For each of the following Boolean expressions, give the truth table, and put the expression in DNF and CNF. a) x'y'z + x(z + yz') c) x(wz + yz'w + yzw') + x'y' b) zy + xy') + y'z d) (xy + x'y')(zw + z'w') + xyw

The probable reason that 37 runners broke the four-minute mile barrier within one year after Roger Bannister originally did in 1954 was due to the psychological barrier being broken.

Before Bannister's accomplishment, many believed that running a mile under four minutes was impossible for a human being. However, once Bannister proved it could be done, it changed people's beliefs about what was possible and opened up new possibilities.

This change in belief and perception likely inspired other runners to push themselves harder and believe that they too could achieve this feat, leading to a rapid increase in the number of people breaking the four-minute mile barrier. Additionally, advances in training techniques and equipment may have also played a role in this increase.

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The expected time taken for the system to fail is dependent on the specific details of the system in question.

However, if we assume that the three modes have equal probabilities of occurring and that the failure occurs when the system reaches a specific mode, we can use the concept of Markov chains to find the expected time until failure.

In this case, the expected time until failure would be the reciprocal of the probability of the system transitioning to the failure mode.

Therefore, if each mode has an equal probability of 1/3, the expected time until failure would be 3 units of time.

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Mr. Krumm paid $9.27 for a tie that had a 3% sales tax added on. So, the tie cost $231.75 before the tax was included.

To find out how much the tie cost before the tax was included, we need to first calculate how much of the total price was due to the tax. We know that the total price Mr. Krumm paid was $9.27, and that this price included a $3% sales tax.

To calculate the amount of tax that was included in the price, we can start by setting up an equation:

0.03x = 9.27 - x

Here, x represents the cost of the tie before the tax was included. We know that the tax is 3% of this cost, which is why we're multiplying it by 0.03.. We're also subtracting x from 9.27 to get the amount of tax that was added on.

Simplifying this equation, we get:

0.04x = 9.27

Dividing both sides by 0.04, we get:

x = 231.75

So the tie cost $231.75 before the tax was included.

In summary, Mr. Krumm paid $9.27 for a tie that had a 3% sales tax added on. To find out how much the tie cost before the tax was included, we set up an equation and solved for the cost of the tie (x) before the tax was added. The answer is that the tie cost $231.75 before the tax was included.

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A one-to-one function is a function in which each input value (x) corresponds to exactly one output value (y) and vice versa. In other words, there are no repeating input values for different output values. Therefore  (h^-1 ◦ h)(-3) = h^-1 (-7) = (-7 - 2)/3 = -3

To find g^-1 (8), we need to find the input value (x) that corresponds to the output value (y) of 8 in the function g. Looking at the given function g, we can see that there is only one input value that corresponds to the output value of 8, which is -3. Therefore, g^-1 (8) = -3.

To find h^-1 (x), we need to solve for x in terms of y. Starting with the function h(x) = 3x + 2, we can rearrange it to get y = 3x + 2. Then, solving for x, we get x = (y - 2)/3. Therefore, h^-1 (x) = (x - 2)/3.

Finally, to find (h^-1 ◦ h)(-3), we need to first find h(-3) and then apply the inverse function h^-1 to the result. Using the function h(x) = 3x + 2, we can see that h(-3) = 3(-3) + 2 = -7. Then, applying the inverse function h^-1, we get (h^-1 ◦ h)(-3) = h^-1 (-7) = (-7 - 2)/3 = -3.

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Caution is needed in interpreting the results of the three significant tests at the 5% level, as they may be due to chance or Type I errors, and may not be generalizable to other contexts or populations.

There are a few reasons why we should exercise caution when interpreting the results of the multiple hypothesis tests conducted in this study:

Type I error: When multiple hypothesis tests are performed, the probability of making at least one Type I error (rejecting a null hypothesis when it is actually true) increases. In this case, since 51 tests were performed, the probability of at least one Type I error is higher than if only one test had been conducted.

Multiple comparisons problem: The more tests that are conducted, the more likely it is that at least one test will produce a significant result purely by chance. This is known as the multiple comparisons problem. Even if a significant result is found, it may not necessarily be meaningful in the broader context of the study.

Replication: The study only surveyed students during the first few semesters that the courses were taught. It is important to replicate the study in other contexts and with different populations to determine whether the results hold up under different conditions.

