Use cylindrical coordinates to find the volume of the solid.
Solid inside x2 + y2 + z2 = 16 and outside z=sq.root (x2+y2)

Answer:

12

Step-by-step explanation:

8×9=6x

x=12

That is the answer

the diagonal length is approximately 0.0686 units.

What are Polynomials?

Polynomials are mathematical expressions consisting of variables, coefficients, and exponents. They are widely used in various fields of mathematics, science, engineering and even computer science.

To determine the dimension of the subspace H, we need to find the number of linearly independent vectors that span the subspace. The dimension of a subspace is equal to the number of vectors in any basis for that subspace.

First, let's check if the vectors in H are linearly independent by setting up a system of equations:

a(10x^2 - 12x - 13) + b(13x - 4x^2 + 9) + c(5x^2 - 7x - 7) = 0

Expanding and collecting like terms:

(5c - 4b + 10a)x^2 + (-7c + 13b - 12a)x + (-7c + 9b - 13a) = 0

For this equation to hold true for all values of x, the coefficients of each power of x must be zero. We can set up a system of equations:

5c - 4b + 10a = 0 (1)

-7c + 13b - 12a = 0 (2)

-7c + 9b - 13a = 0 (3)

We can solve this system of equations to determine if there are any non-trivial solutions. However, we can also observe that the determinant of the coefficient matrix is non-zero:

| 10 -4 5 |

| -12 13 -7 | = 76

| -13 9 -13 |

Since the determinant is non-zero, the system of equations has a unique solution, which means the vectors in H are linearly independent.

Therefore, a basis for the subspace H is {10x^2 - 12x - 13, 13x - 4x^2 + 9, 5x^2 - 7x - 7}.

the diagonal length is approximately 0.0686 units.

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a. The point estimate of the population proportion is 2.4%.

b. The margin of error for a 99% confidence interval estimate is 0.020.

c. The 99% confidence interval for the population proportion is between 0.024 and 0.276

a. The point estimate of the population proportion can be calculated by dividing the number of employees who failed the test (12) by the total number of tests conducted (490) and converting it to a percentage:

Point estimate = (12/490) × 100 = 2.4%

b. The margin of error for a confidence interval estimate can be calculated using the formula:

Margin of error = Z × [tex]\sqrt{(\beta (1 - \beta )/ n) }[/tex]

For a 99% confidence interval, Z is the critical value obtained from the z-distribution table. Since the population proportion is unknown, we use the point estimate as an approximation. n is the sample size, which is 490.

Using the z-distribution table, the critical value for a 99% confidence interval is approximately 2.576.

Plugging in the values, we get:

Margin of error = 2.576 × [tex]\sqrt{0.024 (1 - 0.024) }[/tex] / 490) ≈ 0.020

c. To compute the 99% confidence interval for the population proportion, we use the formula:

Confidence interval = Point estimate ± Margin of error

Substituting the values, we have:

Confidence interval = 2.4% ± 0.020

Confidence interval ≈ (0.024, 0.276)

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VE[X] is -10001999900.

By comparing the values, we can see that E[VF] = E[X] ≥ 1, and E[VE[X]] = -10001999900.

To prove that for a concave function g, we have 9(E[X]) > E[g(X)], we can use Jensen's inequality. Jensen's inequality states that for a concave function g and a random variable X, we have:

g(E[X]) ≥ E[g(X)]

Let's start the proof:

Since g is a concave function, we have g''(x) < 0 for all x.

By Jensen's inequality, we have g(E[X]) ≥ E[g(X)].

Now, let's compare E[X] and E[g(X)]:

E[X] = ∑[x] x * P(X = x) (where ∑[x] denotes the sum over all possible values of X)

E[g(X)] = ∑[x] g(x) * P(X = x)

Since g''(x) < 0 for all x, g(x) is a concave function. By applying Jensen's inequality to g(x), we have:

g(E[X]) ≥ E[g(X)]

Now, we can multiply both sides of the above inequality by 9 (a positive constant):

9 * g(E[X]) ≥ 9 * E[g(X)]

Since g(E[X]) ≥ E[g(X)], we can replace g(E[X]) on the left-hand side:

9 * g(E[X]) ≥ E[g(X)]

Therefore, we have 9(E[X]) > E[g(X)].

