(a) What measures of variation indicate spread about the mean? (Select all that apply.) variance standard deviation coefficient of variation mean (b) Which graphic display shows the median and data spread about the median? 5-number summary histogram box-and-whisker plot frequency chart

The equation of the plane that is equidistant from points A and B is:

-6x - 4y - 2z = 0

To find the equation of the plane equidistant from points A = (3, 2, 1) and B = (-3, -2, -1), we can follow these steps:

1. Find the midpoint M between points A and B:

  M = ((3 + (-3)) / 2, (2 + (-2)) / 2, (1 + (-1)) / 2)

    = (0, 0, 0)

2. Find the vector v from point A to point B:

  v = B - A = (-3 - 3, -2 - 2, -1 - 1)

      = (-6, -4, -2)

3. Since the plane is equidistant from A and B, any vector that lies on the plane must be orthogonal (perpendicular) to the vector v. Therefore, the normal vector of the plane is the same as the vector v:

  n = v = (-6, -4, -2)

4. Now we have the normal vector n = (-6, -4, -2) and a point on the plane M = (0, 0, 0). We can use the point-normal form of the equation of a plane to obtain the equation:

  Ax + By + Cz = D

  Plugging in the values, we have:

  -6x - 4y - 2z = D

  To find the value of D, we substitute the coordinates of the midpoint M into the equation:

  -6(0) - 4(0) - 2(0) = D

  0 = D

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Rounding the result to the nearest whole number, we find that approximately 24,108 bacteria are expected to be present after 7 hours.

To solve this problem, we can use the exponential growth formula for population growth:

N(t) = N₀ * e^(kt),

where:

N(t) is the population at time t,

N₀ is the initial population,

e is the base of the natural logarithm (approximately 2.71828),

k is the growth rate constant,

t is the time in hours.

We are given the initial population N₀ = 70 and the population after 4 hours N(4) = 360.

Using this information, we can solve for the growth rate constant k:

N(4) = 70 * e^(4k) = 360

Dividing both sides of the equation by 70:

e^(4k) = 360/70

Taking the natural logarithm (ln) of both sides:

4k = ln(360/70)

Simplifying:

k = ln(360/70) / 4

Now that we have the value of k, we can find the population N(7) after 7 hours:

N(7) = 70 * e^(7k)

Substituting the value of k:

N(7) = 70 * e^(7 * ln(360/70) / 4)

Using a calculator, we can evaluate this expression:

N(7) ≈ 70 * e^(7 * ln(360/70) / 4) ≈ 70 * 345.828 ≈ 24107.96

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Answer:

  6%

Step-by-step explanation:

You want the simple interest rate that results in an interest charge of $1800 on a $5000 loan for 6 years.

Your list of financial formulas will tell you that simple interest is computed as ...

  I = Prt . . . . interest on principal P at rate r for t years

Using the given information, we can fill in the values like this:

  1800 = 5000 × r × 6

Dividing by the coefficient of r gives ...

  1800/30000 = r = 6/100 = 6%

The annual interest rate for her loan was 6%.

The series is convergent according to the ratio test.

Convergence of the series ∑[n=1 to ∞] (-3)[tex]^n[/tex] / (n² ) can be determined using the ratio test. First, we need to identify the general term (an) of the series, which is given by an = (-3)[tex]^n[/tex] / (n² ).

Now, let's apply the ratio test to evaluate the limit:

lim[n → ∞] |(a_{n+1}) / (a_n)|

Substituting the values:

lim[n → ∞] |((-3)[tex]^(n+1)[/tex]/ ((n+1)^2)) / ((-3)[tex]^n[/tex] / (n² ))|

Simplifying the expression:

lim[n → ∞] |-3(n² ) / ((n+1)² )|

lim[n → ∞] |-3n²  / (n²  + 2n + 1)|

Since both the numerator and denominator are of the same degree, we can divide every term by n²  to simplify further:

lim[n → ∞] |-3 / (1 + 2/n + 1/n² )|

As n approaches infinity, the terms 2/n and 1/n²  approach 0, so we have:

lim[n → ∞] |-3 / (1 + 0 + 0)|

lim[n → ∞] |-3 / 1|

The absolute value of -3 is simply 3, so the limit is:

lim[n → ∞] |-3 / 1| = 3

Now, we can evaluate the limit:

lim[n → ∞] (an+1 / an) = 3

Since the limit is less than 1 (3 < 1), the series converges by the ratio test.

