right triangle $abc$ has one leg of length 6 cm, one leg of length 8 cm and a right angle at $a$. a square has one side on the hypotenuse of triangle $abc$ and a vertex on each of the two legs of triangle $abc$. what is the length of one side of the square, in cm? express your answer as a common fraction.right triangle $abc$ has one leg of length 6 cm, one leg of length 8 cm and a right angle at $a$. a square has one side on the hypotenuse of triangle $abc$ and a vertex on each of the two legs of triangle $abc$. what is the length of one side of the square, in cm? express your answer as a common fraction.

Appeals that focus on projecting the appealing traits of the writer are known as ethos-based appeals.

Ethos is one of the three modes of persuasion identified by Aristotle and refers to the credibility and trustworthiness of the speaker or writer. Ethos-based appeals attempt to establish the author's character and authority on a subject, using various strategies such as emphasizing their expertise, experience, or moral character.

By projecting appealing traits of the writer, such as honesty, intelligence, or likability, ethos-based appeals aim to win the audience's confidence and persuade them to accept the argument.

Ethos is particularly important in situations where the audience may be skeptical or distrustful of the writer, such as in political or advertising contexts. Overall, ethos-based appeals are a powerful tool for writers looking to persuade their audience by establishing their credibility and building trust.

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Let r be the common ratio. The correct answer is Rounding to the nearest dollar, the value of Madame Dumas's art collection 12 years after she started tracking its worth is $9,498,559.

A) Let the value of the collection three years ago be the first term, a = $450,000.

Then we can write:

Second term: [tex]ar = $810,000[/tex]

Third term: [tex]ar^2 = $1,458,000[/tex]

To find the common ratio r, we can divide the second term by the first term and the third term by the second term:

[tex]ar/a[/tex]= [tex]\frac{810,000}{450,000}[/tex]

[tex]r = 1.8[/tex]

[tex]ar^2/ar[/tex] = [tex]\frac{450000}{810000}[/tex]

[tex]r = 1.8[/tex]

So the explicit rule for the value of Madame Dumas's art collection n years after she started tracking its worth is:[tex]a_n = ar^(n-3)[/tex]

B) To find the value of her collection 12 years after she started tracking its worth, we can use the explicit rule:

[tex]a_12 = ar^(12-3) = ar^9[/tex]

We already know that [tex]a = $450,000[/tex] and [tex]r = 1.8[/tex], so we can substitute those values:[tex]a_12 = $450,000(1.8)^9 = $9,498,558.57[/tex]

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It will take 3.6 hours or 3 hours and 36 minutes for Ana and her apprentice to build a wall together.

If constructing a brick wall is one unit of work, Ana may complete one sixth of it in an hour, while her apprentice can complete one ninth. They can do 1/6 + 1/9 of the task in an hour while working jointly. By determining the common denominator of 6 and 9, which is 18, we can determine how much work they can complete in an hour.

1/6 + 1/9

= 3/18 + 2/18

= 5/18

They can complete 5/18 of the task in an hour, according to this. We can build up a percentage to determine how long it would take them to do the assignment collectively.

5/18 = 1/x, the time it takes for them to complete the work together is x. Solving for x, we get,

x = 18/5

x = 3.6 hours

Therefore, it would take Ana and her apprentice 3.6 hours, or 3 hours and 36 minutes, to build the wall together.

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The area of the largest circle is 153.86 yard square.

The measurement that is closest to the area of the largest circle can be calculated as follows:

Therefore,

area of the largest circle = πr²

where

Therefore,

radius = 10 + 4 ÷ 2

radius = 14 / 2

radius = 7 yards

Hence,

area of the largest circle = 3.14 × 7²

area of the largest circle = 3.14 ×49

area of the largest circle = 153.86 yard²

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The slope of the tangent to the curve x² + y³ = 12 at the point when x = 2 is

To find the slope of the tangent to the curve x² + y³ = 12 at the point when x = 2, we need to find the derivative of y with respect to x using implicit differentiation.

