5. Let n be a natural number. Define congruence modn as the following relation on natural numbers: a≡ n b if n divides their difference, i.e. ∃k:Nvnk=∣b−a∣. Prove that this relation is transitive, reflexive, and symmetric. (How could we use the previous question here?)

Answer:

07n + 6

Step-by-step explanation:

Given: The cost of 7 t-shirts and a $6 tax

Let n represent the cost of 1 t-shirt.

Then, the total cost of 7 t-shirts would be 7n.

Adding the $6 tax gives a total cost of 7n + 6.

Therefore, the correct option is:

07n + 6

The answer choice which could represent the cost of 7 t-shirts and a $6 tax as in the task content is: 7n + 6.

It follows that the cost of 7 t-shirts and a $6 tax is the statement which is to be represented algebraically.

On this note, it follows that the if the cost of each t-shirts is taken to be: n.

Therefore, the required representation of the total cost would be:

[tex]\rightarrow\bold{7n + 6}[/tex]

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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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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 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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The median is 133 and the mode is 132.

Median and mode are measures of central tendency. Median is the number that is at the center of a dataset that has been arranged in ascending or descending order.

130, 130, 132, 132, 132, 134, 138, 140, 148, 148

Median = (n + 1) / 2

Where n is the number of observations

(10 + 1) / 2 = 11/5 = 5.5

The median is the 5.5th number - (132 + 134) / 2 = 133

Mode is the number that appears with the highest frequency in the dataset. The mode is 132 that appears 3 times

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Given the functions:f(x) = x² - 3xg(x) = √(2x)h(x) = 5x - 4

To find the value of (hog) (x) for x = 2,

we need to evaluate h(g(x)), which is given by:h(g(x)) = 5g(x) - 4

We know that g(x) = √(2x)∴ g(2) = √(2 × 2) = 2

Hence, (hog) (2) = h(g(2))= h(2)= 5(2) - 4= 6

Therefore, (hog) (2) = 6.

In this problem, we were required to evaluate the composite function (hog) (x) for x = 2,

where g(x) and h(x) are given functions.

The solution involved first calculating the value of g(2),

which was found to be 2. We then used this value to calculate the value of h(g(2)),

which was found to be 6.

Thus, the value of (hog) (2) was found to be 6.

The simplified exact form of √Undefined × X Ś is Undefined,

as the square root of Undefined is undefined.

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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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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 parent function for g(x) = x + 2 is the identity function, f(x) = x, which is a straight line passing through the origin with a slope of 1.

To graph g(x) = x + 2, we start with the parent function and apply the transformation. The transformation for g(x) involves shifting the graph vertically upward by 2 units.

Here's the step-by-step process to graph g(x):

Plot points on the parent function, f(x) = x. For example, if x = -2, f(x) = -2; if x = 0, f(x) = 0; if x = 2, f(x) = 2.

Apply the vertical shift by adding 2 units to the y-coordinate of each point. For example, if the point on the parent function is (x, y), the corresponding point on g(x) will be (x, y + 2).

Connect the points to form a straight line. Since g(x) = x + 2 is a linear function, the graph will be a straight line with the same slope as the parent function.

The transformation of the parent function f(x) = x to g(x) = x + 2 results in a vertical shift upward by 2 units. This means that the graph of g(x) is the same as the parent function, but it is shifted upward by 2 units along the y-axis.

Visually, the graph of g(x) will be parallel to the parent function f(x), but it will be shifted upward by 2 units. The slope of the line remains the same, indicating that the transformation does not affect the steepness of the line.

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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When you are writing a positioning statement, if you do not have real differences and cannot see a way to create them, then you can create a difference based on b) opinion.

A positioning statement is a brief, clear, and distinctive description of who you are and what separates you from your competition when you are competing for attention in the marketplace. A company's position is the set of customer perceptions of its goods and services relative to those of its rivals. A successful positioning strategy places your goods or services in the minds of your customers as better or more affordable than your competitors'. A company's positioning strategy is how it distinguishes itself from its rivals. A strong positioning statement is essential for any company, brand, or product. It communicates to the target audience why a company is unique and distinct from others. Positioning that is based on opinion includes marketing that makes sweeping statements, claims, or guarantees that cannot be validated or demonstrated as fact.