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Therefore, we would need to obtain at least 9604 grade-point averages to ensure that the sample mean is within 0.02 of the population mean with 95% confidence.

To determine the required sample size, we need to use the formula n = [(z * σ) / E]^2, where z is the z-score for the desired level of confidence (e.g. 1.96 for 95%), σ is the standard deviation of the population (which we don't know, so we can use a conservative estimate of 1), and E is the desired margin of error (0.02 in this case). Plugging in these values, we get n = [(1.96 * 1) / 0.02]^2 = 9604. So we would need to obtain at least 9604 grade-point averages to ensure that the sample mean is within 0.02 of the population mean with 95% confidence.
To determine the required sample size for standardizing grade point averages on a scale between 0 and 4 with a margin of error of 0.02, we can use the formula n = [(z * σ) / E]^2, where z is the z-score for the desired level of confidence, σ is the standard deviation of the population, and E is the desired margin of error. Assuming a 95% confidence level and a conservative estimate of σ = 1, we find that we would need at least 9604 grade-point averages to ensure that the sample mean is within 0.02 of the population mean.

Therefore, we would need to obtain at least 9604 grade-point averages to ensure that the sample mean is within 0.02 of the population mean with 95% confidence.

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An integer C that will make the polynomial factorable 32 − 8 + C = 24

To make the polynomial 32 - 8 + C factorable,

we must use a quadratic trinomial - ax²+bx+c.

We can rewrite the polynomial as 24 + C.

For it to be factorable, C   should be equal to -24, so that the expression becomes 0 when x = 2.

Therefore, C = -24.

Proof:

32 - 8 + (-24) = 0

Note;

A polynomial is an expression in mathematics that consists of variables and coefficients and includes only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

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Full Question:

a)  Find an integer C that will make the polynomial below factorable
32 − 8 + C = ____

10b) Show that the integer C you found works by factoring the trinomial using the X that we learned in class.

The stretch of the exponential decay function is y = (0.2)^x

From the question, we have the following parameters that can be used in our computation:

The graph

An exponential function is represented as

y = ab^x

Where

a = initial value i.e. a = y when x = 0

b = growth/decay factor

From the graph, we have

a = 1

Also from the graph, we have

b = 1/5

Evaluate

b = 0.2

This means that

The value of b is less than 1

So this case, the exponential function is a decay function

Recall that

y = ab^x

So, we have

y = 1(0.2)^x

Evaluate

y = (0.2)^x

Hence, the exponential decay function is y = (0.2)^x

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The statement describes the incidence measure of occurrence, which refers to the number of new cases of a disease or condition that occur in a defined population over a specific period of time.

This statement describes the incidence rate of flu among students living in Dunedin hostels.
The incidence rate is a measure of occurrence that calculates the number of new cases (in this case, students getting sick with the flu) in a specific population (150 students in Dunedin hostels) over a specific time period (3 months). This rate helps us understand the frequency at which the flu is affecting this particular group of students

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Observed frequencies are essential in hypothesis testing for comparing population proportions, as they represent the sample data collected from the populations of interest and serve as the basis for calculating test statistics and drawing conclusions about the hypothesis.

In a hypothesis test comparing two or more population proportions, observed frequencies refer to the sample data collected from a population of interest.

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The HDD for the month of January in State College, PA is 930.

To calculate the HDD (Heating Degree Days) for the month of January in State College, PA, we need to subtract the average daily temperature from 65 degrees Fahrenheit, which is considered the standard temperature for indoor heating.

HDD = 65°F - 35°F
HDD = 30°F

Therefore, the HDD for the month of January in State College, PA is 30 degrees Fahrenheit.
Hi! To calculate the Heating Degree Days (HDD) for the month of January in State College, PA with an average outside temperature of 35 degrees F, follow these steps:

1. Determine the base temperature: In the US, the base temperature is commonly 65 degrees F.
2. Subtract the average temperature from the base temperature: 65 - 35 = 30 degrees.
3. Multiply the difference by the number of days in the month: January has 31 days, so 30 * 31 = 930.

So, the HDD for the month of January in State College, PA is 930.

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To calculate the HDD for the month of January in State College, PA, we need to first determine the base temperature. The base temperature is the temperature below which a building needs to be heated in order to maintain a comfortable indoor temperature. In this case, we will use a base temperature of 65 degrees F.