Taking the derivative of both sides with respect to x, we get: 2x + 3y²(dy/dx) = 0

We want to find the slope when x = 2, so we substitute x = 2 into the equation above: 2(2) + 3y²(dy/dx) = 0 4 + 3y²(dy/dx) = 0 3y²(dy/dx) = -4 dy/dx = -4/(3y²)

Now, we need to find the value of y when x = 2. Substituting x = 2 into the original equation, we get: 2² + y³ = 12 y³ = 8 y = 2 So, when x = 2, y = 2. Substituting this into the equation for dy/dx, we get: dy/dx = -4/(3(2²)) = -4/12 = -1/3

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The degrees of freedom would be df = 21 + 20 - 2 = 39. This means we would use the t-distribution with 39 degrees of freedom to find the critical value for our hypothesis test.

To find the critical value for testing the difference between two independent population means with two samples of sizes 21 and 20, we need to use the t-distribution with degrees of freedom equal to the sum of the sample sizes minus two (df = n1 + n2 - 2).

In this case, the degrees of freedom would be df = 21 + 20 - 2 = 39. This means we would use the t-distribution with 39 degrees of freedom to find the critical value for our hypothesis test.

To find the critical value for the difference between two independent population means with unknown but equal standard deviations, you will need to calculate the degrees of freedom. In this case, you have two samples: one of size 21 (n1) and the second of size 20 (n2). The formula for degrees of freedom in this scenario is:

Degrees of Freedom (df) = (n1 - 1) + (n2 - 1)

Plugging in the values:

df = (21 - 1) + (20 - 1)
df = 20 + 19
df = 39

So, there are 39 degrees of freedom used to find the critical value in this case.

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By making the substitution u = a² - x² and using integration by substitution, we can evaluate the integral of (a² - x²[tex])^(^3^/^2^)[/tex] from 0 to a. The resulting value is [tex]-a^5^/^5[/tex].

We can evaluate this integral using the substitution method. Let's substitute u = a² - x². Then du/dx = -2x, which implies dx = -du/(2x). Also, when x = 0, u = a².

Substituting these expressions into the integral, we get:

∫(a² - x²[tex])^(^3^/^2^)[/tex] dx from 0 to a

= ∫(a² - x²[tex])^(^3^/^2^)[/tex] (-du/(2x)) from a² to 0

= (-1/2) ∫[tex]u^(^3^/^2^)[/tex] du from a² to 0

= (-1/2) [2/5 [tex]u^(^5^/^2^)[/tex]] from a² to 0

= (-1/5) [a⁵ - 0⁵]

= [tex]-a^5^/^5[/tex]

Therefore, the value of the integral is [tex]-a^5^/^5[/tex].

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The area of new park will be 419904 yd².

Given that, the area of a park is 46,656 yd², it is planned to build a new park by enlarging the current park by a scale factor of 3.

We need to find the area of the new park,

We know that the ratio of the square of the dimension of similar figure is equal to the ratio of their areas,

Let the area of the new park be x,

So,

x / 46656 = 3²

x = 419904

Hence, the area of new park will be 419904 yd².

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For a loan amount taken by Timmy from the bank on simple interest, the interest rate advertised by the bank is equals to the 2.7% per year.

Simple interest defines to the interest calculated only based on the principal. With simple interest method, a borrower only pays interest on the principal. It is calculated by the principal amount multiplied by the interest rate, multiplied by the number of periods and then resultant is divided by 100. Formula is written as [tex]Simple \: interest = \frac{P \times r \times t}{100}[/tex]

Where, P--> principal amount

t --> time period

r -> simple interest rate

We have Timmy takes out a loan on simple interest. The amount of loan that is principal = $750

Time periods = 15 months

The received amount by him = $725

So, simple interest = 750 - 725 = $25

We have to determine the simple interest rate advertised by the bank. Using the above formula, substitute all known values in formula, 25 = [tex] \frac{ 750 × 15 × r}{12×100}[/tex]

[tex]r = \frac{ 1200× 25}{750× 15}[/tex]

= 2.66% per year

Hence, required interest rate is 2.7 % per year.

