which cards are equevent to 1/2 + 6/9? choose all the correct answers

The weights of the liters of water are 49.75 kg and 4.975 kg

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

1l of water weighs almost 0,995 kg.

This means that

Weight = 0,995 kg

For 50 l, we have

Weight = 50 * Weight  of 1 liter

Substitute the known values in the above equation, so, we have the following representation

Weight = 0,995 * 50 kg

Evaluate

Weight = 49.75 kg

For 0.5 l, we have

Weight = 0.5 * Weight  of 1 liter

So, we have

Weight = 0.995 * 0.5 kg

Evaluate

Weight = 4.975 kg

Hence, the weights are 49.75 kg and 4.975 kg

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Numerous-Choice Exams 14 true-false and two multiple-choice questions make up Professor Easy's final exam. There are 65,536 possible answer sheets.

For each true-false question, there are two possible answers (true or false), so there are [tex]2^{14[/tex] possible answer sheets for the true-false questions.

For each multiple-choice question, there are 4 possible answers. Since there are 2 multiple-choice questions, there are [tex]4^2 = 16[/tex] possible answer sheets for the multiple-choice questions.

To find the total number of possible answer sheets, we need to multiply the number of possible answer sheets for the true-false questions by the number of possible answer sheets for the multiple-choice questions. Therefore, the total number of possible answer sheets is:

[tex]2^{14} \times 16 = 65,536[/tex]

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When a sample of 250 observations is selected at random from an infinite population with a population proportion of 0.25, we can calculate the standard error of the sampling distribution of the sample proportion using a specific formula. The standard error (SE) measures the variability or dispersion of the sample proportion from the true population proportion.

To compute the standard error of the sample proportion, use the following formula:

SE = sqrt[(P * (1 - P)) / n]

where P is the population proportion (0.25 in this case) and n is the sample size (250).

SE = sqrt[(0.25 * (1 - 0.25)) / 250]

SE = sqrt[(0.25 * 0.75) / 250]

SE = sqrt[0.1875 / 250]

SE ≈ 0.027

Thus, the standard error of the sampling distribution of the sample proportion, given a population proportion of 0.25 and a sample size of 250, is approximately 0.027. This value represents the expected variation in the sample proportions from the true population proportion, providing an estimate of the precision of the sample proportion as a representation of the population proportion.

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The probability of each composition is: BBBB = 0.0625, BBGG = 0.375, BGBG = 0.375, BGGB = 0.375, GBBG = 0.375, GBGB = 0.375, GGGB = 0.25, GGGG = 0.0625.

The given composition is not specified, so we will have to calculate the probabilities for all possible compositions of four children: BBBB, BBGG, BGBG, BGGB, GBBG, GBGB, GGGB, GGGG.

The probability of having a boy or a girl for each child is 0.5.

For BBBB, the probability is [tex](0.5)^4[/tex] = 0.0625.

For BBGG, there are 6 possible arrangements (BBGG, BGBG, BGGB, GBBG, GBGB, GGGB), each with a probability of [tex](0.5)^4[/tex] = 0.0625. So, the probability of BBGG is 6 x 0.0625 = 0.375.

For BGBG and BGGB, the probability is the same as BBGG, so each has a probability of 0.375.

For GBBG and GBGB, the probability is also 0.375.

For GGGB, there are 4 possible arrangements (GGGB, GGBG, GBGG, BGGG), each with a probability of [tex](0.5)^4[/tex] = 0.0625. So, the probability of GGGB is 4 x 0.0625 = 0.25.

For GGGG, the probability is [tex](0.5)^4[/tex] = 0.0625.

Therefore, the probability of each composition is: BBBB = 0.0625, BBGG = 0.375, BGBG = 0.375, BGGB = 0.375, GBBG = 0.375, GBGB = 0.375, GGGB = 0.25, GGGG = 0.0625.
Note that these probabilities add up to 1, as they should.

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The integral for the mass of the shell becomes  ∫₀^π ∫₀^{2π} ∫₃^₅ ρ(r, θ, φ) r^2 sin φ dr dθ dφ.

To find the mass of the spherical shell, we need to integrate the density over its volume. Let's assume that the density of the shell is constant, denoted by rho.

Using spherical coordinates, the integral for the mass of the shell can be written as:

M = ∫∫∫ ρ(r, θ, φ) r^2 sin φ dr dθ dφ

where,

ρ(r, θ, φ) is the density of the shell, which is assumed to be constant,

r is the radial distance from the origin,

θ is the azimuthal angle, which measures the angle in the xy-plane from the positive x-axis,

φ is the polar angle, which measures the angle from the positive z-axis.

