The manager of a furniture factory finds that it costs $2,200 to manufacture 100 chairs in one day and $4,800 to produce 300 chairs in one day.
a) Express the cost as a function of the number of chairs produced, assuming that it is linear.
b) Sketch the graph of the function obtained in part (a).
c) What is the slope of the graph and what does it represent?
d) What is the y-intercept of the graph and what does it represent?

The correct translation of the definition of the limit of a real-valued function f(x) of a real variable x at a point a in its domain in predicate logic expressions.

For all ε > 0, there exists δ > 0 such that for all x, if 0 < |x-a| < δ, then |f(x)-L| < ε.

In symbols, this can be written as:

∀ε > 0, ∃δ > 0 such that ∀x, (0 < |x - a| < δ) → (|f(x) - L| < ε).

Note that "ε" and "δ" are the Greek letters epsilon and delta, respectively, and they are used to represent small positive numbers. This definition says that if we want the limit of f(x) to be L, we can find a positive number δ such that the distance between f(x) and L is less than ε whenever x is within δ units of a (but not equal to a). The limit is said to exist if we can find such a δ for any value of ε.

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The price that corresponds to an elasticity of 1.2 for this product is $59.85.

Elasticity is a fundamental concept in economics that measures the responsiveness of demand to changes in price. In this problem, we will use the calculus-based definition of elasticity to find the price that corresponds to a specific elasticity value.

The elasticity of demand measures the percentage change in quantity demanded that results from a 1% change in price. Mathematically, the elasticity of demand is defined as:

E = (%ΔQ / %ΔP) * (P / Q)

where E is the elasticity of demand, %ΔQ is the percentage change in quantity demanded, %ΔP is the percentage change in price, P is the price of the product, and Q is the quantity demanded.

To find the price that corresponds to an elasticity of 1.2, we need to solve for P in the elasticity equation above. Since we know the demand function D(p) = -5p² + 270,000, we can find Q by taking the derivative of the demand function with respect to price:

Q = D'(p) = -10p

Then, we can plug in Q and E = 1.2 into the elasticity equation and solve for P:

1.2 = (%ΔQ / %ΔP) * (P / Q)

1.2 = (%ΔQ / -1%) * (P / (-10p))

1.2 = (%ΔQ / -0.01) * (1 / -10)

%ΔQ = -0.0144

This means that a 1% increase in price will result in a 0.0144% decrease in quantity demanded, given an elasticity of 1.2. Now, we can plug in %ΔQ and Q into the demand function to solve for P:

-0.0144 = (D(p + 0.01P) - D(p)) / D(p)

-0.0144 = (-5(p + 0.01P)² + 270,000 - (-5p² + 270,000)) / (-5p² + 270,000)

-0.0144 = (-0.1P - 0.0005p²) / (-5p² + 270,000)

P = $59.85

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Answer:

x = 12

Step-by-step explanation:

To find the value of x, use logarithmic rules to solve the given equation.

Given logarithmic equation:

[tex]\log_2 (x^2-100) - \log_2 (x + 10) = 1[/tex]

[tex]\textsf{Apply the log quotient law:} \quad \log_ax - \log_ay=\log_a \left(\dfrac{x}{y}\right)[/tex]

[tex]\implies \log_2 \left(\dfrac{x^2-100}{x+10}\right) = 1[/tex]

Factor the numerator x² - 100:

[tex]\implies \log_2 \left(\dfrac{(x-10)(x+10)}{x+10}\right) = 1[/tex]

Cancel the common factor (x + 10):

[tex]\implies \log_2 \left(x-10\right) = 1[/tex]

[tex]\textsf{Apply the log law:} \quad \log_ab=c \iff a^c=b[/tex]

[tex]\implies 2^1=x-10[/tex]

Simplify and solve for x:

[tex]\implies 2=x-10[/tex]

[tex]\implies 2+10=x-10+10[/tex]

[tex]\implies 12=x[/tex]

Surveys suggest that about 7 percent of people in the united states experience a depression during a given year.

The United States experienced a depression during the 1930s, which lasted from 1929 to 1939. This was one of the longest and deepest economic downturns in U.S. history. The Great Depression caused widespread unemployment, poverty, and hardship throughout the country.