To calculate the HDD, we need to subtract the average daily temperature from the base temperature and sum up the values for each day of the month. If we assume that January has 31 days, we can calculate the HDD as follows:

HDD = (65 - 35) x 31
HDD = 930

So the HDD for the month of January in State College, PA is 930. This means that during the month of January, there were 930 heating degree days, which indicates the amount of heating required to maintain a comfortable indoor temperature. It is important to note that the HDD can vary depending on the base temperature used, so it is important to choose a base temperature that is appropriate for the climate and the building being heated.

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The function of 13 is  f(x) = -x -3

The function of 14  is f(X) = 3x-7

The function of table 15 f(x) = -2 + 16

Table 13 is a linear function.

the slope is -1 and the intercept at y-axis is -3

Thus, the function is  f(x) = -x -3

Table 14 the table here has a linear finction. where the slope is 3 and y-intercept is -7 hence,

the function is f(x) = 3x -7

Table 15: This is also a linear function with slope of -2 and y intercept of 16, so

the function f(x) = -2x + 16

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The function f(x) has a relative minimum at x = 1/7 and a relative maximum at x = 0.To find where the function f(x) = 33° - 873 – 210.x2 + 4 is increasing and decreasing, we need to analyze its derivative function, f'(x), which is equal to 12x(1 - 7x). To determine where f'(x) is positive or negative, we use the VD Test and consider the signs of the factors 12, 1 - 7x, and x.

We first find the critical numbers by solving f'(x) = 0:

12x(1 - 7x) = 0
x = 0 or x = 1/7

These critical numbers divide the real line into three intervals: (-∞, 0), (0, 1/7), and (1/7, ∞). We then test a value from each interval in f'(x) to determine its sign:

f'(-1) = 12(-1)(1 + 7) = -96  (negative)
f'(1/14) = 12(1/14)(1 - 1) = 0  (zero)
f'(2) = 12(2)(1 - 14) = -240  (negative)

Using this information, we can construct a chart:

Interval | Test Value | 12 | 1-7x | x | f'(x)
-----------------------------------------------------
(-∞, 0) | -1 | - | - | - | -
(0, 1/7) | 1/14 | + | + | + | Increasing
(1/7, ∞) | 2 | + | - | + | Decreasing

Based on the chart, we can see that f(x) is increasing on the interval (0, 1/7) and decreasing on the interval (1/7, ∞). Therefore, we can conclude that the function f(x) has a relative minimum at x = 1/7 and a relative maximum at x = 0.

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Thus, the cost of the triangular lot land is approximately $81,940 found using Heron's formula.

To determine the cost of the triangular lot, you first need to calculate its area and then convert it to acres.

Given the three sides of the triangle (470 yards, 860 yards, and 1130 yards), you can use Heron's formula to find the area.

Heron's formula for the area of a triangle with sides a, b, and c is:
Area = √(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter, calculated as:
s = (a + b + c) / 2

In this case, a = 470 yards, b = 860 yards, and c = 1130 yards.

Therefore, the semi-perimeter, s, is:

s = (470 + 860 + 1130) / 2 = 1230 yards

Now, plug the values into Heron's formula to calculate the area:

Area = √(1230 * (1230 - 470) * (1230 - 860) * (1230 - 1130))
Area ≈ 198,342.77 square yards

To convert square yards to acres, use the conversion factor:

1 acre = 4,840 square yards
So, the area in acres is:

198,342.77 square yards * (1 acre / 4,840 square yards) ≈ 40.97 acres

Finally, multiply the area in acres by the price per acre to find the cost:
Cost = 40.97 acres * $2000 per acre ≈ $81,940

The cost of the land is approximately $81,940.

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The probability that the sample will contain more than 5% defective units is approximately 0.9525 or 95.25%.

To solve this problem, we can use the binomial distribution formula:

P(X > 5) = 1 - P(X ≤ 5)

where X is the number of defective items in a sample of size n = 100, and p = 0.1 is the probability of an item being defective.

To calculate P(X ≤ 5), we can use the binomial cumulative distribution function (CDF) or a binomial probability table. Alternatively, we can use a normal approximation to the binomial distribution, which is valid when np ≥ 10 and n(1-p) ≥ 10, as is the case here (np = 10 and n(1-p) = 90).