This proves that for a concave function g, we always have 9(E[X]) > E[g(X)].

Moving on to the second part of the question:

a. To determine the distribution of Y = X, we can simply use the given probability mass function of X.

P(Y = y) = P(X = y) (since Y = X)

Therefore, the distribution of Y is the same as the distribution of X.

b. We need to compare E[VF] and E[VE[X]]. Using the given function g(x) = -1, we can see that it is a convex function.

By Jensen's inequality for convex functions, we have:

g(E[X]) ≤ E[g(X)]

Substituting g(x) = -1, we have:

-1 * E[X] ≤ E[-1]

-E[X] ≤ -1

E[X] ≥ 1

This implies that E[VF] = E[X] ≥ 1.

To compare E[VF] and E[VE[X]], we need to compute E[VE[X]]. Using Exercise 8.11 (which is not provided in the question), or by directly calculating, we find:

E[VE[X]] = E[X * X] = ∑[x] (x * x) * P(X = x)

c. To compute VE[X], we need to find the variance of X. Using the formula for variance, we have:

VE[X] = E[X^2] - (E[X])^2

Substituting the given probability mass function of X, we can calculate:

E[X^2] = ∑[x] (x^2) * P(X = x)

E[X^2] = (0^2 * 2) + (1^2 * 100) + (10^2 * 10000)

= 0 + 100 + 1000000

= 1000100

E[X] = ∑[x] x * P(X = x)

E[X] = (0 * 2) + (1 * 100) + (10 * 10000)

= 100010

VE[X] = E[X^2] - (E[X])^2

= 1000100 - (100010)^2

= 1000100 - 10002000100

= -10001999900

Therefore, VE[X] is -10001999900.

By comparing the values, we can see that E[VF] = E[X] ≥ 1, and E[VE[X]] = -10001999900.

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The correct answer is B) 11.30 ounces. To find the mean setting for the soft drink dispenser so that a 12-ounce cup will overflow less than 1% of the time, we need to determine the z-score corresponding to a cumulative probability of 0.99.

Since we assume a normal distribution, we can use the z-score formula:

z = (x - μ) / σ

where:

z is the z-score

x is the value we want to find the z-score for (in this case, 12 ounces)

μ is the mean setting of the dispenser

σ is the standard deviation of the dispenser (0.3 ounce)

We want to find the z-score that corresponds to a cumulative probability of 0.99, which is 1% of the time.

Using a standard normal distribution table or calculator, we can find that the z-score corresponding to a cumulative probability of 0.99 is approximately 2.33.

Now, let's plug in the values into the z-score formula and solve for μ:

2.33 = (12 - μ) / 0.3

Rearranging the formula:

12 - μ = 2.33 * 0.3

12 - μ = 0.699

μ = 12 - 0.699

μ ≈ 11.301

Rounding to two decimal places, the mean setting of the dispenser should be approximately 11.30 ounces.

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The SI for travel to St. Kitts using the ratio to centered moving average method are: Sep 0.12,Oct 0.10,Nov 0.08,Dec 0.06.

The ratio to centered moving average method is a simple moving average method that uses a centered moving average. The centered moving average is calculated by taking the average of the current value and the two values before and after it. The SI is then calculated by dividing the current value by the centered moving average. In the Excel file, the data for travel to St. Kitts is in the range A2:B13. The centered moving average is calculated in the range C2:C13. The SI is calculated in the range D2:D13. The following steps were used to calculate the SI for travel to St. Kitts using the ratio to centered moving average method: The centered moving average was calculated for each month. The SI was calculated for each month by dividing the current value by the centered moving average. The following are the results of the calculation:  Sep 0.12,Oct 0.10,Nov 0.08,Dec 0.06.

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The correct statement is option D: "The p-value is less than or equal to 0.05."

The significance level, also known as the alpha level, is the threshold used to determine whether the results of a statistical test are statistically significant. In this case, the test resulted in the rejection of the null hypothesis at a significance level of 0.05.

The p-value is a measure of the strength of evidence against the null hypothesis. It represents the probability of observing a test statistic as extreme as, or more extreme than, the one obtained if the null hypothesis is true. If the p-value is less than or equal to the chosen significance level (0.05 in this case), it indicates that the evidence is statistically significant and supports the rejection of the null hypothesis.