The stock market crash of 1929 and the subsequent bank failures caused a sharp decline in industrial and agricultural production, as well as a dramatic drop in consumer spending. As a result, millions of Americans were left without jobs, and many lost their homes and other possessions.

The government responded to the crisis with a series of programs and initiatives, such as the New Deal, to stimulate the economy and provide relief for those most affected. These programs helped to create jobs and provide economic stability, but the Great Depression would not end until the U.S. entered World War II in 1941.

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The total number of chairs that the waiter can place on two tables is 10 chairs.

Given, the total number chairs that the waiter can place on 4 tables is 20 chairs.

So, the number of chairs that the waiter can place on 1 chair = 20/4 chairs =  5 chairs.

Applying the concept of ratio and proportion,

Now, the total number of chairs that can be placed on 2 chairs will be =

                                                                          = 2 × 5 chairs = 10 chairs

So, the total number of chairs that the waiter can place on 2 tables is 10 chairs.  

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The value of x in 3v(1/8-x)=-1/2 is 10 2/7

A linear equation only has one or two variables. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.

The highest power of variable in a linear equation is 1

3v(1/8-x)=-1/2

square both sides to eliminate the root

3²v(1/8-x)² =(-1/2)²

= 9(1/8-x) = 1/4

= 9/8 - 9x = 1/4

multiply through by 8

9 - 72x = 2

72x = 9-2

72x = 7

x = 72/7

x = 10 2/7

therefore the value of x is 10 2/7

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The China’s expected GDP is 17lac crore.

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

The relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

Given that China’s GDP was proportional to that of the U.S

Now,

Economic growth is defined as a rise in economic activity and product availability. Real GDP has grown throughout this time period as well proportional to that of the U.S. In light of the current circumstances, the level of prices in the United States would increase, and the GDP would increase, as there is economic growth in China and Europe.

Therefore, by the given proportion the answer will be 17lac crore.

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(d) 95% confidence; n=50 (e) n=100 at any confidence level

What is interval estimate?

Interval estimation in statistics is the use of sample data to estimate an interval of feasible values for a parameter of interest. In contrast to point estimate, which produces a single value.

The margin of error for a confidence interval estimate of the population mean decreases with an increase in the sample size and with an increase in the confidence level.

Therefore, options (b) and (d) are better than (a) and (c) because they have a larger sample size. Between options (b) and (d), a larger sample size would result in a smaller margin of error.

Therefore, option (d), which has a sample size of 50 and a higher confidence level of 95%, would result in the smallest margin of error in estimating the mean salt content μ. Option (e) with a sample size of 100 would also result in a smaller margin of error, but the confidence level is not specified, so it cannot be compared directly to the other options.

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The amount of full sets Alexandra can make would be 3.

When there is no common factor between the fraction's numerator (top) and denominator (bottom), the fraction is said to be in its simplest form.

A fraction is a piece of the entire. In mathematics, the number is represented as a quotient, where the numerator and denominator are divided. Both are integers in a straightforward fraction. In the numerator or denominator of a complex fraction is a fraction.

A fraction is a piece of a whole number and a means to divide a number into pieces that are each equal. The numerator, also known as the number of equal parts being counted, is expressed as being greater than the denominator, also known as the number of parts in the entire.

Multiply both the numerator and denominator of 5 / 2 by 2.

5/2 × 2/2 = 10/4

The sets of bowls you can make, divide by 3 / 4.

(10/4)/(3/4) = 3 and 1/3.

The amount of full sets Alexandra can make is 3.

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The amount of time when the water tank will have 450 gallons will be 16²/₃ minutes. The correct option is D.

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that a partially-filled water tank contains 300 gallons of water. water is then pumped into the tank at a constant rate. After n minutes, the amount of water, w, in the tank is W = 300 + 9n gallons.

The time will be calculated as:-

W = 300 + 9n

450 = 300 + 9n

9n = 150

n = 150 / 9

n = 16²/₃ minutes.

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The measure of angle C in triangle BCD is 39⁰.

option B.

The measure of angle C in triangle BCD is calculated by applying the following formula as shown below.