Since the shell is centered at the origin and has an inner radius of 3 cm and an outer radius of 5 cm, the limits of integration are:

3 ≤ r ≤ 5

0 ≤ θ ≤ 2π

0 ≤ φ ≤ π

Thus, the integral for the mass of the shell becomes:

M = ∫∫∫ ρ(r, θ, φ) r^2 sin φ dr dθ dφ

= ∫₀^π ∫₀^{2π} ∫₃^₅ ρ(r, θ, φ) r^2 sin φ dr dθ dφ

the symmetry of the shell, which means that we are integrating over the entire volume of the shell. If the shell had some symmetry, we could have reduced the domain of the integral by exploiting that symmetry.

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The null hypothesis refers to the population, whereas the research hypothesis refers to the sample.

We have,

In statistics, the null hypothesis is a statement that assumes there is no significant difference between two groups or variables being compared.

It represents the default position that there is no relationship or effect between the variables of interest.

The null hypothesis is typically formulated as a statement about the population parameter.

On the other hand, the research hypothesis (also known as the alternative hypothesis) is a statement that proposes a significant difference or relationship between the variables being studied.

The research hypothesis is typically formulated as a statement about the sample, which is a subset of the population.

Thus,

The null hypothesis refers to the population, whereas the research hypothesis refers to the sample.

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To test the hypothesis that the average time to deliver pizza, once the order is placed, is greater than 25 minutes in the population, the appropriate statistical test to use would be a t-test. This test is used to compare the means of two groups, in this case, the actual average time it takes to deliver the pizza and the hypothesized value of 25 minutes.

The t-test is preferred over a z-test because the population standard deviation is unknown, which is a requirement for a z-test. The t-test, on the other hand, uses the sample standard deviation to estimate the population standard deviation.

To conduct a t-test, we need to collect a random sample of delivery times and calculate the sample mean and standard deviation. Then we would use a one-sample t-test to compare the sample mean to the hypothesized value of 25 minutes. If the calculated t-value is greater than the critical value at a chosen level of significance, we can reject the null hypothesis and conclude that the average time to deliver pizza is indeed greater than 25 minutes in the population.

In conclusion, to test the hypothesis that the average time to deliver pizza, once the order is placed, is greater than 25 minutes in the population, we would use a t-test.

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The endpoints of the other diagonal are ((1/2) * (13 - √146), 11.5) and ((1/2) * (√146 + 13), 11.5).

Let's first find the length and midpoint of the diagonal with endpoints (4, 6) and (9, 17):

Length of diagonal = √[(9 - 4)² + (17 - 6)²] = √(5² + 11²) = √146

Midpoint of diagonal = [((4 + 9) / 2), ((6 + 17) / 2)] = (6.5, 11.5)

Now, we know that the other diagonal is parallel to the x-axis. Let's call the endpoints of this diagonal (a, b) and (c, b), where b is the y-coordinate of the midpoint of the first diagonal, which we just found as 11.5.

Since the rectangle is a right-angled shape, we know that the length of the diagonal with endpoints (a, b) and (c, b) is equal to the length of the diagonal with endpoints (4, 6) and (9, 17):

√[(c - a)² + (b - b)²] = √146

Simplifying this equation, we get:

√[(c - a)²] = √146

Taking the square of both sides, we get:

(c - a)² = 146

We also know that the midpoint of this diagonal is (6.5, 11.5). So we can write:

(a + c) / 2 = 6.5

Solving these two equations simultaneously, we get:

c - a = √146 ... (1)

a + c = 13 ... (2)

Adding equations (1) and (2), we get:

2c = √146 + 13

c = (1/2) * (√146 + 13)

Substituting this value of c in equation (2), we get:

a = 13 - c

a = (1/2) * (13 - √146)

Therefore, the endpoints of the other diagonal are ((1/2) * (13 - √146), 11.5) and ((1/2) * (√146 + 13), 11.5).

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The coordinates of the ends of one diagonal of a rectangle are (4,6) and (9,17) . If its other diagonal is parallel to the x-axis, find its ends coordinates.

In order to compute eigenvalues of symmetric matrices, Householder reflections are indeed used to transform the matrix into a tridiagonal form.

This is an important step because it simplifies the computation and takes advantage of the inherent properties of symmetric matrices.