Using the normal approximation, we can standardize the distribution of X as follows:

[tex]z = (X - np) / \sqrt{(np(1-p))}[/tex]

Then, we can use a standard normal table or calculator to find the probability of z ≤ z0, where z0 is the standardized value corresponding to X = 5.

Let's use the normal approximation method to solve the problem:

np = 100 x 0.1 = 10

σ = [tex]\sqrt{(np(1-p))} = \sqrt{(9)} = 3[/tex]

z0 = (5 - 10) / 3 = -1.67 (rounded to two decimal places)

Using a standard normal table or calculator, we find that P(Z ≤ -1.67) = 0.0475 (rounded to four decimal places).

Therefore, P(X > 5) = 1 - P(X ≤ 5) = 1 - 0.0475 = 0.9525 (rounded to four decimal places).

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We would need approximately 51.75 pounds of seed (0.069 x 750) should be in the blend.

In order to determine how many pounds of each seed should be in the blend, we need to consider the relative importance of each factor - shade tolerance, traffic, and drought resistance. If all factors are equally important, we could simply divide the total weight of the blend (let's assume 100 pounds for simplicity) by the total number of points required (1450 points) to get the amount of each seed needed for one point. This would be approximately 0.069 pounds per point.

To calculate the amount of each seed needed for the specific score requirements, we would then multiply the required points for each factor by the amount needed per point. For shade tolerance (300 points), we would need approximately 20.7 pounds of seed (0.069 x 300).

For traffic (400 points), we would need approximately 27.6 pounds of seed (0.069 x 400). And for drought resistance (750 points), we would need approximately 51.75 pounds of seed (0.069 x 750).

Of course, it's possible that certain factors may be more important than others in a particular situation. In that case, we would need to adjust the amounts of each seed accordingly.

Additionally, other factors such as cost, availability, and compatibility with existing vegetation may also need to be considered when choosing the specific seeds to include in the blend.

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The probability density function for a uniform distribution is given by f(x) = 1/(b-a). The height of the uniform distribution is 1/11.

a) The random variable in the context of this problem is Orv X - the waiting time for a randomly selected commuter on the Red Line during peak rush hours, which follows a uniform distribution between 0 minutes and 11 minutes.
b) The height of the uniform distribution can be computed as follows:
The probability density function for a uniform distribution is given by:
f(x) = 1/(b-a)
where a and b are the lower and upper limits of the distribution, respectively.
In this case, a = 0 and b = 11, so:
f(x) = 1/(11-0) = 1/11
a) In the context of this problem, the random variable (X) represents the waiting time for a randomly selected commuter on the Red Line during peak rush hours.
b) To compute the height of the uniform distribution, you'll need to use the formula:
Height = 1 / (b - a)
where 'a' represents the minimum waiting time (0 minutes) and 'b' represents the maximum waiting time (11 minutes).
Height = 1 / (11 - 0)
Height = 1 / 11
So, the height of the uniform distribution is 1/11.

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The equivalent expression is 15a(-2a + 3).

To factor out the common factor of -15a from the given expression -30a^2 + 45a, we can first factor out -15, then factor out a:

-30a^2 + 45a = -15a(2a - 3)

Therefore, the expression -30a^2 + 45a is equivalent to -15a(2a - 3).

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According to data from the Pew Research Center, there were approximately 10.5 million unauthorized immigrants living in the United States in 2017. This number represents around 25% of the total foreign-born population in the country. In conclusion, out of the 41.3 million foreign-born individuals in the U.S., about 10.5 million are considered unauthorized immigrants.

Unauthorized immigrants, also known as undocumented immigrants, are individuals who enter the United States without legal permission or overstay their visas. They are not eligible for most government benefits and are often subject to deportation if caught.

The Pew Research Center estimates that there were 41.3 million foreign-born individuals living in the United States in 2017, which includes both authorized and unauthorized immigrants. Of this population, around 10.5 million were unauthorized immigrants.

Unauthorized immigration is a complex and contentious issue in the United States, with many different opinions on how to address it. Understanding the size and characteristics of this population is an important part of any discussion or policy debate.