Therefore, the correct statement is that the p-value is less than or equal to 0.05. Option A is not necessarily true because the results may not be statistically significant at the 10% level. Option B is also not necessarily true because the results may not be statistically significant at the 1% level. Option C is incorrect as it contradicts the fact that the null hypothesis was rejected at the 0.05 significance level.

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Answer:

-3, -2, 2, 3

Step-by-step explanation:

integers are any whole numbers (so not fractions), including negative and positive numbers, as well as zero

so therefore the first option shown is incorrect because there are fractions included which are not integers

(2a² + 3a - 5) - (3a² + 3a + 7)

Distributing the negative sign, we have:

2a² + 3a - 5 - 3a² - 3a - 7

Combining like terms, we get:

(2a² - 3a²) + (3a - 3a) + (-5 - 7)

= -a² - 12

The equation (x - 2)² = 13 is not equivalent because it represents a perfect square, not the original quadratic equation.

The equation (x - 2)² = 17 is also not equivalent because the constant term is different from the original equation.

The equation (x - 4)² = 13 is equivalent to the original equation because it represents a perfect square with the correct constant term.

The equation (x - 4)² = 17 is not equivalent because the constant term is different from the original equation.

(2x - 1)(3x + 4)

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Step-by-step explanation:

- 1/2 x + 6 > - 12        add 12 to both sides of the equation

 - 1/2x + 18 > 0          add 1/2 x to both sides

        18 > 1/2 x          multiply both sides by two

         36 > x        or     x < 36          Done.

a. The probability P(X < 10) is U(0, 50) is 0.20.

b. The probability P(X > 595) is U(0, 1,000) is 0.405.

c. The probability P(21 < X < 49) is U(19, 68) is 0.4762.

a. For a uniform continuous distribution U(0, 50), the probability of an event X < 10 can be calculated by dividing the length of the interval [0, 10] by the length of the entire interval [0, 50]. Since the lengths of both intervals are equal, the probability is 10/50 = 0.20.

b. Similarly, for a uniform continuous distribution U(0, 1,000), the probability of an event X > 595 can be calculated by dividing the length of the interval [595, 1,000] by the length of the entire interval [0, 1,000]. The length of the interval [595, 1,000] is 1,000 - 595 = 405, and the length of the entire interval is 1,000 - 0 = 1,000. Thus, the probability is 405/1,000 = 0.405.

c. For a uniform continuous distribution U(19, 68), the probability of an event 21 < X < 49 can be calculated by dividing the length of the interval [21, 49] by the length of the entire interval [19, 68]. The length of the interval [21, 49] is 49 - 21 = 28, and the length of the entire interval is 68 - 19 = 49. Therefore, the probability is 28/49 = 0.5714.

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To construct a box plot from the given data, which represents the diameters of cans in an assembly line, we need to determine the five-number summary and plot the corresponding box and whisker plot.

The five-number summary consists of the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value.

To construct the box plot, we start by arranging the data in ascending order: 5.1, 5.2, 5.2, 5.2, 5.3, 5.5, 5.5, 5.5, and 5.6. The minimum value is 5.1, and the maximum value is 5.6. The median is the middle value, which in this case is 5.3.

To find the first quartile (Q1) and the third quartile (Q3), we divide the data into two halves. Q1 is the median of the lower half, which consists of 5.1, 5.2, 5.2, and 5.2. Q3 is the median of the upper half, which consists of 5.5, 5.5, 5.5, and 5.6. The box plot will show the minimum value, Q1, Q2 (median), Q3, and the maximum value, giving us a visual representation of the distribution and variability of the data.

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False, if an overall F-test in an ANOVA model is significant, it is important to conduct follow-up comparisons to test for differences between pairs of means.

When the overall F-test in an ANOVA model is found to be significant, it indicates that there is evidence of at least one significant difference among the group means. However, it does not provide specific information about which particular group means are different from each other. Therefore, follow-up comparisons, such as post hoc tests or pairwise comparisons, are necessary to determine the specific pairs of means that are significantly different.

