What is the measure of angle x?

Angles are not necessarily drawn to scale.

B

35°

A

Response

So the correct answer is: A. fog(x) = 4x and go f(x) = 2x² + 2 for the given function.

In mathematics, a function is a rule that assigns a unique output value to each input value in a specified set. The input values are often called the domain of the function, while the output values are called the range. Functions are often represented using function notation, such as f(x), where f is the name of the function and x is the input value. The value of f(x) is the output value that corresponds to the input value x. Functions can be defined using a variety of methods, including algebraic equations, tables of values, graphs, and verbal descriptions. Some common types of functions include linear functions, quadratic functions, exponential functions, trigonometric functions, and logarithmic functions. Functions are an important concept in mathematics and have many practical applications in fields such as physics, engineering, economics, and computer science. They are used to model and analyze real-world phenomena, to make predictions, and to solve problems.

Here,

We have:

f(x) = x² - 1

g(x) = -2x

To find fog(x), we first substitute g(x) into f(x) wherever we see an "x":

fog(x) = f(g(x))

= f(-2x)

= (-2x)² - 1

= 4x² - 1

To find go f(x), we first substitute f(x) into g(x) wherever we see an "x":

go f(x) = g(f(x))

= g(x² - 1)

= -2(x² - 1)

= 2x² + 2

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a)The standard deviation of the productivity statistics for each plant must be calculated in order to determine which facility is more productively consistent overall.

Plant A's standard deviation is 2.84, while Plant B's is 2.01. Since plant B's standard deviation is lower than plant A's, we can infer that plant B's productivity is more stable.

b) Because it examines the absolute values of the data points, the standard deviation is typically a more accurate indicator of dispersion than the coefficient of variation. The coefficient of variation, which quantifies relative variability, can be helpful when assessing the variability of data sets with different measurements or scales.The coefficient of variation may be helpful information in addition to the standard deviation in this case as we are comparing production statistics for two different plants.

However, the standard deviation would be a more accurate indicator of variability if the data sets used the same scales and units.

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The company accrues a profit of $908,404 each month, assuming 1 month is 4 weeks.

To calculate the profit accrued each month, we need to first calculate the total cost of producing 55728 shirts. The supplies cost $13.5 per shirt, so the total cost of supplies is 55728 x $13.5 = $752,736.

The fixed monthly costs are $700. We also need to factor in employee wages and other monthly supply costs, but we don't have enough information to calculate those. So, we'll assume they amount to $10,000 per month.

The total cost of production each month is $752,736 + $700 + $10,000 = $763,436.

Now, to calculate the profit, we need to subtract the total cost of production from the revenue generated from selling the shirts. The revenue generated from selling 55728 shirts at $30 per shirt is 55728 x $30 = $1,671,840.

So, the profit accrued each month is $1,671,840 - $763,436 = $908,404.

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The amount that must be paid back at the end of the [tex]10[/tex]-year period is approximately $[tex]42,748.37[/tex].

To calculate the amount that must be paid back for a loan of $[tex]19,000[/tex] borrowed for [tex]10[/tex] years at [tex]7.5[/tex]% interest, compounded annually, we can use the formula for compound interest:

[tex]A = P(1+\frac{r}{n} )^{nt}[/tex]

Where:

A = the amount to be paid back

P = the principal amount (initial loan amount) = $[tex]19,000[/tex]

r = annual interest rate (as a decimal) = [tex]7.5[/tex]% or [tex]0.075[/tex]

n = number of times interest is compounded per year = [tex]1[/tex] (compounded annually)

t = time period in years = [tex]10[/tex]

According to the problem,

A = [tex]19000(\frac{1+0.075}{1} )^{1*10}[/tex]

A =[tex]= 19000(1.075)^{10}[/tex]

A ≈ $[tex]42,748.37[/tex]

Therefore, the amount that must be paid back at the end of the [tex]10[/tex]-year period is approximately $[tex]42,748.37[/tex].

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g(x) = -4cosx + (1/3)x⁶ - [tex]2x^(3/2)[/tex] + C, where C is any constant, satisfies g'(x) = 4sinx + (2x⁵-√x/x).