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Answer: 7.5 years

Step-by-step explanation: First, we need to calculate your annual savings by multiplying your gross pay by 15% and then subtracting the taxes:

$40,000 x 15% = $6,000 (annual savings before taxes)

$6,000 - $4,000 = $2,000 (annual savings after taxes)

Next, we can calculate how many years it will take to save $15,000 by dividing the goal by the annual savings:

$15,000 ÷ $2,000 = 7.5 years

Therefore, it will take you 7.5 years to achieve your goal of creating a $15,000 emergency fund by saving 15% of your gross pay.

Using the dual simplex method to find a solution to the linear programming problem formed by adding the constraint 3xi 5x3> 15 to the problem. The final solution is:
xi = 20
x2 = 0
x3 = 15
x4 = 0
x5 = 40

Adding the constraint 3xi + 5x3 > 15 does not affect the optimal solution, as none of the variables involved in the new constraint are in the basis. Therefore, the final solution remains the same.

To use the dual simplex method to find a solution to the linear programming problem formed by adding the constraint 3xi + 5x3 > 15 to the problem in example 2, we need to follow these steps:

1. Rewrite the problem in standard form by adding slack variables:
Maximize 4xi + 3x2 + 5x3
Subject to:
2xi + 3x2 + 4x3 + x4 = 60
3xi + 2x2 + x3 + x5 = 40
xi, x2, x3, x4, x5 >= 0

2. Calculate the initial feasible solution by setting all slack variables to 0:
xi = 0
x2 = 0
x3 = 0
x4 = 60
x5 = 40

3. Calculate the reduced costs of the variables:
c1 = 4 - 2/3x4 - 3/2x5
c2 = 3
c3 = 5 - 2/3x4 - 1/2x5
c4 = -2/3x1 - 1/2x2
c5 = -3/2x1 - 1/2x2

4. Choose the entering variable with the most negative reduced cost. In this case, it is x1.

5. Calculate the minimum ratio test for each constraint to determine the leaving variable:
For the first constraint: x4/2 = 30, x1/2 = 0, so x4 is the leaving variable.
For the second constraint: x5/3 = 40/3, x1/3 = 0, so x5 is the leaving variable.

6. Update the solution by performing the pivot operation:
- Pivot on x1 and x4 in the first constraint: x1 = 20, x4 = 0, x2 = 0, x3 = 15, x5 = 40/3
- Pivot on x1 and x5 in the second constraint: x1 = 0, x4 = 0, x2 = 0, x3 = 15, x5 = 40

7. Repeat steps 3-6 until all reduced costs are non-negative or all minimum ratio tests are negative.

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For a group of 32 pets including 16 puppies and 16 kittens is lined up in random order, the probability that the pet in the 15-th position is a kitten is equals to the 0.20.

Probability is calculated by dividing the favourable outcomes to the total possible outcomes. There is a group of 16 puppies and 16 kittens. It is lined up in random order. Total number of pets = 32

Which means 32 out of 32 are selected for line up and order of selection is important. So, using the permutation, total possible outcomes = ³²P₃₂ = 32!

We have to determine probability that the pet in the 15ᵗʰ position is a kitten.

When 15ᵗʰ position is a kitten, there is 16 ways to select a kitten to be 15ᵗʰ place and there are ³¹P₃₁ ways to line up the remaining 31 pets. So, favourable outcomes = 16.³¹P₃₁ = 16 × 31!

The required probability = [tex]\frac{ 16 × 31! }{32!} [/tex]

= [tex]\frac{ 16 × 31! }{32×31!} [/tex]

= 0.20

Hence, required probability is 0.20.

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The calculated value of x in the expression x^(1/12) = 49^(1/24) is 7

From the question, we have the following parameters that can be used in our computation:

x^(1/12) = 49^(1/24)

Express 49 as 7^2

So, we have

x^(1/12) = 7^(2 * 1/24)

Evaluate the products

This gives

x^(1/12) = 7^(1/12)

When both sides of the equations are compared, we have

x = 7

Hence, the value of x in the expression is 7

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Answer:99

Step-by-step explanation:

The value of the test statistic is approximately 2.8

To determine the effectiveness of the advertising campaign at the fast-food restaurant, we can use the sample data and perform a hypothesis test using the test statistic. In this case, the terms you want me to include are: average daily sales, sample size, population standard deviation, and the test statistic.