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The 98% confidence interval for the mean usage of water is (15.6, 15.8) gallons per day. This means that we are 98% confident that the true mean household usage of water for the town falls within this interval.

In this case, we are trying to estimate the mean household usage of water for a small town. We know the population standard deviation, which is 1.7 gallons, and we have a sample mean of 15.7 gallons per day based on a sample of 1454 families.

To construct a 98% confidence interval for the mean usage of water, we can use the following formula:

CI = x ± zα/2 * (σ/√n)

where CI is the confidence interval, x is the sample mean, zα/2 is the critical value from the standard normal distribution for a 98% confidence level (which is 2.33), σ is the population standard deviation, and n is the sample size.

Substituting in the given values, we get:

CI = 15.7 ± 2.33 * (1.7/√1454)

CI = 15.7 ± 0.11


In summary, we used the confidence interval formula to estimate the mean household usage of water for a town with a 98% level of confidence. We also explained why we used this formula and how we obtained the critical value and sample size.

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Triple integration is a powerful tool for computing volumes of complex regions in three-dimensional space and is widely used in mathematical modeling, physics, and engineering.

For the first problem, the volume of the region U can be computed using triple integration in cylindrical coordinates

The bounds of integration for r, θ and z must be determined based on the shape of the region.

For the second problem, the volume of the region U can also be computed using triple integration in cylindrical coordinates, but with different bounds of integration due to the different shape of the region.

In both cases, cylindrical coordinates are used because the regions have cylindrical symmetry, making it easier to integrate over the region.

Triple integration is a powerful tool for computing volumes of complex regions in three-dimensional space and is widely used in mathematical modeling, physics, and engineering.

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To repeat the previous problem with a reliability of 98 percent, we need to find the value of C10 for which the Weibull distribution function gives a probability of 0.98 when X is at its 10th percentile. Using the Weibull parameters from example 11-3, we have a shape parameter (beta) of 2.5 and a scale parameter (eta) of 5.

To find C10, we can use the inverse Weibull distribution function, which is given by:

X = eta * (-ln(1 - p))^1/beta

where p is the probability (0.98), eta is the scale parameter (5), and beta is the shape parameter (2.5).

Substituting the values, we get:

C10 = eta * (-ln(1 - 0.98))^1/beta = 5 * (-ln(0.02))^0.4 = 13.86

Therefore, we expect to obtain a larger value for C10 with a reliability of 98 percent compared to the result in the previous problem, where the reliability was 90 percent. This is because the higher the reliability requirement, the more reliable the system needs to be, which means it needs to have a longer life. Hence, the value of C10, which represents the life at which 10 percent of the units fail, will be larger for a higher reliability requirement.

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Step-by-step explanation:

To answer the question, we can use algebra. Let's assume that the total number of voters is "x". If 40% of the voters chose Shane, then 60% of the voters chose the other candidates. We can set up an equation:

0.6x = 540

Solving for x, we get:

x = 900

Therefore, there were 900 voters in total.

Answer: 900

If two poles are connected by a wire that is also connected to the ground then The wire should be anchored to the ground at a distance of 20 feet from the first pole A to minimize the amount of wire needed.

To minimize the length of the wire, we need to find the point P that minimizes the length of the wire APB.

Let's assume that the wire is perfectly straight, which means that the line segment connecting A and P and the line segment connecting B and P are both perpendicular to the ground.

Let's also call the distance from point P to the first pole A as x. Then the distance from point P to the second pole B is 96 - x.