To find all capabilities function g that fulfill g'(x) = 4sinx + (2x⁵-√x/x), we really want to coordinate the two sides of the situation concerning x. In the first place, we can coordinate the right-hand side by separating it into two sections:

∫[4sinx + (2x⁵-√x/x)]dx = ∫4sinxdx + ∫(2x⁵-√x/x)dx

We can incorporate the initial segment involving the reconciliation recipe for sine:

∫4sinxdx = -4cosx + C₁,

where C₁ is a consistent of coordination.

For the subsequent part, we can utilize the power rule and the chain rule to incorporate the articulation:

∫(2x⁵-√x/x)dx = (2/6)x⁶ - [tex]2x^(3/2)[/tex]+ C₂,

where C₂ is one more steady of incorporation.

Assembling the two sections, we get:

g(x) = - 4cosx + (1/3)x⁶ - [tex]2x^(3/2)[/tex]+ C,

Where C = C₁ + C₂ is the steady of joining.

Hence, the arrangement of all capabilities g that fulfill g'(x) = 4sinx + (2x⁵-√x/x) is given by g(x) = - 4cosx + (1/3)x⁶ - [tex]2x^(3/2)[/tex] + C, where C is any steady.

As such, there are limitlessly many capabilities g that fulfill g'(x) = 4sinx + (2x⁵-√x/x), and they contrast exclusively by a consistent. This is on the grounds that the reconciliation of a capability generally presents a consistent of coordination, and any two capabilities that vary by a steady have a similar subordinate.

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The given equation has exactly one real root.

We have,

Assuming the equation is:

x² + e^(-x) = 1

Let f(x) = x² + e^(-x) - 1.

We have f(0) = 0² + e^(0) - 1 = 0, and f(-1) = (-1)² + e^(1) - 1 = e - 1 > 0.

Therefore, we have:

f(0) = 0 < 0 < e - 1 = f(-1)

Since f(x) is continuous and differentiable for all x, we can use the Intermediate Value Theorem to conclude that there exists a value of x in the interval (-1, 0) where f(x) = 0.

Suppose there exist two distinct real roots a and b with a < b.

Then, since f(x) is differentiable, there must exist a value of x between a and b for which f'(x) = 0 (by Rolle's Theorem).

However, we have:

f'(x) = 2x - e^(-x)

For x < 0, we have e^(-x) > 1, so f'(x) < 0.

For x > 0, we have e^(-x) < 1, so f'(x) > 0.

Therefore, f'(x) cannot be 0 for any x, which contradicts our assumption that there exist distinct real roots a and b.

Thus,

The given equation has exactly one real root.

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The value of C in the area of the region enclosed by the parabolas is 6.206.

We have,

The two parabolas intersect at points (c,0) and (-c,0).

We can find the area enclosed by them by integrating the difference of their equations with respect to x from -c to c:

A = ∫[-c,c] (x² - c² - c² + x²) dx

= 2 ∫[0,c] (2x² - 2c²) dx

= 4 ∫[0,c] (x² - c²) dx

= 4 [x³/3 - c² x] [0,c]

= 4 (c³/3 - c³)

= 4c³/3

We want this area to be 400, so we have:

4c³/3 = 400

Solving for c, we get:

c³ = 300

c = (300)^(1/3)

Therefore,

C ≈ 6.206.

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The  frequency table for the data is as follows:

Number of Concerts Frequency Cumulative Frequency

   1                                       1                                 1

   3                                       1                                 2

   5                                       1                                3

   6                                       1                                  4

   7                                        1                                  5

  8                                          1                                 6

  9                                          1                                  7

  11                                           1                                 8

  12                                          1                                  9

  15                                           1                                  10

In order to get table proceed,

Arrange the data in ascending order. The given data is: 5, 8, 6, 7, 11, 15, 9, 3, 12, 1 Arranging it in ascending order: 1, 3, 5, 6, 7, 8, 9, 11, 12, 15

Create a table with three columns: "Number of Concerts", "Frequency" and "Cumulative Frequency."

Fill in the "Number of Concerts" column with the sorted data.

Count the number of times each number appears in the data set and fill in the "Frequency" column accordingly.