In summary, the series ∑[n=1 to ∞] (-3)[tex]^n[/tex] / (n² ) is convergent, as determined by the ratio test.

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Answer:

The steps to solve the equation x^2 - 18x + 8 = 0 by completing the square are:

1. Move the constant term to the right side: x^2 - 18x = -8

2. Take half of the coefficient of x, square it, and add it to both sides of the equation:

x^2 - 18x + (-18/2)^2 = -8 + (-18/2)^2

x^2 - 18x + 81 = -8 + 81

3. Simplify the left side and the right side of the equation:

(x - 9)^2 = 73

4. Take the square root of both sides of the equation, remembering to include both the positive and negative square roots:

x - 9 = ± √73

5. Add 9 to both sides of the equation:

x = 9 ± √73

Therefore, the solutions to the equation x^2 - 18x + 8 = 0 are x = 9 + √73 and x = 9 - √73.

Step-by-step explanation:

The distance traveled by a particle with position (x, y) as t varies in a given time interval can be determined by integrating the speed of the particle with respect to time over that interval. This integration involves calculating the derivatives of x and y with respect to t, representing the rates of change in the x and y coordinates over time. By integrating the speed function, we obtain the total distance traveled by the particle within the given time interval.

To compare this distance with the length of the curve, we need to consider the curve traced by the particle's path in the (x, y) plane. The length of the curve represents the total distance along the curve, taking into account its shape and curvature. Comparing the distance traveled by the particle with the length of the curve allows us to assess if the particle deviates from a straight path or follows a more complex trajectory, providing insights into the particle's motion characteristics.

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The distance traveled by a particle with position (x, y) as t varies in a given time interval, we need to integrate the speed of the particle with respect to time over that interval. This will give us the length of the curve traced by the particle's position.

The distance traveled by a particle is determined by integrating its speed over a given time interval. The speed of the particle is the magnitude of its velocity, which can be calculated using the derivatives of the position function.

Let's assume the particle's position is given by a parametric equation x = f(t) and y = g(t), where t represents time. The velocity of the particle can be found by taking the derivatives of f(t) and g(t) with respect to t.

The speed of the particle is then calculated as the magnitude of the velocity vector, which is the square root of the sum of the squares of the derivatives: sqrt((dx/dt)^2 + (dy/dt)^2).

To find the distance traveled by the particle over a specific time interval, we integrate the speed function over that interval with respect to t. This will give us the length of the curve traced by the particle's position.

In summary, to find the distance traveled by a particle, we calculate the speed as the magnitude of its velocity and integrate the speed function over the given time interval. The result will be the length of the curve traced by the particle's position.

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Answer:

To create a square snowflake, use the following steps. (See figure below.) 1. Draw a square. 2. Divide each of the square's straight lines into fourths. In a clockwise order: • Leave the first fourth alone. • Replace the second fourth with a square that's on the outer side of the line. • Replace the third fourth with a square that's on the inner side of the line. . Leave the last fourth alone. Apply the above procedure to each of the square's straight lines. Be sure to do the above in a clockwise order! 3. Apply the procedure from step 2 to each of the straight lines in step 2. 4. Apply the procedure from step 2 to each of the previous step's straight lines. Continue this process indefinitely. step 1 15 Dividing the lines into fourths Replacing the 2nd and 3rd fourths on the top line Replacing the 2nd and 3rd fourths on two lines Replacing the 2nd and 3rd fourths on all lines step 2 (a) Find a formula for the total perimeter P of step n of the process described above if the original square has sides of length 1 ft. Po = ft (b) Use the formula from part (a) to find the perimeter of a square snowflake. (Enter INFINITY for co if needed.) P= ft (c) Find a formula for the total area A of step n of the above process if the original square has sides of length 1 ft. HINT: It's an incredibly easy formula. Don't make it hard. A = sq ft (d) Use the formula from part (c) to find the area of a square snowflake. (Enter INFINITY for co if needed.) A = sq ft

Step-by-step explanation:

To find the nullity of matrix a, we need to determine the number of linearly independent vectors in the null space of matrix a.