This is often referred to as 'puffery.' Puffery is a technique used by advertisers to promote a product in a way that does not make a factual statement but instead generates a feeling in the consumer that their product is superior to others on the market. Opinion-based positioning requires a great deal of creativity and should be combined with strong marketing, advertising, and public relations to ensure that the message is communicated successfully to the target audience.

Therefore, the correct answer is b) opinion.

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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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(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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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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i) The polar form of Z is:[tex]Z = 3\sqrt 2 \left( {\cos \frac{\pi }{4} + i\sin \frac{\pi }{4}} \right),[/tex]

ii) [tex]Z^7 = -2187 - 2187i[/tex] and is expressed in the form a + bi

Polar Form of Z = -3 -3i.

In order to express the complex number -3-3i in polar form, we use the formula:

r = \sqrt {a^2 + b^2 }

where a = -3 and b = -3,

hence;[tex]r &= \sqrt {a^2 + b^2 } \\&= \sqrt {{\left( { - 3} \right)^2} + {\left( { - 3} \right)^2}} \\&= \sqrt {18} \\&= 3\sqrt 2 \[/tex]

We can calculate the argument [tex]\theta of Z as:\theta = \tan ^{ - 1} \left( {\frac{b}{a}} \right)[/tex]

where a = -3 and b = -3,

hence;

  [tex]\theta &= \tan ^{ - 1} \left( {\frac{b}{a}} \right) \\&= \tan ^{ - 1} \left( {\frac{{ - 3}}{{ - 3}}} \right) \\&= \tan ^{ - 1} \left( 1 \right) \\&= \frac{\pi }{4} \[/tex]

Therefore, the polar form of Z is:

Z = [tex]3\sqrt 2 \left( {\cos \frac{\pi }{4} + i\sin \frac{\pi }{4}} \right)[/tex]

ii)  Z^7 = -2187 - 2187i and is expressed in the form a + bi

Since we already have Z in polar form we can now easily find

Z^7.Z^7 = [tex]{\left( {3\sqrt 2 } \right)^7}{\left( {\cos \frac{\pi }{4} + i\sin \frac{\pi }{4}} \right)^7}[/tex]

We can expand [tex]\left( {\cos \frac{\pi }{4} + i\sin \frac{\pi }{4}} \right)^7[/tex] using De Moivre's theorem:

[tex]\left( {\cos \theta + i\sin \theta } \right)^n = \cos n\theta + i\sin n\ \\theta\\Therefore; \\Z^7 &= {\left( {3\sqrt 2 } \right)^7}\left( {\cos \frac{{7\pi }}{4} + i\sin \frac{{7\pi }}{4}} \right) \\&= 3^7\left( {2\sqrt 2 } \right)\left( {\cos \left( {\frac{{6\pi }}{4} + \frac{\pi }{4}} \right) + i\sin \left( {\frac{{6\pi }}{4} + \frac{\pi }{4}} \right)} \right) \\&= 2187\sqrt 2 \left( { - \frac{1}{{\sqrt 2 }}} \right) + 2187i\left( { - \frac{1}{{\sqrt 2 }}} \right) \\&=  - 2187 - 2187i \[/tex]

Thus, Z^7 = -2187 - 2187i and is expressed in the form a + bi

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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 solution for the expression are A=0, B=1, C=0 and D=3. The solution for x=5/2 and y=√15/2.

Given expression is:

\frac{x^2+x+3}{(x^2+1)^2}=\frac{Ax+B}{x^2+1}+\frac{Cx+D}{(x^2+1)^2}

Comparing the two sides, we get:

(x^2+x+3)=(Ax+B)(x^2+1)+(Cx+D)

Expanding the right side, we get:

(x^2+x+3)=Ax^3+(A+B)x^2+(B+C)x+(C+D)

For the coefficients of x^3 on both sides to be equal, we must have A=0.

For the coefficients of x^2 on both sides to be equal, we must have A+B=1.

Substituting A=0, we get B=1.

For the coefficients of x on both sides to be equal, we must have B+C=1.

Substituting B=1, we get C=0.

For the constants on both sides to be equal, we must have C+D=3.

Substituting C=0, we get D=3.