However, Householder reflections alone cannot fully diagonalize the matrix. After obtaining the tridiagonal matrix using Householder reflections, other numerical methods, such as the QR algorithm or the Lanczos method, are required to perform the diagonalization. These methods iteratively transform the tridiagonal matrix into an even more simplified form, ultimately converging to a diagonal matrix.

The reason Householder reflections cannot fully diagonalize the matrix is because they are primarily designed to introduce zeros below the main diagonal, without altering the eigenvalues. The method aims to maintain orthogonality during the process, preserving the properties of symmetric matrices.

To complete the diagonalization, further iterative methods are necessary to isolate the eigenvalues along the main diagonal of the matrix.

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At a speed of 85 miles per hour, the bus takes 7 hours to cover the distance of 595 miles.

Given that, the speed of the bus is 85 miles per hour.

The total distance covered by the bus is 595 miles.

Now, we need to find the time taken by the bus to cover a distance of 595 miles at a speed of 85 miles per hour.

The formula to find the time when distance and speed are known is,

Time = Distance / Speed.

Now, we can substitute the values in the above formula.

Time = 595 / 85.

595 / 85 = 7.

Time = 7 hours.

Therefore, the bus covers a distance of 595 miles at a speed of 85 miles per hour in 7 hours.

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The width of the flower bed is approximately 1.4m.

To solve this problem, we need to use the formula for the area of a rectangle: A = L x W, where A is the area, L is the length, and W is the width.

Let's start by finding the area of the pond:
A pond = 10m x 7m = 70m^2

Next, we need to find the total area of the pond and the flower bed combined. We are given that this area is 180m^2:
A total = A pond + A flower bed
[tex]180m^2 = 70m^2 + A flower bed[/tex]
[tex]110m^2 = A flower bed[/tex]

Now we can use the formula for the area of a rectangle again to find the width of the flower bed:
A flower bed = L x W
[tex]110m^2 = (10m + 2x) (7m + 2x)[/tex]
[tex]110m^2 = 70m^2 + 20xm + 14xm + 4x^2[/tex]

Simplifying and rearranging, we get:
[tex]4x^2 + 34xm + 40m^2 - 110m^2 = 0[/tex]
[tex]4x^2 + 34xm - 70m^2 = 0[/tex]

Dividing both sides by 2, we get:
[tex]2x^2 + 17xm - 35m^2 = 0[/tex]

Now we can use the quadratic formula to solve for x:
[tex]x= \frac{-b±\sqrt{(x^{2})-4ac } }{2a}[/tex]
Where a = 2, b = 17m, and [tex]c = -35m^2[/tex].

Plugging these values in, we get:
[tex]x =\frac{ (-17m ± \sqrt{17(m)^{2}+280(m)^{2}  } }{4}[/tex]
[tex]x= \frac{(-17m ±\sqrt{697} )}{4}[/tex]

Since the width of the flower bed can't be negative, we take only the positive root:
[tex]x= \frac{(-17m +\sqrt{697} )}{4}[/tex]
x = 1.4m

Therefore, the width of the flower bed is approximately 1.4m.

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The Discussion section of a report is an essential part where you can delve into the theories related to your research topic, irrespective of the results.

In this section, you have the opportunity to analyze and interpret your findings and draw conclusions from them.

You can also discuss the implications of your research and suggest future directions for further investigation.

Additionally, it is crucial to discuss relevant literature in this section to provide context for your findings and demonstrate your knowledge of the subject area.

However, it is essential to avoid merely restating points already made in previous sections but instead reformulate them and provide additional insights.

In summary, the Discussion section is a crucial part of the report, providing a space for reflection and critical thinking about your research findings.

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The probability of finding a sample mean (Y¯) of 1.5 ounces or more when the liquor tax has no impact on consumption (µ = 0). To determine this probability, we would use the Central Limit Theorem and the z-score formula.

Since the population mean (µ) is 0, we'll need to know the population standard deviation (σ) and the sample size (n) to proceed. Without these values, it's impossible to provide an exact probability. However, I can explain the general process.

First, you would calculate the standard error (SE) using the formula SE = σ / √n. Next, you would find the z-score, which is the difference between the sample mean (Y¯) and the population mean (µ) divided by the standard error: z = (Y¯ - µ) / SE.

Once you have the z-score, you can look it up in a standard normal distribution table or use a calculator to find the probability associated with it. In this case, you're looking for the probability of finding a sample mean of 1.5 ounces or more, which corresponds to the area to the right of the z-score in the distribution.

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To calculate the number of MADs that the means are apart, we need to find the difference between the two means and then divide that by the MAD.