Angle D = 92 ⁰ ( vertically opposite angles are equal )

∠CDB = 180 - 92⁰ = 88⁰ ( sum of angles on a straight line )

∠AED = 180 - (35 + 92 ) ( sum of angles in a triangle)

∠AED = 53⁰

In the quadrilateral FBDE,

angle B = 180 - 53 ( opposite angles of a cyclic quadrilateral are supplementary)

angle B = 127

∠CBD = 180 - 127 ( sum of angles on a straight line)

∠CBD = 53⁰

Then, the value of ∠BCD is calculated as;

∠BCD + ∠CBD + ∠CDB = 180 ( sum of angles in a triangle )

∠BCD + 53 + 88 = 180

∠BCD + 141 = 180

∠BCD = 180 - 141

∠BCD = 39⁰

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The central angle of the pie chart representing the moderate conditions is 80°

Central angle is the angle subtended by an arc of a circle at the center of a circle. The radius vectors form the arms of the central angle.

Given that, a pictogram which was displayed in a pie chart, we need to find the central angle of the pie chart,

Sea conditions is shown,

Calm = 3 blue dots

Moderate = 2 and a half dot

Rough = 5 and 3 parts of the dot

Each dot = 4 points

Therefore,

Calm = 3×4 = 12

Moderate = 2×4 + 1/2 × 4 = 10

Rough = 5×4 + 3/4 ×4 = 23

The central angle = 10 / 12+10+23 × 360° = 80°

Hence, the central angle of the pie chart representing the moderate conditions is 80°

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The complete question is:-

If the information in this pictogram was displayed in a pie chart, what would the central angle of the sector representing rough conditions be? Give your answer in degrees  Sea conditions

Calm Key

Moderate

Rough

All the linear functions are,

⇒ x = 3

⇒ y = 2x - 5

⇒ 3x - 6 = 4

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We know that;

Degree of variables in a linear function is always one and the graph shows the straight line.

So, By all options;

All the linear functions are,

⇒ x = 3

⇒ y = 2x - 5

⇒ 3x - 6 = 4

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The sequence with nth term an = (5√n)/(5√n + 6) diverges.

To determine the convergence or divergence of the sequence with the given nth term, we can use the limit comparison test by comparing it with the divergent series 1/n.

We have

lim n→∞ an/(1/n) = lim n→∞ (5√n)/(5√n + 6) * n = 5/5 = 1

Since the limit is a positive finite number, and the series 1/n diverges, the series with the nth term also diverges by the limit comparison test.

Therefore, the sequence with the nth term an = (5√n)/(5√n + 6) diverges.

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Answer:

8 Units

Step-by-step explanation:

The volume of a cone can be calculated using the formula:

V = (1/3) * π * r^2 * h

where V is the volume, π is pi, r is the radius, and h is the height of the cone. We are given the volume and the diameter, so we can use the diameter to find the radius:

d = 2r

14 = 2r

r = 7

Now we can substitute the values for the volume and radius into the formula and solve for the height:

V = (1/3) * π * r^2 * h

1230.88 = (1/3) * 3.14 * 7^2 * h

1230.88 = (1/3) * 3.14 * 49 * h

1230.88 = (49/3) * 3.14 * h

1230.88 = 153.86 * h

h = 1230.88 / 153.86

h = 8

So, the height of the cone is approximately 8 units

The inverse of the function f(x) = 64x³ - 1 is [tex]f^{-1}(x) = (\frac{3\sqrt{(x + 1)}}{4})[/tex].

First to be an inverse function that function needs to one to one function, meaning every different preimage must correspond to a different image.

We can obtain the inverse of a function by switching the variables x and y with their respective positions and solving for y in terms of x.

Given, A function f(x) = 64x³ - 1.

Let, y = 64x³ - 1.

64x³ = y + 1.

x³ = (y + 1)/64.

[tex]x = (\frac{(y + 1)}{64})^{\frac{1}{3}[/tex].

[tex]x = (\frac{3\sqrt{(y + 1)}}{4})[/tex].

[tex]y = (\frac{3\sqrt{(x + 1)}}{4})[/tex].

[tex]f^{-1}(x) = (\frac{3\sqrt{(x + 1)}}{4})[/tex].

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The coordinates of the point on y sin(x) that is closest to the point (4, y) is (3.579, 0.323).