These follow-up comparisons allow for a more detailed understanding of the group differences and help identify which specific groups are driving the significant overall F-test result. By conducting these additional tests, researchers can gain insights into the specific pairwise differences and make more accurate and informed interpretations of their data.

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A Poisson random variable represents the number of occurrences of an event in a fixed interval of time or space. It is a discrete random variable, which means that it can only take on integer values, starting from zero. Therefore, the smallest numerical value that a Poisson random variable can be is zero.

This means that there is a possibility that the event will not occur at all during the given interval. For example, if we are counting the number of customers who visit a store in an hour, it is possible that no customers show up during that hour, resulting in a Poisson random variable of zero.

However, the probability of this occurring depends on the average rate of the event occurring, which is denoted by the parameter λ in the Poisson distribution. The larger the value of λ, the smaller the probability of a Poisson random variable being zero.

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The total surface area of the given triangular prism is 204 cm².

To find the total surface area of a triangular prism, we need to calculate the areas of each individual face and then sum them up.

Given that the dimensions are not to scale, we'll consider the following measurements:

Base of the triangular face: 10 cm

Height of the triangular face: 6 cm

Length of the prism: 8 cm

First, let's find the area of the triangular faces:

Area of one triangular face = (1/2) × base × height

= (1/2) × 10 cm × 6 cm

= 30 cm²

Since there are two triangular faces, the total area of the triangular faces is 2 × 30 cm² = 60 cm².

Next, let's find the area of the rectangular faces:

Area of one rectangular face = length * height

= 8 cm × 6 cm

= 48 cm²

Since there are three rectangular faces, the total area of the rectangular faces is 3 × 48 cm² = 144 cm².

Finally, to find the total surface area of the prism, we add the areas of the triangular and rectangular faces:

Total surface area = Area of triangular faces + Area of rectangular faces

= 60 cm² + 144 cm²

= 204 cm²

Therefore, the total surface area of the given triangular prism is 204 cm².

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Question

The diagram shows the sketch of a net of a triangular prism . 10 cm not to scale  6 cm 8 cm 15 work out the total surface area of the prism. X10-Dhx L xb2%x.

The Cartesian product of two C¹ surfaces, denoted as M x M1 by M2, where M1 and M2 are surfaces of dimensions m1 and m2 in [tex]R^{n_{1} }[/tex]and [tex]R^{n_{2} }[/tex]respectively, is a C¹ surface of dimensions m1 * m2 in [tex]R^{n_{1} +n_{2} }[/tex]. The tangent space of M x M1 by M2 at a point can be expressed in terms of the tangent space.

Consider two C¹ surfaces, M1 in [tex]R^{n_{1} }[/tex] and M2 in [tex]R^{n_{2} }[/tex], with dimensions m1 and m2 respectively. The Cartesian product of these surfaces, denoted as M x M1 by M2, is obtained by taking every point (p, q) where p belongs to M1 and q belongs to M2. This results in a new surface of dimensions m1 * m2.

To understand the tangent space of M x M1 by M2 at a specific point, we need to consider the tangent spaces of M1 and M2 at their respective points. Let's denote the tangent space of M1 at a point p as Tp(M1), and the tangent space of M2 at a point q as Tq(M2).

The tangent space of M x M1 by M2 at a point (p, q) can be expressed as the Cartesian product of Tp(M1) and Tq(M2). In other words, it can be written as Tp(M1) x Tq(M2). This means that the tangent space of the Cartesian product surface is obtained by taking every combination of tangent vectors from Tp(M1) and Tq(M2).

Overall, the Cartesian product of two C¹ surfaces, M x M1 by M2, is a C¹ surface of dimensions m1 * m2 in [tex]R^{n_{1} +n_{2} }[/tex] . The tangent space of M x M1 by M2 at a point (p, q) is expressed as the Cartesian product of the tangent spaces of M1 and M2 at points p and q, respectively.

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Answer:

That's 43 cents ( D ).

Therefore, the volume of the solid bounded by the coordinate planes and the plane 6x + 4y + z = 24 is 96 cubic units.

To find the volume of the solid bounded by the coordinate planes (xy-plane, xz-plane, and yz-plane) and the plane 6x + 4y + z = 24, we need to determine the region in space enclosed by these boundaries.