The difference between the means is 20 - 16 = 4.

Dividing that by the MAD of 2.5, we get:

4 / 2.5 = 1.6

Therefore, the means are 1.6 MADs apart.

1. Find the difference between the two mean ages: 20 years (mean age of second sample) - 16 years (mean age of first sample) = 4 years.

2. Divide the difference by the MAD: 4 years (difference between means) / 2.5 (MAD) = 1.6.

So, the means are 1.6 MADs apart.

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Thus, the possible values for d are 10, 11, and 12. The sum of all possible values of d is 10 + 11 + 12 = 33.

The given integer has 4 digits when written in base 8, meaning its general form can be represented as: N = a * 8^3 + b * 8^2 + c * 8^1 + d * 8^0, where 0 ≤ a, b, c, d ≤ 7.

Since N has 4 digits, a must be nonzero, so 1 ≤ a ≤ 7.

Now, we need to find the number of digits (d) when this integer is written in base 2. To do so, we first express the integer in terms of powers of 2.

N = a * (2^3)^3 + b * (2^3)^2 + c * (2^3)^1 + d * (2^3)^0
N = a * 2^9 + b * 2^6 + c * 2^3 + d

Since 1 ≤ a ≤ 7, the minimum value for a is 1 and the maximum value is 7. Therefore, the smallest possible value for N in base 2 is 1 * 2^9 (which has 10 digits in base 2) and the largest possible value is 7 * 2^9 + 7 * 2^6 + 7 * 2^3 + 7 (which has 12 digits in base 2).

Thus, the possible values for d are 10, 11, and 12. The sum of all possible values of d is 10 + 11 + 12 = 33.

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2/3  is the probability that the spinner will land on a number greater than 4 or on a shaded section.

The probability of a given occurrence is a number used in science to describe how likely it is that the event will take place. In terms of percentage notation, it is represented as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the greater the probability it will be that the occurrence will take place. A certain occurrence has a chance of 1, while an impossible event has a probability of 0.

P(Greater than 4) = 2/6

P(Shaded) = 3/6

P(Shaded section and greater than 4) = 1/6

P = 2/6 + 3/6 - 1/6

P = 4/6

P = 2/3

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When comparing the means between two independent samples, the alternative hypothesis should be stated as: "There is a significant difference between the means of the two independent samples."

In this context, the alternative hypothesis (often denoted as H1 or Ha) proposes that there is a meaningful or notable difference between the means of the two groups being compared, suggesting that the observed difference is not due to chance or sampling error.

The alternative hypothesis is tested against the null hypothesis (H 0), which states that there is no significant difference between the means of the two independent samples. In other words, the null hypothesis suggests that any observed difference is simply due to random chance or sampling variability.

When conducting a hypothesis test, researchers use statistical tests, such as the t-test or ANOVA, to determine the likelihood of the observed difference between the means occurring by chance alone. If the probability of obtaining the observed difference under the null hypothesis is sufficiently low (typically less than 0.05), the null hypothesis is rejected in favor of the alternative hypothesis, indicating that there is a significant difference between the means of the two independent samples.

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The probability of getting at least an Ace, King, Queen, Jack, and 10 in a 13-card hand from a standard 52-card deck is approximately 0.740 or 74.0%.

To calculate the probability of getting at least an Ace, King, Queen, Jack, and 10 in a 13-card hand from a standard 52-card deck, we can use the principle of inclusion-exclusion.

There are C(52,13) ways to choose a 13-card hand from the 52 cards in the deck.

The number of ways to choose a hand that does not contain any of the desired cards is:

C(48,13)

Therefore, the number of ways to choose a hand that contains at least one of the desired cards is:

C(52,13) - C(48,13)

The probability of getting at least one of the desired cards can be calculated by dividing this number by the total number of possible hands:

P(at least one of the desired cards) = [tex]$\frac{{52\choose 13}-{48\choose 13}}{{52\choose 13}}$[/tex]

[tex]$1 - \frac{{48\choose 13}}{{52\choose 13}}$[/tex]

= 1 - 0.260

= 0.740

The probability of getting at least one of the desired cards is quite high, as it is more likely than not that a 13-card hand will contain at least one of these five cards.

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Answer:

We can factor out a 4x from the numerator and a 2 from the denominator, which gives:

(4x(x+1)) / 2( x^2 - 1)

We can then factor the denominator further using the difference of squares formula, which gives:

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

Simplifying this expression further, we can cancel out the (x+1) terms in the numerator and denominator, which gives:

2x / (x-1)

Therefore, 4x^2 + 4x / 2x^2 - 2 simplifies to 2x / (x-1).

