we can solve for z:

2z = -6

z = -6/2

z = -3

Substituting z = -3 into the first equation:

x + 50y = 7/2

From here, we have an equation with two variables. To find a unique solution, we would need another equation or constraint. Without additional information, we cannot determine the specific values of x and y.

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