First, let's consider the plane equation 6x + 4y + z = 24. To find the x-intercept, we set y = 0 and z = 0:

6x + 4(0) + 0 = 24

6x = 24

x = 4

So, the plane intersects the x-axis at (4, 0, 0).

Similarly, to find the y-intercept, we set x = 0 and z = 0:

6(0) + 4y + 0 = 24

4y = 24

y = 6

So, the plane intersects the y-axis at (0, 6, 0).

To find the z-intercept, we set x = 0 and y = 0:

6(0) + 4(0) + z = 24

z = 24

So, the plane intersects the z-axis at (0, 0, 24).

We can visualize that the solid bounded by the coordinate planes and the plane 6x + 4y + z = 24 is a tetrahedron with vertices at (4, 0, 0), (0, 6, 0), (0, 0, 24), and the origin (0, 0, 0).

To find the volume of this tetrahedron, we can use the formula:

Volume = (1/3) * base area * height

The base of the tetrahedron is a right triangle with sides of length 4 and 6. The area of this triangle is (1/2) * base * height = (1/2) * 4 * 6 = 12.

The height of the tetrahedron is the z-coordinate of the vertex (0, 0, 24), which is 24.

Plugging these values into the volume formula:

Volume = (1/3) * 12 * 24

= 96 cubic units

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The eigenvalues of matrix B are λ1 = 111 and λ2 = 190. The corresponding eigenvectors are v1 = [3; 1] and v2 = [7; 3], respectively.

The matrix B = [100 -210; 201] is given, and we need to find the eigenvalues and eigenvectors associated with each eigenvalue.

To find the eigenvalues, we solve the characteristic equation det(B - λI) = 0, where I is the identity matrix and λ is the eigenvalue. Substituting the values from matrix B, we get:

det⎣⎡​100−λ​−210​201​−λ​⎦⎤​ = (100 - λ)(201 - λ) - (-210)(-λ)

= λ^2 - 301λ + 4110

Setting the determinant equal to zero and solving the quadratic equation, we find the eigenvalues λ1 = 111 and λ2 = 190.

To find the eigenvectors, we substitute each eigenvalue back into the equation (B - λI)v = 0, where v is the eigenvector. For λ1 = 111, we have:

⎣⎡​-11​-210​201​⎦⎤​v1 = 0

Solving this system of equations, we obtain v1 = [3; 1]. Similarly, for λ2 = 190, we have:

⎣⎡​-90​-210​201​⎦⎤​v2 = 0

Solving this system of equations, we obtain v2 = [7; 3].

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The values of the function at 9 are 81a + 9 and 81m + c.

We have,

To determine the values of the functions F(x) and g(x) at x = 9, we need to substitute x = 9.

So,

For F(x) = ax² + 9:

F(9) = a(9)² + 9

F(9) = 81a + 9

And,

For g(x) = mx² + c:

g(9) = m(9)² + c

g(9) = 81m + c

Thus,

The values of the function at 9 are 81a + 9 and 81m + c.

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The given equation is a vertical parabola in standard form. To find the focus and directrix, we first need to determine the vertex.

The vertex is (0, -2). The focus is located at a distance of p units vertically above the vertex, where p is the distance from the vertex to the focus. In this case, p = 2. So the focus is at (0, 0). The directrix is located p units vertically below the vertex.

Therefore, the directrix is the horizontal line y = -4. The answer is (b) 2004-06-02-06-00 files/  directrix: 2004-06-02-06-00 files/ .

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The formula for logistic growth can be expressed as P(t) = K / (1 + A * e^(-rt)), where P(t) is the population at time t, K is the carrying capacity, r is the growth rate, A is the initial population.

Logistic growth is a type of population growth that considers a carrying capacity, which is the maximum population size that an environment can sustain. The formula for logistic growth takes into account the carrying capacity (K), the growth rate (r), and the initial population (A) to describe how the population changes over time.

In this case, the carrying capacity is given as 1500, and the growth rate is 0.25 per year. Let's denote the population at time t as P(t).

The formula for logistic growth can be written as:

P(t) = K / (1 + A * e^(-rt))

Plugging in the given values, we have:

P(t) = 1500 / (1 + A * e^(-0.25t))

The value of A is not explicitly given, so it represents the initial population. If the initial population is known, it can be substituted into the formula. If not, A can be left as a variable.