Given that the set {u1, u2, u3} is a basis for [tex]R^3\\[/tex], it means that these vectors span the entire space and are linearly independent.

Matrix a is formed by arranging these basis vectors as columns:

a = [u1 u2 u3]

Since the vectors in the basis {u1, u2, u3} are linearly independent, the null space of matrix a will contain only the zero vector. In other words, the only vector that satisfies the equation a * x = 0 is the zero vector, where x is a column vector.

Therefore, the nullity of matrix a is zero, as there are no linearly independent vectors in the null space.

In summary, nullity(a) = 0.

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The area of the domain enclosed by the curve with parametric equations x = tsin t, y = cost, t ∈ [0, 2π] is 2π + 2.

The parametric equations given are:

x = t sin t
y = cos t

To find the area of the domain enclosed by the curve, we can use the formula for the area of a region bounded by a curve given in parametric form:

A = ∫[a,b] y dx

where a and b are the limits of the parameter t that describe the domain of the curve.

In this case, we have:

a = 0
b = 2π

So, we need to compute:

A = ∫[0,2π] cos(t) (t sin(t)) dt

Using integration by parts with u = t and dv = sin(t) dt, we get:

A = [t cos(t)]|[0,2π] - ∫[0,2π] cos(t) dt + ∫[0,2π] sin(t) dt

The first integral evaluates to:

[t cos(t)]|[0,2π] = 2π

The second integral evaluates to:

∫[0,2π] cos(t) dt = [sin(t)]|[0,2π] = 0

The third integral evaluates to:

∫[0,2π] sin(t) dt = [-cos(t)]|[0,2π] = 2

Therefore, the area of the domain enclosed by the curve is:

A = 2π - 0 + 2 = 2π + 2

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The next number in the given series 6 18 20 10 30 32 16 should be. 48

A series is produced by sequence, which is also known as progression. One of the fundamental ideas in mathematics is sequence and series.

A series is the total of the elements in a sequence, whereas sequences are groups of numbers arranged in an ordered and specific manner.

As an illustration, the series that corresponds to the four-element sequence 2, 4, 6, and 8 is 2 + 4 + 6 + 8;

the total of the series, or its value, is 20.

The next series follows the order as

6 × 3 = 18

18 + 2 = 20

20 / 2 = 10

10 × 3 = 30

30 + 2 = 32

32 / 2 = 16

The following figure will be 16 x 3 = 48.

Hence, the required answer number is 48.

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The monetary costs for Job A are the cost to commute and the affording the formal work attire. For Job B, they are the low entry - level pay.

The non - monetary costs for Job A include the long commute and limited opportunities for advancement.  For Job B it is the repetitive work.

The opportunity cost that will be encountered with Job A is, other than commuting, individuals could utilize the time for various activities including spending quality time with loved ones, engaging in hobbies, or unwinding.

Also, working additional hours beyond regular schedules could result in burnout and reduced efficiency.

For Job B, the opportunity cost is instead of filling out and submitting paperwork, valuable time could be utilized for networking or acquiring new skills.

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For the function, f(x)=x, the most appropriate way to describe function is

(b) f is a bijection, because it is one-to-one and onto.

The function f(x) = x, is well-defined,

We let x ∈ R (real-numbers), then f(x) = x ,∈ R;

One-to-One : We have f(x) = f(y) ⇒ x = y,

So, the function "f" is one-to-one,

Onto : For all y∈ R, there exist x ∈ R, such that f(x) = y, which is  x = y,

Hence, the function "f" is one-to-one and onto, so, it can be called as an bijective function.

Therefore, the correct option is (b).