Calculate the cumulative frequency by adding the frequency of each number to the frequency of all the numbers that come before it in the sorted data set and put it on the "Cumulative Frequency" column.

Here is the resulting cumulative frequency table:

Number of Concerts Frequency Cumulative Frequency

   1                                       1                                 1

   3                                       1                                 2

   5                                       1                                 3

   6                                       1                                  4

   7                                        1                                  5

  8                                          1                                 6

  9                                          1                                  7

  11                                           1                                 8

  12                                          1                                  9

  15                                           1                                  10

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The total number of eggs Dale collected is 187 eggs.

To find the total number of eggs, you would use operations, specifically, multiplication and addition. First, multiply the number of cartons (15) by the number of eggs per carton (12). This gives you a total of 180 eggs (15 x 12) in the cartons.

In addition to the eggs in the cartons, Dale also has 7 eggs left over. To find the total number of eggs Dale collected, you simply add the number of leftover eggs (7) to the total number of eggs in the cartons (180). This results in a grand total of 187 eggs (180 + 7) that Dale collected from his chickens.

In summary, Dale collected 187 eggs in total: 180 eggs from filling 15 cartons with 12 eggs each and 7 additional leftover eggs.

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The magnitude of the resultant vector is 96.77 and its direction is -78.95°.

To find the magnitude and direction of the resultant vector, we can use vector addition. We can break down each vector into its horizontal and vertical components using trigonometry:

For the first vector with magnitude 35 and direction 55°:

Horizontal component = 35 cos(55°) = 19.77

Vertical component = 35 sin(55°) = 28.25

For the second vector with magnitude 80 and direction 130°:

Horizontal component = 80 cos(130°) = -39.06

Vertical component = 80 sin(130°) = 66.68

To find the horizontal and vertical components of the resultant vector, we can add the corresponding components of the two vectors:

Horizontal component of the resultant vector = 19.77 - 39.06 = -19.29

Vertical component of the resultant vector = 28.25 + 66.68 = 94.93

The magnitude of the resultant vector is the square root of the sum of the squares of its horizontal and vertical components:

Magnitude of the resultant vector = sqrt((-19.29)^2 + (94.93)^2) = 96.77

The direction of the resultant vector can be found using trigonometry:

Direction of the resultant vector = arctan(94.93/-19.29) = -78.95° (measured counterclockwise from the positive x-axis)

Therefore, the magnitude of the resultant vector is 96.77 and its direction is -78.95°.

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To minimize the total cost, about 4,286 units should be produced on machine A and 18,364 units on machine B.

To minimize the total cost of producing x+y=22,650 units on two machines, A and B, we need to find the optimal distribution of units produced on each machine. Let's start by setting up the equations:

Machine A: C(x) = 20x^2
Machine B: C(y) = 240y^3 + 79y

Total cost: C(x, y) = C(x) + C(y)

Since x + y = 22,650, we can express y in terms of x: y = 22,650 - x.

Now, let's substitute this expression for y into the total cost equation:

C(x, y) = 20x^2 + 240(22,650 - x)^3 + 79(22,650 - x)

Now, to find the minimum total cost, we need to find the critical points by taking the derivative of C(x, y) with respect to x and set it to zero:

dC(x)/dx = 40x - 240(3)(22,650 - x)^2 + 79 = 0

Solve for x to get the optimal number of units to be produced on machine A:

x ≈ 4,286

Now, find the corresponding number of units to be produced on machine B using the relationship y = 22,650 - x:

y ≈ 22,650 - 4,286 = 18,364

Thus, to minimize the total cost, about 4,286 units should be produced on machine A and 18,364 units on machine B.

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The mean is 13 and the mean absolute deviation (MAD) is 3.9

From the box plot, we have the following readings that can be used in our computation:

4  8  10  10  10  10  12  14  14  16  16  18  20  20

The mean is calculated as

Mean = Sum/Count

So, we have

Mean = (4 + 8 + 10 + 10 + 10 + 10 + 12 + 14 + 14 + 16 + 16 + 18 + 20 + 20)/14

Evaluate

Mean = 13

The mean absolute deviation is calculated as

MAD = 1/n * ∑|xi - mean|

So, we have

MAD = 1/14 * (|4 - 13| + |8 - 13| + |10 - 13| + |10 - 13| + |10 - 13| + |10 - 13| + |12 - 13| + |14 - 13| + |14 - 13| + |16 - 13| + |16 - 13| + |18 - 13| + |20 - 13| + |20  - 13|)

Evaluate

MAD = 3.9

Hence, the mean is 13

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The answer to the student's question falls under the criteria of a result being undefined, because division of any number by 0 proceeds to the number being labelled indeterminate. Then the required answer to the question is Option D.