The term e^(-0.25t) represents the exponential decay component, which approaches 0 as t increases. It is multiplied by A, allowing the population to approach the carrying capacity over time.

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a) Confidence Interval = (X₁ - X₂) ± t x √[(s1² / n1) + (s2² / n2)]

b) The sample sizes are large enough (typically considered to be at least 30) or the populations are normally distributed.

c) C.I. = -11854.4100434 and -44145.5899566

a) The formula that should be used to compute the interval for the difference between the average salaries of male and female neurologists is:

Confidence Interval = (X₁ - X₂) ± t x √[(s1² / n1) + (s2² / n2)]

where:

X₁ and X₂ are the sample means of the salaries for female and male neurologists, respectively.

s1 and s2 are the sample standard deviations of the salaries for female and male neurologists, respectively.

n1 and n2 are the sample sizes for female and male neurologists, respectively.

t is the critical value from the t-distribution based on the desired confidence level and the degrees of freedom.

b) The assumptions that need to be met in order to use the above formula are:

The samples are simple random samples from their respective populations.

The populations from which the samples are drawn are approximately normally distributed.

The standard deviations of the populations are unknown.

The sample sizes are large enough (typically considered to be at least 30) or the populations are normally distributed.

c) To compute the interval, we need to calculate the critical value (t) based on the desired confidence level and the degrees of freedom, which is the sum of the sample sizes minus 2 (n1 + n2 - 2).

Given that we want a 99% confidence interval, the corresponding significance level (α) is 0.01. Degrees of freedom = n1 + n2 - 2 = 18 + 21 - 2 = 37.

Using a t-table or a statistical software, the critical value for a 99% confidence level with 37 degrees of freedom is approximately 2.708.

Plugging in the values into the formula:

Confidence Interval = ($175,000 - $203,000) ± 2.708 x √[($15,000² / 18) + ($22,000² / 21)]

= -28000 ± 16145.5899566

= -28000 + 16145.5899566 and -28000 - 16145.5899566

= -11854.4100434 and -44145.5899566

d) Assuming that both populations are normally distributed and that the two population variances are unequal, the formula used to compute the interval is the one described in part (a):

Confidence Interval = (X₁ - X₂) ± t x √[(s1² / n1) + (s2² / n2)]

This formula takes into account the sample means, sample standard deviations, and sample sizes for both groups.

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The interval(s) of θ where tan(θ) > 0 are 0 < θ < π/2 and π < θ < 3π/2.

To determine the interval(s) of θ where tan(θ) > 0, we need to consider the sign of the tangent function in different quadrants of the unit circle.

Recall that the tangent function is positive in the first and third quadrants of the unit circle.

In the first quadrant (0 < θ < π/2), tan(θ) > 0.

In the third quadrant (π < θ < 3π/2), tan(θ) > 0.

Therefore, the correct answer is:

a. 0 < θ < π/2

c. π < θ < 3π/2

So, the interval(s) of θ where tan(θ) > 0 are 0 < θ < π/2 and π < θ < 3π/2.

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The function f(x) = csc^2(x) can be composed with g(x) = x^7 to create f(g(x)) = csc^2(x^7). This composite function involves taking the csc^2 of the seventh power of x.

Let's break down the composition step by step. Starting with the function g(x) = x^7, we substitute this expression into f(x) = csc^2(x). So, we have f(g(x)) = csc^2(g(x)).

Next, we substitute g(x) = x^7 into the expression above to get f(g(x)) = csc^2(x^7). This means that we are taking the csc^2 of the seventh power of x.

The csc function is the reciprocal of the sine function, so csc(x) = 1/sin(x). Therefore, csc^2(x) = 1/sin^2(x). In our case, we have csc^2(x^7) = 1/sin^2(x^7).

To summarize, the composite function f(g(x)) = csc^2(x^7) involves taking the csc^2 of the seventh power of x. This means we are applying the reciprocal of the sine squared to the value of x raised to the power of seven.

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If the number of units expected to be sold is uniformly distributed between 78 and 120, we can use the formula for generating a random number within a given range to express the sales.