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The given question is incomplete, the complete question is

The domain and target of the following function is the set of real numbers. f(x)=x,

Which is the most appropriate way to describe this function?

(a) f is one-to-one but not onto,

(b) f is a bijection,

(c) f is onto but not one-to-one,

(d) f is not well defined.

If a curve is defined by parametric equations with x-values in the range [2, 5] and y-values in the range [2, 5], then nothing can be said about the curve. The answer is D.

Given the parametric equations x = f(t) and y = g(t), where the range of f is [2, 5] and the range of g is [2, 5], we can't make any definitive statements about the shape or position of the curve without further information about the functions f(t) and g(t).

The ranges of f and g only provide constraints on the possible values of x and y, but they don't determine the specific behavior of the curve.

Without knowing the explicit forms of f(t) and g(t) or any additional information about the curve, we cannot determine its shape, whether it lies inside or outside certain regions, or whether it corresponds to a specific geometric figure like a circle or a line.

Hence, the correct option is D. Nothing can be said about the curve.

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The total area of the composite figure is 198.54 square cm

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

The composite figure

The total area of the composite figure is the sum of the individual shapes

So, we have

Area = circle + rectangle

This gives

Area = π * (10/2)² + 10 * (22 - 10)

Evaluate

Area = 198.54

Hence, the total area of the figure is 198.54 square cm

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The population declines by approximately 2.96% each decade.

To determine the percentage decline of a population each decade, we need to consider the compounding effect of the annual decline over a 10-year period.

Given that the population declines by 0.3% each year, we can calculate the overall decline over a decade as follows:

(1 - 0.003)^10

Using a calculator, we find that (1 - 0.003)^10 ≈ 0.970439.

To express this as a percentage decline, we subtract this value from 1 and multiply by 100:

(1 - 0.970439) * 100 ≈ 2.96.

Therefore, the population declines by approximately 2.96% each decade.

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To maintain equilibrium in the position shown, we need to determine the magnitude of the vertical force P at point C and the reactions at points A and B.

(a) To find the magnitude of the vertical force P at point C, we can use the principle of equilibrium. The sum of the vertical forces must be equal to zero. Since the weight of the load is 120 lbs acting downwards, the force P at point C should be equal to the weight of the load to maintain equilibrium. Therefore, the magnitude of force P is 120 lbs.

(b) Assuming that the bearing at point B does not exert any axial thrust, we can analyze the reactions at points A and B. Since the system is in equilibrium, the sum of the forces in the horizontal and vertical directions at points A and B should be equal to zero.

In the horizontal direction, the reaction at point B will balance the horizontal component of force P. However, without additional information or constraints, we cannot determine the specific value of the reaction at point B.

In the vertical direction, the reaction at point A will balance the weight of the load plus the vertical component of force P. Therefore, the magnitude of the reaction at point A is the sum of the weight of the load and the vertical component of force P, which is 120 lbs plus the vertical component of force P.

In summary, the magnitude of the vertical force P at point C should be 120 lbs, and the reactions at points A and B depend on additional information or constraints that are not provided in the given problem statement

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To maintain equilibrium in the given position with a small winch raising a 120-lb load, the magnitude of the vertical force P at point C needs to be determined. Additionally, the reactions at points A and B can be calculated, assuming the bearing at B does not exert any axial thrust.

(a) To maintain equilibrium, the vertical force P at point C should be equal to the weight of the load, which is 120 lbs. This is because for an object to be in equilibrium, the sum of the forces acting on it must be zero. In this case, the force P must balance out the weight of the load, ensuring there is no net force in the vertical direction.

(b) Assuming the bearing at B does not exert any axial thrust, the reactions at points A and B can be determined. The reaction at point A, denoted as RA, will be equal in magnitude and opposite in direction to the vertical force P. This is because the weight of the load is transmitted through the cable to point A. As for point B, the vertical reaction, denoted as RB, will be equal to the weight of the load (120 lbs) since there is no axial thrust from the bearing at B.