Using the Pythagorean theorem, we can express the length of the wire AB as: AB^2 = (20 - x)^2 + 12^2

Simplifying this expression, we get: AB^2 = x^2 - 40x + 784

To minimize AB, we need to find the value of x that minimizes AB^2. To do that, we take the derivative of AB^2 with respect to x and set it equal to 0: d/dx (AB^2) = 2x - 40 = 0

Solving for x, we get: x = 20

Therefore, the wire should be anchored to the ground at a distance of 20 feet from the first pole A to minimize the amount of wire needed.

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The Cartesian equation for the curve is y = 2x/(1+x^2), which is the equation of a limacon.'

The cartesian form of equation of a plane is ax + by + cz = d, where a, b, c are the direction ratios, and d is the distance of the plane from the origin.

To find the Cartesian equation for the curve, we need to use the relationships between polar and Cartesian coordinates:

x = r cos(theta) and y = r sin(theta)

Substituting r = 2 tan(theta) sec(theta), we get:

x = 2 tan(theta) sec(theta) cos(theta) = 2 sin(theta)

y = 2 tan(theta) sec(theta) sin(theta) = 2 tan(theta)

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To find My and Nx, we need to take the partial derivatives of M and N with respect to y and x, respectively.

My = 2cos(y) - 6sin(x)

Nx = 6sin(x) - 2ysin(y)

To check if the differential equation is exact, we need to check if My = dN/dx and Nx = dM/dy.

dN/dx = 6cos(x) - 2ysin(y)

dM/dy = 2cos(y) - 6sin(x)

Since My = dN/dx and Nx = dM/dy, the differential equation is exact.

To find the function F(x,y), we need to integrate M with respect to x and add a function of y only, which will be the constant of integration.

F(x,y) = ∫(2sin(y) - 6ysin(x))dx + g(y)

F(x,y) = 2sin(y)x + 3ycos(x) + g(y)

To find g(y), we take the partial derivative of F with respect to y and set it equal to N.

dF/dy = 2cos(y)x + g'(y)

2cos(y)x + g'(y) = 6cos(x) + 2xcos(y) - 2y

g'(y) = 2y - 2xcos(y) + 6cos(x) - 2cos(y)x

g(y) = y^2 - 2xsin(y) + 6sin(x) + h(x)

where h(x) is a function of x only.

Thus, the general solution to the differential equation is given by the implicit function F(x,y) = 2sin(y)x + 3ycos(x) + y^2 - 2xsin(y) + 6sin(x) + h(x) = C.
Hi! Let's analyze the given differential equation and find the required terms and function.

The given equation is:
(2sin(y)−6ysin(x))dx+(6cos(x)+2xcos(y)−2y)dy=0

Here, we have:
M = 2sin(y)−6ysin(x)
N = 6cos(x)+2xcos(y)−2y

Now, we need to find My and Nx.

My = ∂M/∂y = 2cos(y) - 6sin(x)
Nx = ∂N/∂x = -6sin(x) + 2cos(y)

Since My = Nx, the given differential equation is exact. Now, we need to find a function F(x,y) such that dF(x,y) is the left hand side of the differential equation.

To find F(x,y), we can integrate M with respect to x, and N with respect to y, and then combine the results.

∫M dx = ∫(2sin(y)−6ysin(x)) dx = 2xsin(y) - 6y∫sin(x) dx = 2xsin(y) + 6ycos(x) + g(y)

∫N dy = ∫(6cos(x)+2xcos(y)−2y) dy = 6cos(x)y + 2x∫cos(y) dy - ∫2y dy = 6ycos(x) + 2xsin(y) - y^2 + h(x)

Comparing the two integrals, we find:
F(x,y) = 2xsin(y) + 6ycos(x) - y^2 + C

Here, C is the constant of integration. The level curves F(x,y)=C give implicit general solutions to the differential equation.

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The letters of the word "are" can be arranged in 6 different ways. These arrangements are: are, aer, rae, rea, ear, era. To calculate the number of arrangements, we use the formula for permutations of n objects, which is n!. In this case, n = 3, so there are 3! = 6 ways to arrange the letters.