Let us take this into consideration that whenever we associate with a number being divided by 0 it automatically gets into a phase where it is no longer able to project its value like before, this phase of being unnoticed is refered to as being indeterminate.

Since, in this case we are given to perform division of  0 ÷ 0 and derive result, it is not possible to do so because of the above reason.

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a. To find the probability that a mountain bike is brown, we need to add up the production volumes for both models that come in brown and divide it by the total production volume. So, the total production volume is:
302 (XB-50 in blue) + 40 (YZ-99 in blue) + 105 (XB-50 in brown) + 205 (YZ-99 in brown) + 200 (XB-50 in white) + 130 (YZ-99 in white) = 982
The production volume for brown mountain bikes is 105 (XB-50) + 205 (YZ-99) = 310. So, the probability that a mountain bike is brown is:
310 / 982 = 0.316 or 31.6%

The production volume for brown mountain bikes is 31.6%.

b. To find the probability that the mountain bike is a YZ-99, we need to add up the production volumes for YZ-99 in all three colors and divide it by the total production volume. So, the production volume for YZ-99 is:
40 (blue) + 205 (brown) + 130 (white) = 375
The total production volume is still 982. So, the probability that the mountain bike is a YZ-99 is:
375 / 982 = 0.382 or 38.2%

The probability that the mountain bike is a YZ-99 is 38.2%.

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After expressing 10.1818181818...as a rational number, in the form p/q where p and q are positive integers with no common factors, we have p = 1008 and q = 99.

We can express the repeating decimal 10.1818181818... as follows:

Let x = 10.1818181818...

Then, 100x = 1018.18181818...

Subtracting x from 100x gives:

100x - x = 1018.18181818... - 10.1818181818...

Simplifying, we get:

99x = 1008

Dividing both sides by 99, we obtain:

x = 1008/99

Therefore, we have expressed the repeating decimal 10.1818181818... as the rational number 1008/99. Since 1008 and 99 are positive integers with no common factors, this fraction is in its simplest form.

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The equation of the tangent to the circle at the point where x=4 and y is negative is y = (4/3)x - 16/3, where m = 4/3 and c = -16/3.

The equation of a circle with center (a,b) and radius r is (x-a)² + (y-b)² = r². In our problem, the equation of the circle is x² + y² = 65. This means that the center of the circle is at the origin (0,0) and the radius is √65.

We can find the derivative of x² + y² = 65 using implicit differentiation. Taking the derivative with respect to x, we get:

2x + 2y(dy/dx) = 0

Simplifying this equation, we get:

dy/dx = -x/y

At the point where x=4 and y is negative, we have x=4 and y=-3. Plugging these values into the equation above, we get:

dy/dx = -4/(-3) = 4/3

This means that the slope of the tangent to the circle at the point (4,-3) is 4/3.

To find the equation of the tangent, we can use the point-slope form of a line, which is y - y₁ = m(x - x₁), where (x₁,y₁) is a point on the line and m is the slope. Plugging in the values we found, we get:

y - (-3) = (4/3)(x - 4)

Simplifying this equation, we get:

y = (4/3)x - 16/3

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The term that is used to describe a situation where two variables are correlated but there is no apparent connection between them is "spurious correlation."

Spurious correlation refers to a situation where two variables appear to be correlated or associated, but in reality, there is no actual causal relationship or meaningful connection between them. This can happen due to various reasons, such as coincidence, sampling errors, or confounding variables that are not accounted for in the analysis.

In other words, the observed correlation between the variables is merely coincidental or occurs by chance, and there is no true underlying relationship between them. It is important to be cautious when interpreting correlations, as spurious correlations can lead to incorrect or misleading conclusions.