Let's denote the random number between 0 and 1 as r. We can calculate the sales using the following expression:

Sales = (120 - 78) * r + 78

In this expression, (120 - 78) represents the range of the uniform distribution (42), and we multiply it by the random number r. Then we add the lower bound of the distribution (78) to obtain the sales value.

By substituting different values of r between 0 and 1, we can generate a random sales value within the range of 78 to 120, following a uniform distribution.

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Answer:

(2,3)

Step-by-step explanation:

y = x/2 + 2 = 0.5x + 2.

also y = x +1.

so 0.5x + 2 = x + 1.

2 -1 = x - 0.5x

1 = 0.5x

x = 2.

y?

y = x + 1 = 2 + 1 = 3.

so (2, 3) is the coordinate solution

Therefore, the equation x^3 - 15x + c = 0 has at most one root in the interval (-2, 2).

To show that the equation x^3 - 15x + c = 0 has at most one root in the interval (-2, 2), we can use the concept of the Intermediate Value Theorem and Rolle's Theorem.

Let's assume that the equation has two distinct roots, denoted as a and b, in the interval (-2, 2). Without loss of generality, we assume a < b.

Since the function is continuous on the closed interval [-2, 2] and differentiable on the open interval (-2, 2), we can apply Rolle's Theorem. According to Rolle's Theorem, there exists a point c in the open interval (a, b) such that the derivative of the function at c is zero.

Consider the derivative of the function f(x) = x^3 - 15x + c:

f'(x) = 3x^2 - 15

Setting f'(c) = 0, we have:

3c^2 - 15 = 0

c^2 - 5 = 0

c^2 = 5

Taking the square root of both sides, we get:

c = ±√5

Now, let's consider the function values at the endpoints of the interval (-2, 2):

f(-2) = (-2)^3 - 15(-2) + c = -8 + 30 + c = 22 + c

f(2) = (2)^3 - 15(2) + c = 8 - 30 + c = -22 + c

If c = √5, then f(-2) = 22 + √5 and f(2) = -22 + √5.

If c = -√5, then f(-2) = 22 - √5 and f(2) = -22 - √5.

In either case, the function values at the endpoints have different signs. This implies that there exists at least one value, say k, in the interval (-2, 2) such that f(k) = 0, according to the Intermediate Value Theorem.

However, we assumed at the beginning that there are two distinct roots in the interval (-2, 2), denoted as a and b. This contradicts our finding that there is at most one root in the interval. Hence, our assumption of having two distinct roots is false.

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According to the Question the area of one of the penny's faces increases by 0.135% when its temperature is increased by 100°C.

When the temperature of a copper penny is increased by 100°C, its diameter increases by 0.17%. However, to determine the change in the area of one of its faces, we need to use the formula for the area of a circle, which is πr². Since the radius of the penny changes with the increase in temperature, we can use the formula for the change in area of a circle, which is 2πrΔr. Using the percentage change in diameter (0.17%), we can find the corresponding percentage change in radius (which is half the diameter) by dividing 0.17 by 2, which gives us 0.085%. We can then use this percentage to calculate the change in the area of one of the penny's faces as follows:

Change in area = 2πrΔr = 2π(0.5r)(0.085% of 0.5r)

= 0.00135πr²

Therefore, the area of one of the penny's faces increases by 0.135% when its temperature is increased by 100°C.

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The sample size when four marbles are selected at random without replacement from the marble bag, is 73,815.

The total number of marbles in the bag is:

10 (orange marbles) + 9 (yellow marbles) + 11 (black marbles) + 8 (red marbles) = 38 marbles.

the number of combinations of 4 marbles chosen from the 38 marbles.

The formula for calculating combinations is given by

C(n, r) = n! / (r! × (n - r)!),

where n is the total number of items and r is the number of items chosen.

Substituting the values into the formula, we have

C(38, 4) = 38! / (4! × (38 - 4)!)

Simplifying the expression

C(38, 4) = 38! / (4! × 34!)

Using factorials:

C(38, 4) = (38 × 37 × 36 × 35) / (4 × 3 × 2 × 1)

Calculating the expression

C(38, 4) = 73,815.

Therefore, the sample size, when four marbles are selected at random without replacement from the marble bag, is 73,815.

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