When you roll a six sided die two times. You know the sum of the two rolls is 4. The probability that you rolled two 2s in a row (2, 2) is 1/3.

To find the probability of rolling two 2s in a row given that the sum of the two rolls is 4, we first need to find all the possible combinations of two rolls that add up to 4. These combinations are (1, 3), (2, 2), and (3, 1).

However, we only want to consider the probability of rolling two 2s in a row, so we can eliminate the other two combinations. This means we are left with only one possible outcome, which is rolling two 2s in a row.

Therefore, the probability of rolling two 2s in a row given that the sum of the two rolls is 4 is 1/3.

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The rotor has turned approximately 0.0906 revolutions or 9.06% of a full revolution.

To determine the number of revolutions the rotor has turned, we need to know the angle traversed by the rotor in one revolution. This value is dependent on the motor and can vary based on its design and the gear ratio used.

Assuming we have this information, we can use the angle traversed by the rotor in one revolution to calculate the number of revolutions it has made based on the angle rotated by the probe. Let's assume that the angle traversed by the rotor in one revolution is 360 degrees. In this case, we can calculate the number of revolutions as follows:

Number of revolutions = Angle rotated by probe / Angle traversed in one revolution

= 32.6 degrees / 360 degrees

= 0.0906 revolutions

It's important to note that this calculation assumes that the rotation of the probe is directly proportional to the rotation of the rotor. If there is any slippage or other factors that affect the relationship between the two, the result may not be accurate.

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So, Bob is 12 years old and his father is 36 years old solved by using the substitution method.

To determine the ages of Bob and his father, let's use the given information and set up two equations.

Let Bob's age be represented as B and his father's age as F. The terms provided are:

1. Bob's father is three times as old as he is: F = 3B
2. Four years ago, Bob's father was four times as old as he is: F - 4 = 4(B - 4)

Now, we can solve these equations simultaneously. Substituting the first equation into the second equation, we get:

3B - 4 = 4(B - 4)

Simplify the equation:

3B - 4 = 4B - 16

Add 4 to both sides:

3B = 4B - 12

Subtract 4B from both sides:

-B = -12

Finally, divide by -1:

B = 12

Now that we know Bob's age, we can find his father's age using the first equation:

F = 3B
F = 3(12)
F = 36

So, Bob is 12 years old and his father is 36 years old.

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We need to add 4 blue marbles to the bag to make the probability of drawing a blue marble 1/2.

Currently, there are two blue marbles out of a total of eight marbles in the bag, so the probability of drawing a blue marble is 2/8 or 1/4.

Let x be the number of blue marbles we need to add to the bag. After adding x blue marbles, there will be a total of 2 + x blue marbles in the bag, out of a total of 8 + x marbles.

We want the probability of drawing a blue marble to be 1/2, so we can set up the equation:

(2 + x) / (8 + x) = 1/2

Multiplying both sides by (8 + x), we get:

2 + x = (8 + x) / 2

Multiplying both sides by 2, we get:

4 + 2x = 8 + x

Subtracting x from both sides, we get:

4 + x = 8

Subtracting 4 from both sides, we get:

x = 4

Therefore, we need to add 4 blue marbles to the bag to make the probability of drawing a blue marble 1/2.

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Answer:

your answer would be 121

Step-by-step explanation:

put the equation as 100(1+10/100)^2

100(11/10)^2

100(121/100)

cancel 100 on both sides

and your final answer would be 121 :)

Judy bought 25 pens in packages of 5 for $0.80 per package and sold 600 pens in packages of 3 for $0.60 per package.

Judy bought the pens in packages of 5 for $0.80 per package, which means she paid $0.16 per pen (0.80/5=0.16). If she sold them in packages of 3 for $0.60 per package, she received $0.20 per pen (0.60/3=0.20). This means that her profit per pen was $0.20 - $0.16 = $0.04.

If her profit was $8.00, we can use the formula profit = revenue - cost to calculate how many pens she bought and sold. Let's call x the number of packages she bought and y the number of packages she sold.

Judy's cost was:

cost = x * 5 * 0.16 = 0.8x

Judy's revenue was:

revenue = y * 3 * 0.20 = 0.6y

Her profit was:

profit = revenue - cost = 0.6y - 0.8x = 8

We can simplify this equation by dividing both sides by 0.2:

0.3y - 4x = 40

Now we need to find two integers x and y that satisfy this equation. We can use trial and error or substitution to find them. For example, if we try x=5, we get:

0.3y - 4(5) = 40

0.3y = 60

y = 200

This means that Judy bought 5 packages of pens (25 pens) and sold 200 packages of pens (600 pens).