Therefore, the correct term used for this phenomenon is "spurious correlation."

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(a)  The formula for the squirrel population is:

P(t) = 1500 / (1 + 1125 e^(-0.7t))

(b) It will take about 2.5 years for the squirrel population to double.

(a) Using separation of variables, we have:

dP/P(1500-P) = 0.7 dt

Integrating both sides, we get:

ln|P| - ln|1500 - P| = 0.7t + C

where C is the constant of integration.

Applying the initial condition P(0) = 375, we get:

ln|375| - ln|1125| = C

C = ln(3)

Therefore, the formula for the squirrel population is:

P(t) = 1500 / (1 + 1125 e^(-0.7t))

(b) To find the doubling time, we need to solve the equation:

2P(0) = P(d)

where P(0) = 375 and P(d) is the population after time d. Substituting the formula for P(t) from part (a), we get:

2(375) = 1500 / (1 + 1125 e^(-0.7d))

Simplifying and solving for d, we get:

d = ln(3) / 0.7

d ≈ 2.5 years

Therefore, it will take about 2.5 years for the squirrel population to double.

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The two ways of expressing the null hypothesis for a hypothesis test of the difference of means for two independent populations are: H0: μ1 - μ2 = 0, assuming no significant difference between the means.

H0: μ1 - μ2 = d, assuming a specific difference between the means equal to a constant value d.

Null Hypothesis H0: μ1 - μ2 = 0: This form of the null hypothesis assumes that there is no significant difference between the means of the two populations x1 and x2. In other words, the null hypothesis states that the difference between the means is exactly zero, indicating that the two populations have identical means.

Null Hypothesis H0: μ1 - μ2 = d: This form of the null hypothesis assumes that the difference between the means of the two populations x1 and x2 is equal to a specified value d. This specified value could be based on prior knowledge, theoretical expectations, or practical considerations. For example, if you are comparing the effectiveness of two different medications, you might specify d to be the minimum clinically important difference, which represents the smallest difference between the means that would be considered clinically meaningful.

Therefore, the two ways of expressing the null hypothesis for a hypothesis test of the difference of means for two independent populations are:

H0: μ1 - μ2 = 0, assuming no significant difference between the means.

H0: μ1 - μ2 = d, assuming a specific difference between the means equal to a constant value d.

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To find out after how many weeks the siblings will have the same amount of money in their savings accounts, we need to solve the system of linear equations. We can do this by setting the two equations equal to each other and solving for x:

7x + 40 = 5x + 60 7x - 5x = 60 - 40 2x = 20 x = 10

So, after 10 weeks, the siblings will have the same amount of money in their savings accounts. The correct answer is 10 weeks

After 10 weeks, their savings accounts will have the same amount of money. The correct answer is option (C).

A linear equation is defined as an equation in which the highest power of the variable is always one.

To find when their savings accounts will have the same amount of money, we need to solve for x in the equation:

7x + 40 = 5x + 60

Subtracting 5x from both sides, we get:

2x + 40 = 60

Subtracting 40 from both sides, we get:

2x = 20

Dividing both sides by 2, we get:

x = 10

Therefore, after 10 weeks, their savings accounts will have the same amount of money.

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Lanquaye should take 1200/50 = 24 steps before their footprint match again.

What is meant by steps?

A step refers to a single operation or action taken to solve a problem or prove a statement. Each step follows logically from the previous one and leads to the next, until the desired outcome is achieved. Steps may involve calculations, algebraic manipulation, logical deductions, or other techniques depending on the problem at hand.

According to the given information

The minimum number of steps that Lanquaye should take before their footprint match again is 50 steps.

Here’s how to calculate it:

The difference between Abu’s footprints is 48cm.

The difference between Lanquaye’s footprints is 50cm.

To find the minimum number of steps that Lanquaye should take before their footprint match again, we need to find the least common multiple (LCM) of 48 and 50.

The LCM of 48 and 50 is 1200.

Therefore, Lanquaye should take 1200/50 = 24 steps before their footprint match again.

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The sticker price for the food processors should be $45 to achieve a 60% gross margin.