Hence, we get:\frac{x^2+x+3}{(x^2+1)^2}=\frac{1}{x^2+1}+\frac{3}{(x^2+1)^2}

Solving the system of equations x^2-y^2=-5 and 3x^2+2y^2=30:

Multiplying the first equation by 2, we get:

2x^2-2y^2=-10\implies x^2-y^2+2x^2= -5+2x^2

Substituting 3x^2+2y^2=30, we get:

(3x^2+2y^2) + x^2-y^2 = 30-5\implies 4x^2 = 25\implies x = \pm\frac{5}{2}

Substituting in x^2-y^2=-5, we get:

y^2 = \frac{15}{4}\implies y = \pm\frac{\sqrt{15}}{2}

Therefore, the solutions are:(x,y) = \left(\frac{5}{2},\frac{\sqrt{15}}{2}\right), \left(\frac{5}{2},-\frac{\sqrt{15}}{2}\right), \left(-\frac{5}{2},\frac{\sqrt{15}}{2}\right), \left(-\frac{5}{2},-\frac{\sqrt{15}}{2}\right).

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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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A formula involving integrals for a particular solution of the differential equation y"' - 27y" + 243y' - 729y = g(t) is given by Y(t) = ∫[∫[∫g(t)dt]dt]dt.

To find a particular solution Y(t) of the given differential equation, we can use an integral formula.

The formula is Y(t) = ∫[∫[∫g(t)dt]dt]dt, which involves multiple integrals of the function g(t) with respect to t.

By repeatedly integrating g(t) with respect to t, we perform three successive integrations, representing the third, second, and first derivatives of the function Y(t), respectively.

This allows us to obtain a particular solution that satisfies the given differential equation.

It is important to note that the integral formula provides a general approach to finding a particular solution.

The specific form of g(t) will determine the integrals involved and the limits of integration, which need to be considered during the integration process.

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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 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 national people meter sample has 4,000 households, and 250

of those homes watched program A on a given Friday Night. In other

words 6.25% of all households watched program A.

To determine the fraction of all households that watched program A, we divide the number of households that watched program A by the total number of households in the sample.

Fraction of households that watched program A = Number of households that watched program A / Total number of households in the sample

Fraction of households that watched program A = 250 / 4000

Fraction of households that watched program A ≈ 0.0625

Therefore, approximately 6.25% of all households watched program A.

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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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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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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 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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The solution of the higher-order differential equation y⁽⁴⁾ - y‴ + 2y = 0 is: y = C₁e^t + C₂cos(√2t) + C₃sin(√2t) + C₄t * e^t

where C₁, C₂, C₃, and C₄ are arbitrary constants.

Given the higher-order differential equation y⁽⁴⁾ - y‴ + 2y = 0.

To solve this equation, assume a solution of the form y = e^(rt). Substituting this form into the given equation, we get:

r⁴e^(rt) - r‴e^(rt) + 2e^(rt) = 0

⇒ r⁴ - r‴ + 2 = 0

This is the characteristic equation of the given differential equation, which can be solved as follows:

r³(r - 1) + 2(r - 1) = 0

(r - 1)(r³ + 2) = 0

Thus, the roots are r₁ = 1, r₂ = -√2i, and r₃ = √2i.

To find the solution, we can use the following steps:

For the root r₁ = 1, we get y₁ = e^(1t).

For the root r₂ = -√2i, we get y₂ = e^(-√2it) = cos(√2t) - i sin(√2t).

For the root r₃ = √2i, we get y₃ = e^(√2it) = cos(√2t) + i sin(√2t).

For the double root r = 1, we need to find a second solution, which is given by t * e^(1t).

The general solution of the differential equation is:

y = C₁e^t + C₂cos(√2t) + C₃sin(√2t) + C₄t * e^t

The above solution contains four arbitrary constants (C₁, C₂, C₃, and C₄), which can be evaluated using initial conditions or boundary conditions. Therefore, the solution of the higher-order differential equation y⁽⁴⁾ - y‴ + 2y = 0 is:

y = C₁e^t + C₂cos(√2t) + C₃sin(√2t) + C₄t * e^t

where C₁, C₂, C₃, and C₄ are arbitrary constants.

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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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