If the letters of "are" are arranged in a random order, the probability that the result will be "era" is 1/6. This is because there is only one way to get "era" out of the 6 possible arrangements, and each arrangement is equally likely to occur.

In other words, the probability of an event happening is equal to the number of ways that event can occur, divided by the total number of possible outcomes. In this case, the event is getting the word "era" and the total number of outcomes is 6.

I hope this helps answer your question. Let me know if you have any more questions!
Hello! It seems that you've missed providing the specific letters and the result you're looking for in your question. However, I can explain the process using a general example.

Let's say you have the letters A, B, and C. To determine the number of different arrangements, you can use the formula for permutations, which is n! (n-factorial), where n represents the number of unique items.

For this example:

n! = 3!
  = 3 × 2 × 1
  = 6

So, there are 6 different ways to arrange the letters A, B, and C.

Now, if you're looking for the probability of getting a specific arrangement (for example, "ABC"), you can calculate it by dividing the number of desired outcomes by the total number of possible outcomes. Since there's only 1 way to get the "ABC" arrangement and there are 6 possible arrangements in total:

Probability = (Desired outcomes) / (Total outcomes)
           = 1 / 6
           ≈ 0.1667

This means there's approximately a 16.67% chance of getting the "ABC" arrangement when arranging these letters randomly.

Please provide the specific letters and the result you want to calculate the probability for, and I'd be happy to help you with your question.

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The given equation is a curve in the Cartesian plane. Therefore, the  exact length of the curve [tex]y^2= 4(x+5)^3 , 0 \leq x \leq 3, y > 0[/tex]  is  [tex]2(3 \sqrt{3} - \sqrt{6} )[/tex] units

To find its length, we can use the formula for the arc length of a curve in terms of its parameterization.

First, we need to rewrite the equation in terms of a parameterization. Let's use x as the parameter, so we have [tex]y = 2\sqrt{(x+5)^3}[/tex]. Then, taking the derivative of y with respect to x, we get:

dy/dx = √(x+5)

Using this, we can calculate the arc length of the curve as:

[tex]L = \int_0^3 \sqrt{(1 + (dy/dx)^2) dx}[/tex]

Substituting dy/dx, we get:

[tex]L = \int_0^3 \sqrt{(1 + x+5) dx}[/tex]

Simplifying the inside of the square root, we get:

[tex]L = \int_0^3 \sqrt{(x+6) dx}[/tex]

Making the substitution u = x+6, we get:

[tex]L = \int_6^9 \sqrt{u \;du}[/tex]

Using the power rule of integration, we get:

[tex]L = (2/3)u^{(3/2)} |_6^9[/tex]

[tex]L = (2/3)(9\sqrt{9} - 6\sqrt{6} )[/tex]

[tex]L = 2(3\sqrt{3} - \sqrt{6})[/tex]

Therefore, the exact length of the curve is [tex]2(3 \sqrt{3} - \sqrt{6} )[/tex] units

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Part A: The factored expression is 8x(5x - 2)

Part B: The expression is 4x - 16

Note that algebraic expressions are described as expressions that consists of coefficients, factors, constants, terms and variables.

They are also made up of arithmetic operations such as addition, subtraction, bracket, parentheses, multiplication and division

From the information given, we have that;

40x² and 16x

40x² - 16x

factorize the values

8x(5x - 2)

To add the expressions;

(10x+8) - 3(2x + 8)

expand the bracket, we have;

10x + 8 - 6x - 24

collect the like terms

4x - 16

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In summary, the equation for the tangent plane can be written as z - z0 = ∂f/∂x(x0, y0)(x - x0) + ∂f/∂y(x0, y0)(y - y0), where (x0, y0, z0) represents the coordinates of the given point. This equation represents a linear approximation of the surface near the point of tangency.

To understand the equation for the tangent plane, we start by considering the first-order partial derivatives of the function f(x, y) with respect to x and y. These partial derivatives, denoted as ∂f/∂x and ∂f/∂y, represent the rates of change of the surface with respect to x and y, respectively. At the point (x0, y0), the tangent plane approximates the behavior of the surface near that point.