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There are 5040 different committees that can be formed from a class of 10 students where each member is assigned a unique position in the committee.

In this problem, we are asked to find the number of possible committees that can be formed from a class of 10 students, where each committee has 4 members who are assigned unique positions. This means that the order in which the students are selected and assigned positions matters. Therefore, we need to use the permutation formula to solve this problem.

To find the number of possible committees, we need to calculate the number of ways we can select 4 students from a class of 10 and assign each of them a unique position. We can do this in two steps:

Step 1: Selecting 4 students from a class of 10
The number of ways we can select 4 students from a class of 10 is given by the combination formula:

C(10,4) = 10!/(4!6!) = 210

Step 2: Assigning unique positions to the selected students
Once we have selected the 4 students, we need to assign each of them a unique position. The first student can be assigned any of the 4 positions (President, Vice President, etc.). The second student can then be assigned any of the remaining 3 positions, the third student can be assigned any of the remaining 2 positions, and the fourth student will be assigned the last remaining position. Therefore, the number of ways we can assign unique positions to the selected students is given by:

4 x 3 x 2 x 1 = 24

Putting these two steps together, we get the total number of possible committees as:

210 x 24 = 5040

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The equation r^2 = sin(2θ) can be rewritten as: r = ± √(sin(2θ))

Since r is always non-negative, we only need to consider the positive square root:

r = √(sin(2θ))

The shaded region is given by the area inside the curve r = √(sin(2θ)) and outside the curve r = 0. This region is symmetric about the polar axis, so we can find the area of one half and multiply by 2.

Using the formula for the area of a polar region, we have:

A = 2∫[0,π/4] 1/2 (r(θ))^2 dθ

Substituting r = √(sin(2θ)), we get:

A = 2∫[0,π/4] 1/2 sin(2θ) dθ

Using the identity sin(2θ) = 2sin(θ)cos(θ), we can simplify the integral:

A = 2∫[0,π/4] 1/2 (2sin(θ)cos(θ)) dθ

A = ∫[0,π/4] sin(θ)cos(θ) dθ

Using the double angle formula, we have:

A = 1/2 ∫[0,π/4] sin(2θ) dθ

Integrating with respect to θ, we get:

A = 1/4 [-cos(2θ)]|[0,π/4]

A = 1/4 (-cos(π/2) + cos(0))

A = 1/4 (0 + 1)

A = 1/4

Therefore, the area of the shaded region is 1/4 square units.

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To calculate the area of a shaded region defined by a polar curve, use the formula Area = 1/2 ∫ (from α to β) ([tex]r(\theta)^2[/tex]) dθ. The polar function is the square root of the given function, and the boundaries of the shaded region mark limits of integration. Without those values, we can't provide a numerical answer.

The question asks us to calculate the area of a shaded region defined by the polar equation [tex]r^2[/tex]=sin 2(theta). This equation falls under the category of a polar curve. To find the area of a region defined by a polar curve, we use the formula Area = 1/2 ∫ (from α to β) ([tex]r(\theta)^2[/tex]) dθ. Here, r(θ) is the polar function (in this case, since [tex]r^2[/tex] =[tex]sin^2(theta)[/tex], r(θ) = sqrt([tex]sin^2(theta)[/tex])), and α and β are the boundaries of the shaded region.

Without knowing the exact values for α and β, we can't provide a numerical answer, but you would integrate the resulting equation from the lower bound to the upper bound. Before integrating, it is vital to ensure that the function is only taking positive values, otherwise, it could lead to miscalculations. Hence, it's important that when you square root sin2(theta), you use the absolute value of sin(theta).

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The cases are explained in the solution.

We know that, the isosceles triangle has two equal side.

Ist case =

perimeter=22 cm

Let's suppose that the known side of 6 cm is one of the two equal sides

perimeter=6+6+x

22=6+6+x

x=22-12

x=10 cm

The possible lengths of the other two sides are

6 cm 

10 cm

IInd case -

Let's suppose that the known side of 6 cm is the side that is not equal

perimeter=22 cm

perimeter=6+x+x

22=6+x+x

2x=22-6

2x=16

x=8 cm

the possible lengths of the other two sides are

8 cm 

8 cm

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