The average daily sales before the campaign were $6,000 per day. After introducing the advertising campaign, a sample of 49 days of sales was taken, showing an average of $6,400 per day. The population standard deviation is $1,000.

To calculate the test statistic, we can use the following formula:

Test statistic = (Sample mean - Population mean) / (Population standard deviation / sqrt(Sample size))

Test statistic = ($6,400 - $6,000) / ($1,000 / sqrt(49))

Test statistic = $400 / ($1,000 / 7)

Test statistic = $400 / $142.86

Test statistic ≈ 2.8

So, the value of the test statistic is approximately 2.8. This test statistic can be used to determine the effectiveness of the advertising campaign by comparing it to a critical value or finding the p-value, which will help us understand if the observed increase in daily sales is statistically significant.

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If the base of the solid is bounded by the curves f(x) = x2 and g(x) = x + 2 and the cross-sections perpendicular to the x-axis are rectangles of height 3 then the volume of the given solid is 9 cubic units.

The volume of the given solid can be found using the method of slicing, where we first determine the area of each cross-sectional rectangle and then integrate it over the specified region.

The base of the solid is bounded by the curves f(x) = x^2 and g(x) = x + 2. To find the region between these curves, we can set them equal to each other and solve for x:

x^2 = x + 2
x^2 - x - 2 = 0
(x - 2)(x + 1) = 0

This gives us two points of intersection: x = 2 and x = -1.

Now, let's find the length of the base of each rectangle, which is the difference between the y-values of the two curves:

Base length = g(x) - f(x) = (x + 2) - x^2

Since the height of each rectangle is given as 3, the area of each rectangle can be calculated as:

Area = Base length * Height = [(x + 2) - x^2] * 3

To find the volume of the entire solid, we integrate the area of the rectangles along the x-axis, between the intersection points -1 and 2:

Volume = ∫[3((x + 2) - x^2)] dx from -1 to 2

Evaluating this integral, we get:

Volume = 3[(x^2/2 + 2x - x^3/3)] from -1 to 2 = 9 cubic units

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The number of people on x tables is 2n + mx

From the question, we have the following parameters that can be used in our computation:

Tables = x

Represent the number of people on the middle tables with m

So, we have

Middle table = (x - 2) * m

Considering the first and the last tables have more people

Represent the additional number of people with n

So, we have

First and last = 2(m + n)

The number of people on x tables is

People = First and last + Middle table

This gives

People = 2(m + n) + (x - 2) * m

Expand

People = 2m + 2n + mx - 2m

Evaluate the like terms

People = 2n + mx

Hence, the number of people is 2n + mx

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The angles in DD and DMS

(a) sin−1 (0.5432) = 32° 36' 0"

(b) cos−1 (0.3165) = 71° 12' 0"

(c) tan−1 (1.1111) = 46° 24' 0"

(d) cot−1 (4) = 14° 0' 0"

(e) sec−1 (2.5) = 66° 24' 0

(f) csc−1 (1.25) = 51° 6' 0"

(a) The sine of an angle is opposite/hypotenuse. So, sinθ = 0.5432. Using the inverse sine function on a calculator, we get θ ≈ 32.6°. In DMS notation, this would be 32° 36' 0".

(b) The cosine of an angle is adjacent/hypotenuse. So, cosθ = 0.3165. Using the inverse cosine function on a calculator, we get θ ≈ 71.2°. In DMS notation, this would be 71° 12' 0".

(c) The tangent of an angle is opposite/adjacent. So, tanθ = 1.1111. Using the inverse tangent function on a calculator, we get θ ≈ 46.4°. In DMS notation, this would be 46° 24' 0".

(d) The cotangent of an angle is adjacent/opposite. So, cotθ = 4. Using the inverse cotangent function on a calculator, we get θ ≈ 14.0°. In DMS notation, this would be 14° 0' 0".

(e) The secant of an angle is hypotenuse/adjacent. So, secθ = 2.5. Using the inverse secant function on a calculator, we get θ ≈ 66.4°. In DMS notation, this would be 66° 24' 0".

(f) The cosecant of an angle is hypotenuse/opposite. So, cscθ = 1.25. Using the inverse cosecant function on a calculator, we get θ ≈ 51.1°. In DMS notation, this would be 51° 6' 0".

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