The equation for the tangent plane is derived by using the point-slope form of a linear equation, where the slope of the plane in the x-direction is given by ∂f/∂x(x0, y0) and in the y-direction by ∂f/∂y(x0, y0). The equation is then written as z - z0 = ∂f/∂x(x0, y0)(x - x0) + ∂f/∂y(x0, y0)(y - y0), which relates the changes in x and y coordinates to the change in z-coordinate on the tangent plane.

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Answer:

Step-by-step explanation:

4x2=8

8x8=64 cm cube

The sequence can be written as {n}. As n approaches infinity, the sequence grows without bound and the limit does not exist.

The limit of the sequence {n^2/n} can be found by simplifying the expression. n^2/n can be written as n, since one of the n terms cancels out. Given the sequence {n^2/n}, we can simplify it as follows:

{n^2/n} = {n}

Now, to find the limit of this sequence as n approaches infinity, we can express it mathematically:

lim (n -> ∞) {n}

In this case, the limit does not exist because as n approaches infinity, the value of n will continue to grow without bound.

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A 95% confidence interval estimate of , the mean procrastination scale for first-year students at this college is between 34.881 and 38.919.

We know that:

Sample size (n) = 65

Sample mean (x) = 36.9

Sample standard deviation (s) = 6.41

Confidence level = 95%

Degrees of freedom = n - 1 = 64

To calculate the confidence interval, we use the formula:

CI = x ± tα/2 * (s/√n)

where tα/2 is the t-score with (n-1) degrees of freedom and α/2 = (1 - confidence level)/2.

Using a t-table or a calculator, we find that tα/2 for a 95% confidence level and 64 degrees of freedom is 1.997.

Plugging in the values, we get:

CI = 36.9 ± 1.997 * (6.41/√65)

Simplifying the expression, we get:

CI = (34.881, 38.919)

Therefore, we can be 95% confident that the true mean procrastination scale for first-year students at this college falls between 34.881 and 38.919.

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The predicted population of the dying town in 2023 is approximately 8,200

To predict the population in 2023 using the exponential law, p(t) = P0e^(kt) or p(t) = P0b^t, we first need to find the constants P0 and k (or b). We know the population was 10,000 in 2016 and 9,500 in 2018.

Step 1: Set up the equations using the given information.
For the year 2016 (t=0), p(0) = P0e^(k*0) = 10,000
For the year 2018 (t=2), p(2) = P0e^(k*2) = 9,500

Step 2: Solve for P0 and k.
From the first equation, P0 = 10,000.
Substitute P0 in the second equation: 9,500 = 10,000e^(2k)

Step 3: Solve for k.
Divide both sides by 10,000: 0.95 = e^(2k)
Take the natural logarithm of both sides: ln(0.95) = 2k
Divide by 2: k = ln(0.95) / 2 ≈ -0.0253

Step 4: Predict the population in 2023 (t=7).
p(7) = P0e^(kt) = 10,000e^(-0.0253*7) ≈ 8,200

So, the predicted population of the dying town in 2023 is approximately 8,200, rounded to the nearest whole number.

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The probability that both students chosen forgot their lunch is 1/45. Therefore, the probability that both students chosen forgot their lunch is 1/45.

To find the probability that both students chosen forgot their lunch, we need to use the formula for calculating probability:

P(A and B) = P(A) x P(B|A)

where P(A) is the probability of event A occurring, and P(B|A) is the probability of event B occurring given that event A has already occurred.

In this case, event A is the first student being chosen as someone who forgot their lunch (which has a probability of 2/10), and event B is the second student also being chosen as someone who forgot their lunch (which has a probability of 1/9, since there is one less student left to choose from).

So, putting it all together:

P(both students forgot their lunch) = P(A and B) = P(A) x P(B|A)
= (2/10) x (1/9)
= 1/45

Therefore, the probability that both students chosen forgot their lunch is 1/45.