In summary, to maintain equilibrium, the magnitude of the vertical force P at point C should be 120 lbs. The reactions at points A and B are equal to 120 lbs and 120 lbs, respectively, assuming no axial thrust from the bearing at B.

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A solid S has a circular base with radius r, and its cross-sections perpendicular to the base are squares. The task is to determine the volume of S by finding the area of each cross-section and integrating those areas.

To determine the volume of S, we can start by finding the area of each cross-section. Since the cross-sections are squares, the area can be found using the formula A = s^2, where s is the side length of the square. The side length of each square can be found by considering the radius of the circular base and the fact that the diagonal of each square is equal to the diameter of the circular base. Using the Pythagorean theorem, we can find that s = sqrt(2)r. The area of each cross-section is therefore A = (sqrt(2)r)^2 = 2r^2.

To find the volume of S, we need to integrate the areas of the cross-sections. Since the cross-sections are perpendicular to the base, the integral can be set up as ∫2r^2 dx, where x represents the distance from the base. The limits of integration are 0 to the height of S, which is not given in the problem. However, we can still find the general formula for the volume of S by integrating the expression for the area of each cross-section. This gives us V = ∫2r^2 dx = 2r^2x + C, where C is the constant of integration. The volume of S can be found by evaluating this expression at the limits of integration.

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Answer:

23.6 m

Step-by-step explanation:

In a right-angled triangle, a ² + b ² = c ²

b² = 10² - 8.8²

100 - 77.44

b = 4.75

perimeter = 4.75 + 8.8 + 10 = 23.55 = 23.6 m to nearest tenth

Answer:

The answer is 23.5m to 1 d.p

Step-by-step explanation:

c²=a²+b²

b²=c²-a²

b²=10²-8.8²

b²=100-77.44

b²=22.56

√b²=√22.56

b=4.7m

Perimeter =a+b+c

P=8.8+4.7+10

P=23.5

P=23.5m to 1 d.p

To find E(X1X10X22), the expected value of the product of the first, tenth, and twenty-second picks from an urn containing 30 chips numbered from 1 to 30, we can use the concept of linearity of expectation.

The expected value of the product is the product of the expected values of the individual picks.

The expected value of a single pick can be computed by taking the sum of all possible values multiplied by their respective probabilities. In this case, the expected value of a single pick is (1+2+3+...+30)/30 = 15.5.

Using the linearity of expectation, the expected value of the product X1X10X22 is E(X1) * E(X10) * E(X22) = 15.5 * 15.5 * 15.5 = 3759.875. Therefore, the expected value of the product of the first, tenth, and twenty-second picks is approximately 3759.875.

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These are the computed values of beta and p-value for the given alternative value p = 0.21 and sample sizes.

To compute the beta (β) for the given alternative value p = 0.21 and sample sizes n = 81, 900, 14,400, 40,000, and 90,000, we need to determine the probability of Type II error. Beta is the probability of failing to reject the null hypothesis (H₀) when the alternative hypothesis (Hₐ) is true.

To calculate beta, we need to find the corresponding z-value for the given significance level α = 0.01, which corresponds to a z-value of 2.33 (approximate value).

Using the formula for the standard deviation of the sample proportion:

σ = sqrt((p₀ * (1 - p₀)) / n)

where p₀ is the null hypothesis value, p = 0.2, and n is the sample size.

Then we can calculate the z-score for the alternative value p = 0.21 using the formula:

z = (p - p₀) / σ

Finally, we can compute beta using the standard normal distribution table or a statistical calculator.