Gross margin is a financial metric that represents the difference between the cost of goods sold (COGS) and the revenue earned from selling those goods. It is calculated by subtracting the COGS from the revenue and dividing the result by the revenue. The gross margin is usually expressed as a percentage.

To solve the problem is to use the gross margin formula, which relates the cost of an item to its selling price and gross margin percentage:

selling price = cost / (1 - gross margin percentage)

In this case, the cost of each food processor is $18 and the gross margin percentage is 60%, so we have:

selling price = $18 / (1 - 0.60) = $18 / 0.40 = $45

Therefore, the sticker price for the food processors should be $45 to achieve a 60% gross margin.

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The mean, variance and coefficient of variation of the PME are 3.3, 1.61 and 48.99%.

To find the mean of X, also known as the expected value, we calculate the weighted average of all possible values that X can take on, where the weights are the probabilities of each value. Mathematically, the mean can be expressed as:

E(X) = Σ k x P(X=k)

where Σ is the summation operator that goes from k=0 to k=5. Therefore, we can calculate the mean of X as:

E(X) = 0x0.05 + 1x0.05 + 2x0.15 + 3x0.20 + 4x0.25 + 5x0.30

= 3.3

So the mean of X is 3.3. This means that if we repeat this experiment many times, we expect the average outcome to be around 3.3.

Next, let's find the variance of X, which measures how much the values of X vary around the mean. Mathematically, the variance of X can be expressed as:

Var(X) = Σ (k - E(X))² x P(X=k)

where Σ is the summation operator that goes from k=0 to k=5, and E(X) is the mean of X that we calculated earlier. Therefore, we can calculate the variance of X as:

Var(X) = (0-3.3)²x0.05 + (1-3.3)²x0.05 + (2-3.3)²x0.15 + (3-3.3)²x0.20 + (4-3.3)²x0.25 + (5-3.3)²x0.30

= 1.61

So the variance of X is 1.61. This means that the values of X are scattered around the mean by an average of 1.27 (the square root of the variance).

Lastly, let's find the coefficient of variation (CV) of X, which is a measure of the relative variability of X compared to its mean. The CV is calculated by dividing the standard deviation (which is the square root of the variance) by the mean, and then multiplying by 100% to express the result as a percentage. Mathematically, the CV of X can be expressed as:

CV(X) = (Var(X)⁰°⁵ / E(X)) x 100%

Therefore, we can calculate the CV of X as:

CV(X) = (1.61⁰°⁵ / 3.3) x 100%

= 48.99%

So the CV of X is 48.99%. This means that the standard deviation of X is almost half of the mean. In other words, the values of X are relatively tightly clustered around the mean, with little variation.

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The discount amount was $232.80, and the new selling price of the computer system after the discount was applied was $543.20.

First, we need to calculate the amount of the discount. We can do this by multiplying the original price of the computer system by the percentage discount:

Discount = 30% x $776

Discount = 0.30 x $776

Discount = $232.80

So, the discount offered by the store is $232.80.

Next, we need to calculate the new selling price of the computer system after the discount has been applied. To do this, we need to subtract the discount amount from the original price:

New Selling Price = Original Price - Discount

New Selling Price = $776 - $232.80

New Selling Price = $543.20

Therefore, the new selling price of the computer system after the discount is $543.20.

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Complete Question:

A computer store buys a computer system at a cost of $465.60. The selling price was first at $776, but then the store advertised a 30% markdown on the system.

A) Find the new selling price. Round to the nearest cent if necessary.

1. What two critical things will the hypothesis test allow you to evaluate for this problem

The two critical things that a hypothesis test permits you to assess are the null hypothesis and the alternative hypothesis. The null hypothesis is a declaration about the population parameter that assumes there's no significant distinction or effect,

Whilst the alternative hypothesis is a announcement that contradicts the null hypothesis and suggests that there is a significant difference or impact.

2. What does the p-value represent ?

The p-value represents the probability of acquiring a test statistic as extreme or more excessive than the discovered value, assuming that the null hypothesis is true. In other words, it measures the proof against the null hypothesis supplied by the pattern statistics.

A small p-value shows robust proof against the null hypothesis and supports the alternative hypothesis, while a huge p-value suggests weak evidence towards the null hypothesis and fails to guide the alternative hypothesis.