To find the point on the curve y = sin(x) that is closest to the point (4, y), we can use the distance formula between two points. The distance between two points (x1, y1) and (x2, y2) is given by:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

We want to minimize this distance, so we can minimize the square of the distance:

d^2 = (x2 - x1)^2 + (y2 - y1)^2

Let (x, sin(x)) be a point on the curve y = sin(x). Then the distance squared between this point and (4, y) is:

d^2 = (x - 4)^2 + (sin(x) - y)^2

To minimize this distance, we can take the derivative of d^2 with respect to x, set it equal to zero, and solve for x:

d^2 = (x - 4)^2 + (sin(x) - y)^2

d^2/dx = 2(x - 4) + 2(sin(x) - y)cos(x) = 0

Simplifying this expression, we get:

x - 4 + (sin(x) - y)cos(x) = 0

We can solve this equation numerically using a numerical method such as Newton's method or the bisection method. Once we have found the value of x that minimizes the distance, we can find the corresponding value of y = sin(x) and the closest point on the curve is (x, sin(x)).

Using a numerical method, we can find that the value of x that minimizes the distance is approximately 3.579. Therefore, the closest point on the curve is (3.579, sin(3.579)) which is approximately (3.579, 0.323).

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For the given five number summary, the interquartile range is 7.

The five number summary is a set of descriptive statistics that provides a concise summary of a data set. It consists of the following five values:

In this problem, the five number summary is:  53 59 63 66 88

Hence,

Q1 = 59

Q3 = 66

Interquartile range = Q3 - Q1

                               = 66 - 59

                               = 7

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The value of the equation 7.4×10⁻⁸=6.7×10⁻⁹ is 6.73×10⁻⁸

A number system is defined as a system of writing to express numbers.

The given expression is 7, point, 4, dot, 10, start superscript, minus, 8, end superscript, minus, 6, point, 7, dot, 10, start superscript, minus, 9, end superscript, equals

7.4×10⁻⁸=6.7×10⁻⁹

Now let us take 6.7×10⁻⁹ to left hand side

7.4×10⁻⁸-6.7×10⁻⁹

Factor 10⁻⁸ from both terms

10⁻⁸(7.4-0.67)

6.73×10⁻⁸

Hence, 6.73×10⁻⁸ is the value of the equation 7.4×10⁻⁸=6.7×10⁻⁹

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The complete question will be as below

Find the simplest form of the equation 7.4×10⁻⁸=6.7×10⁻⁹?

The perimeter of the triangle FGH is 7.

Perimeter is defined as space around the any two dimensional figure. It is given by the sum of all the sides of the figure.

Given the triangle FGH which is formed by connecting the midpoints of the side of the triangle CDE.

The length of the sides of the triangle are given as, CD=6, DE=4, EC=4.

To find the perimeter of the triangle FGH,

Since FGH are the midpoints of CD,DE and EC respectively.

Then, FH=1/2 DE, FG=1/2EC, GH=1/2DC.

Perimeter of a triangle = sum of the length of sides.

                                   = 1/2(DE+EC+DC)

                                   = 1/2(4+6+4)

Perimeter of a triangle = 7

Hence, the perimeter of the triangle FGH is 7.

Question:

Triangle FGH is formed by connecting the midpoint of the sides of the triangle CDE. The length of the sides is CD=6, EC=4, DE=5. find the perimeter of the triangle FGH.

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(a) The​ least-squares regression line treating the commute​ time, x, as the explanatory variable and the index​ score, y, as the response variable is y = 70.22 - 0.075x

(b) The slope and​ y-intercept, if appropriate is the slope of the regression line, -0.075, indicates that for every additional minute of commute time, the well-being index score decreases by an average of 0.075.

(c) The​ well-being index of a person whose commute is 35 minutes is the slope of the regression line, -0.075, indicates that for every additional minute of commute time, the well-being index score decreases by an average of 0.075.

(d) Yes, she is. Barbara more​ "well-off" than the typical individual who has a ​15-minute ​commute.

(a) To find the least-squares regression line, we need to find the slope and y-intercept of the line that minimizes the sum of the squared vertical distances between the actual data points and the predicted values on the line. Using a calculator or software, we get:

Slope: b = -0.075

y-intercept: a = 70.22

Therefore, the least-squares regression line is:

y = 70.22 - 0.075x

(b) The slope of the regression line, -0.075, indicates that for every additional minute of commute time, the well-being index score decreases by an average of 0.075. The y-intercept, 70.22, represents the predicted well-being index score for someone with a commute time of 0 minutes (which is not a meaningful value in this context).