Let's calculate beta for the given sample sizes:

For n = 81:

σ = sqrt((0.2 * (1 - 0.2)) / 81) ≈ 0.0447

z = (0.21 - 0.2) / 0.0447 ≈ 0.2494

beta = P(Z > 0.2494) ≈ 0.4013

For n = 900:

σ = sqrt((0.2 * (1 - 0.2)) / 900) ≈ 0.0143

z = (0.21 - 0.2) / 0.0143 ≈ 0.7692

beta = P(Z > 0.7692) ≈ 0.2206

For n = 14,400:

σ = sqrt((0.2 * (1 - 0.2)) / 14,400) ≈ 0.0071

z = (0.21 - 0.2) / 0.0071 ≈ 1.5493

beta = P(Z > 1.5493) ≈ 0.0606

For n = 40,000:

σ = sqrt((0.2 * (1 - 0.2)) / 40,000) ≈ 0.0050

z = (0.21 - 0.2) / 0.0050 ≈ 2.2000

beta = P(Z > 2.2000) ≈ 0.0139

For n = 90,000:

σ = sqrt((0.2 * (1 - 0.2)) / 90,000) ≈ 0.0033

z = (0.21 - 0.2) / 0.0033 ≈ 3.0303

beta = P(Z > 3.0303) ≈ 0.0012

Next, let's calculate the p-value for the alternative value P = 0.21 and sample sizes n = 81, 900, 14,400, and 40,000.

The z-score for the given P can be calculated using the formula:

z = (P - p₀) / σ

Using the standard normal distribution table or a statistical calculator, we can find the area under the curve beyond the calculated z-score to obtain the p-value.

For n = 81:

σ = sqrt((0.2 * (1 - 0.2)) / 81) ≈ 0.0447

z = (0.21 - 0.2) / 0.0447 ≈ 0.2494

p-value = P(Z > 0.2494) ≈ 0.4013

For n = 900:

σ = sqrt((0.2 * (1 - 0.2)) / 900) ≈ 0.0143

z = (0.21 - 0.2) / 0.0143 ≈ 0.7692

p-value = P(Z > 0.7692) ≈ 0.2206

For n = 14,400:

σ = sqrt((0.2 * (1 - 0.2)) / 14,400) ≈ 0.0071

z = (0.21 - 0.2) / 0.0071 ≈ 1.5493

p-value = P(Z > 1.5493) ≈ 0.0606

For n = 40,000:

σ = sqrt((0.2 * (1 - 0.2)) / 40,000) ≈ 0.0050

z = (0.21 - 0.2) / 0.0050 ≈ 2.2000

p-value = P(Z > 2.2000) ≈ 0.0139

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The correct answer is B. t-distribution.

In the testing of hypothesis about the population mean when the population standard deviation is unknown, the critical values are determined using the t-distribution.

When the population standard deviation is unknown, we use the t-distribution to account for the uncertainty in estimating the population standard deviation from the sample data.

The t-distribution is similar to the standard normal (z) distribution but has thicker tails, which allows for more variability in the data.

The critical values, also known as the cutoff values, are the boundary values that determine the rejection region for the hypothesis test. These values are obtained from the t-distribution table or using statistical software.

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The value of tan θ for the given angle is option (b): tan θ = negative four times square root of thirty-three divided by thirty-three.

To determine the value of tan θ, we need to evaluate the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

By examining the options, we can see that option (b) matches this criteria: tan θ = negative four times square root of thirty-three divided by thirty-three.

The negative sign indicates that the angle is in the third quadrant or the second quadrant, where the tangent function is negative.

The square root of thirty-three represents the length of the side opposite the angle, and the denominator, thirty-three, represents the length of the side adjacent to the angle.

Therefore, the correct answer is option (b): tan θ = negative four times square root of thirty-three divided by thirty-three.

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The probability that Lice has at least two books in her backpack that cover the same subject is 56/330, which can be simplified to 28/165.

To calculate the probability that Lice has at least two books in her backpack that cover the same subject, we can consider the different cases where she has duplicates.

Case 1: Two books of the same subject and one book of a different subject.

The probability of choosing two books of the same subject is given by:

(4/12) * (3/11) * (8/10) * ³C₂ (choose 2 subjects out of 3)

Case 2: Three books of the same subject.

The probability of choosing three books of the same subject is given by:

(4/12) * (3/11) * (2/10) * ³C₁ (choose 1 subject out of 3)

To find the total probability, we sum up the probabilities of these two cases:

P(at least two books of the same subject) = P(case 1) + P(case 2)

P(at least two books of the same subject) = [(4/12) * (3/11) * (8/10) * ³C₂] + [(4/12) * (3/11) * (2/10) * ³C₁]

Simplifying the equation, we have:

P(at least two books of the same subject) = [32/330] + [24/330]

P(at least two books of the same subject) = 56/330=28/165.