3. How do you determine which hypothesis test to use (z or t)?

the choice among a z-check and a t-test depends at the pattern size and whether the population standard deviation is known or unknown. If the sample length is large (commonly greater than 30) and the population standard deviation is thought, a z-check may be used.

If the population standard deviation is unknown or the pattern size is small (normally less than 30), a t-test should be used.

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Now, as n approaches infinity, the term (2n - 7) in the denominator will dominate, and the limit will approach 0.

So, lim Cn = 0 as n → ∞.

To determine the limit of the sequence [tex]C_n = \frac{ln(2n - 7)} { (7n + 4)}[/tex] as n approaches infinity, we can use L'Hôpital's Rule since it is an indeterminate form of type ∞/∞.

First, we find the derivatives of the numerator and the denominator with respect to n:
[tex]\frac{d}{dn}(ln(2n - 7)) = \frac{2 }{(2n - 7)}\\d/dn(7n + 4) = 7[/tex]

Now, we apply L'Hôpital's Rule by taking the limit of the ratio of these derivatives as n approaches infinity:
lim (n → ∞)[tex]\frac{ [2 / (2n - 7)] }{ 7}[/tex]

Dividing by 7 is the same as multiplying by 1/7:
lim (n → ∞)[tex][2 / (2n - 7)] * (1/7)[/tex]

Now, as n approaches infinity, the term (2n - 7) in the denominator will dominate, and the limit will approach 0.

So, lim Cn = 0 as n → ∞.

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Answer:

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a) Use the following formula:

The total cost is the cost of the copier plus the cost of the materials for each order. So, we have:

We can simplify this equation to:

Hence the average cost per order is $40.

b) As more orders are printed, the average cost per order will decrease because the fixed cost of the copier is spread over a larger number of orders. In other words, the more orders that are printed, the lower the average cost per order will be.

The solution to the given integral is ∫ t⁶ / √(1 - t¹⁴) dt  is -2 / (13t¹⁴√(1 - t¹⁴)) + C

The given integral is:

∫ t⁶ / √(1 - t¹⁴) dt

To solve this integral, we need to use a technique called substitution. Let u = 1 - t¹⁴. Then du/dt = -14t¹³, and dt = -1/(14t¹³) du.

Substituting these values in the integral, we get:

∫ t⁶ / √(1 - t¹⁴) dt = -1/14 ∫ (1 - u)¹/₂ / u^(7/14) du

Now, let's simplify the integrand. We have:

(1 - u)¹/₂ = (u - 1)-¹/₂

And,

u^(7/14) = (u¹/₂)⁷ = (1 - t¹⁴)¹/₂)⁷

Substituting these values in the integral, we get:

∫ t⁶ / √(1 - t¹⁴) dt = -1/14 ∫ (u - 1)-¹/₂ / (1 - t¹⁴)^(7/2) du

Using the power rule of integration, we get:

-2v¹/₂ / (13(1 - t¹⁴)^(5/2)) + C

Substituting back the value of v, we get:

-2(1 - t¹⁴)¹/₂ / (13(1 - t¹⁴)^(5/2)) + C

Simplifying this expression, we get:

-2 / (13t¹⁴√(1 - t¹⁴)) + C

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This can be answered by the concept of Integration. Answer is f ''(4) = 3/4.

To find f, we first integrate f ''(t) = 3/t with respect to t. This gives us:

f'(t) = 3ln(t) + C1

To find C1, we use the initial condition f'(4) = 4:

4 = 3ln(4) + C1
C1 = 4 - 3ln(4)

Now we integrate f'(t) = 3ln(t) + C1 with respect to t to get f(t):

f(t) = 3tln(t) - 3t + C2

To find C2, we use the initial condition f(4) = 9:

9 = 3(4)ln(4) - 3(4) + C2
C2 = 9 - 3(4)ln(4) + 12
C2 = 21 - 3ln(256)

So the solution is:

f(t) = 3tln(t) - 3t + 21 - 3ln(256)

To find f ''(t), we take the second derivative of f(t):

f ''(t) = 3/t

Therefore, f ''(4) = 3/4.

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