In a class of 10 students, there are 2 students who forgot their lunch. If the teacher chooses 2 students, the probability that both of them forgot their lunch is calculated as follows:

First, determine the total number of ways to choose 2 students out of 10. This can be done using combinations:
C(10,2) = 10! / (2! * (10-2)!) = 45 combinations

Now, consider the 2 students who forgot their lunch. There's only 1 way to choose both of these students:
C(2,2) = 2! / (2! * (2-2)!) = 1 combination

The probability that both chosen students forgot their lunch is the ratio of the favorable combinations to the total combinations:
P = 1/45

So, the probability that both students chosen forgot their lunch is 1/45.

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Description: The analytical solution you have given is a second-order linear homogeneous differential equation with a forcing function. DSolve[{m*x''[t] + b*x'[t] + k*x[t] =  v{0}, x[t], t].

It describes the motion of a mass-spring-damper system subjected to an external force. The solution to this equation depends on the initial conditions and the parameters of the system.

To obtain the analytical solution using Mathematica, you can use the DSolve function, which is a built-in function for solving differential equations. Here's an example code:

DSolve[{m*x''[t] + b*x'[t] + k*x[t]

== Sin[56*t], x[0]

== x0, x'[0]

== v{0}, x[t], t]

In this code, m, b, and k are the mass, damping coefficient, and spring constant, respectively. x[t] is the displacement of the mass at time t, and x''[t] and x'[t] are its second and first derivatives with respect to time. Sin[56*t] is the external force, and x0 and v0 are the initial displacement and velocity, respectively.

The output of the DSolve function will give the analytical solution for x[t]. You can simplify the solution to two terms using trigonometric identities for the sum or difference of a sine or cosine. The observed graphical output will depend on the values of the parameters and initial conditions, and it will relate to the algebraic form of the simplified analytical solution by showing the displacement of the mass as a function of time.

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The distributive law for sets A, B, C can be stated as follows: A union (B intersection C) = (A union B) intersection (A union C).

To prove the distributive law for sets, we need to show that any element x that belongs to A union (B intersection C) also belongs to (A union B) intersection (A union C), and vice versa.

Let x be an arbitrary element in A union (B intersection C). Then, x must belong to either A or (B intersection C) or both.

Case 1: If x belongs to A, then x must belong to A union B and A union C, since A is a subset of both sets. Therefore, x belongs to (A union B) intersection (A union C).

Case 2: If x belongs to B intersection C, then x belongs to both B and C. Therefore, x belongs to A union B and A union C, since A is a subset of both sets. Therefore, x belongs to (A union B) intersection (A union C).

Hence, we have shown that A union (B intersection C) is a subset of (A union B) intersection (A union C), and vice versa. Therefore, the distributive law holds.

To prove the second part, we will use a proof by contradiction.

Assume that A and C are not disjoint, and B and D are not disjoint. Then, there exist elements a and c such that a belongs to both A and C, and there exist elements b and d such that b belongs to both B and D.

Therefore, (a,b) belongs to both A times B and C times D, which contradicts the assumption that A times B and C times D are disjoint.

Hence, either A and C are disjoint, or B and D are disjoint.

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There is no algebraic solution to this equation.

We have,

There is no algebraic solution to this equation, as it is a fifth-degree polynomial equation, which cannot be solved exactly using algebraic methods.

However, it is possible to find an approximate solution using numerical methods, such as graphing the equation and finding the point of intersection with the line y=2, or using iterative methods such as the Newton-Raphson method.

Thus,

There is no algebraic solution to this equation.

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The slope (m) of the line y = x + 3 is 1, while the y-intercept of the line is 3.

If an equation of a line is expressed in slope-intercept form as y = mx + b, we can easily determine its slope and the y-intercept which are:

m is the slope

b is the y-intercept.

Given the equation y = x + 3, therefore:

the coefficient of x is the slope (m), which is 1.

the y-intercept (b) of the line is 3.

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