Therefore, the probability that Lice has at least two books in her backpack that cover the same subject is 56/330, which can be simplified to 28/165.

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The Giapetto problem is a linear programming problem that involves maximizing profit from producing two types of wooden toys. In response to the questions:

a. The dual of the Giapetto problem can be obtained by interchanging the roles of the variables and constraints. The objective of the dual problem is to minimize the sum of the dual variables (representing the costs) subject to the constraints defined by the coefficients of the original primal problem.

b. To determine the optimal dual solution, we can examine the optimal tableau of the Giapetto problem. The dual solution is obtained by considering the dual variables associated with the constraints. These variables represent the shadow prices or the marginal values of the resources in the primal problem. By analyzing the optimal tableau, we can identify the values of the dual variables and determine the optimal dual solution.

c. In this instance, we can verify that the Dual Theorem holds. The Dual Theorem states that the optimal value of the dual problem is equal to the optimal value of the primal problem. By comparing the optimal solutions obtained in parts (a) and (b), we can confirm whether they are equal. If the optimal values match, it confirms the validity of the Dual Theorem, indicating a duality relationship between the primal and dual problems. The dual of the Giapetto problem involves minimizing costs instead of maximizing profit. By examining the optimal tableau, we can determine the optimal dual solution. Lastly, by comparing the optimal solutions of the primal and dual problems, we can verify the Dual Theorem's validity, which states that the optimal values of both problems are equal, demonstrating their duality relationship.

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The value of  x of chord = 5

By definition of circle,

The chord of a circle is defined as the line segment connecting any two locations on the circle's perimeter; nevertheless, the diameter is the longest chord of a circle that goes through the centre of the circle.

The chord is one of the several line segments that may be made in a circle, and its endpoints are on the circumference.

⇒ 6 (6 + x) = 7 (7 + 11)

Solve for x;

⇒ 36 + 6x = 7 × 18

⇒ 36 + 6x = 126

⇒ 6x = 126 - 36

⇒ 6x = 90

⇒ x = 15

Thus, The value of x = 5

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Answer:

Density = Mass / Volume

3 g/cm^3 = 25 g / Volume

Volume = 25 g / 3 g/cm^3 = 8.33 cm^3

Round to two decimal places: 8.33 cm^3

Therefore, the volume of the block of wood is 8.33 cm^3.

Step-by-step explanation:

Composite area:

   Consists of a rectangle,    area  12 x 10 = 120 cm^2

        and two half-circles ( making a whole circle)    Area = pi r^2

                          = pi (5^2) = 78.5 cm^2

Total area = 198.5 cm^2

The option that is NOT equivalent to 52% is Option B: 5.2.

To determine which option is not equivalent to 52%, we need to compare the given options with the value of 52%. Let's examine each option and determine if it is equivalent to 52%.

Option A: 0.52

Option B: 5.2

Option C: 0.052

Option D: 520

To find the equivalent decimal value of a percentage, we divide the percentage by 100.

Let's calculate the equivalent decimal value for 52%:

52% = 52/100 = 0.52

Now, let's compare this with the given options:

Option A: 0.52

This option is equivalent to 52%, as we calculated earlier.

Option B: 5.2

This option is not equivalent to 52%. It represents 520%, which is ten times greater than 52%.

Option C: 0.052

This option is not equivalent to 52%. It represents 5.2%, which is one-tenth of 52%.

Option D: 520

This option is not equivalent to 52%. It represents the whole number 520, which is a hundred times greater than 52%.

Therefore, the option that is NOT equivalent to 52% is Option B: 5.2.

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Answer: 175cm^2

Step-by-step explanation:

area of parallelogram=base*height

a=14*5=60

area of trapezium=1/2*sum of parallel side*height

a(trapzium)=105

total area=a+a(trapzium)=60+105=175