(c) To predict the well-being index of a person with a commute time of 35 minutes, we can substitute x = 35 into the regression equation and solve for y:

y = 70.22 - 0.075(35) = 67.97

Therefore, the predicted well-being index score for someone with a commute time of 35 minutes is 67.97.

(d) To determine whether Barbara is more "well-off" than the typical individual with a 15-minute commute, we can compare her actual well-being index score of 68.5 to the predicted score based on the regression equation:

y = 70.22 - 0.075(15) = 68.47

Barbara's score of 68.5 is slightly higher than the predicted score of 68.47, which suggests that she is somewhat better off than the typical individual with a 15-minute commute. However, we should note that this comparison is based on a single data point and may not be representative of the larger population.

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The length of PQ is (1 + √(3))/2, which is answer choice (C) equilateral triangle, angle BAC measures 60 degrees.

Since ABC is an equilateral triangle, angle BAC measures 60 degrees. Also, since A' is the midpoint of BC and A is folded onto A', angle BAA' and CAA' are both 30 degrees.

Now, let's use the Law of Cosines to find the length of PQ. Let x be the length of PQ. Then, applying the Law of Cosines to triangles ABP and AQC, we have:

BP² = x² + 1 - 2x cos30

CQ² = x² + 4 - 4x cos30

Since BP = CQ, we can set the two expressions equal to each other and simplify:

x² - 2x cos30 + 3 = 0

Solving for x using the quadratic formula, we get:

x = cos30 +/- √(cos²30 - 3)

Since cos30 = √(3)/2, we have:

x = 1/2 +/- √(3)/2

Since x must be positive, we take the positive root and get:

x = 1/2 + √(3)/2 = (1 + √(3))/2

Therefore, the length of PQ is (1 + √(3))/2, which is answer choice (C).

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The question is -

Equilateral triangle ABC has P on AB and Q on AC. The triangle is folded along PQ so that vertex A now rests at A' on side BC. If BA'=1 and A'C=2 then the length of the crease PQ is

[tex]\text{(A) } \frac{8}{5} \text{(B) } \frac{7}{20}\sqrt{21} \text{(C) } \frac{1+\sqrt{5}}{2} \text{(D) } \frac{13}{8} \text{(E) } \sqrt{3}[/tex]

Answer: To find the volume of the tank, we need to determine the length and width of the base. We can use the area formula for rectangles:

Area = length * width

Since the area of the base is given as 485 square inches, we can rearrange this equation to find the length or width:

length * width = 485

Let's call the length "l". Then, the width is 485/l. The volume of the tank is the product of the area of the base and the height:

Volume = Area * height = (l * 485/l) * 16 = 485 * 16 = 7760 cubic inches.

So the volume of Tiny's tank is 7760 cubic inches.

Step-by-step explanation:

The required height of the kite to the nearest tenth of a meter is 25.2 m.

In a right-angled triangle, the ratio of the perpendicular (with respect to an angle) to the hypotenuse gives the sine value of that angle. This relationship is used to solve problems regarding height and distance chapter.

Draw the figure as per the given instructions.

The position of the kite is at A and the string is AC which is 37 m long. The angle of elevation is ∠ACB, which is 43°. The height of the kite is AB.

So, apply the formula of sine ratio (perpendicular/hypotenuse) = sinα, where α be the opposite angle of that perpendicular.

AB/AC = sin∠ACB

AB/37 = sin43°

AB = 37sin43°

≈37×0.68

= 25.16

≈ 25.2

Therefore, the obtained answer is 25.2 m.

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The value of the function j(7) will be 1 / 10.

An inverse function is defined as the function in which the two variables are changed by each other and the function is solved for the value of an independent variable.

F(x) = 2x

y = 2x

The inverse function is calculated as:-

x = 2y

y = x / 2

F'(x) = x / 2

Given that Functions hand j are inverses. at the value of x = -10, the value of h(x) is 7, or h (-10) = 7.

The value of the function j(7) will be equal to 1 / 10 for the given data.

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The area covered by the airport in 2000000 m² and 200 Ha.

A scale factor is a numerical value that can be used to alter the size of any geometric figure or object in relation to its original size. It is used to find the missing length, area, or volume of an enlarged or reduced figure as well as to draw the enlarged or reduced shape of any given figure. It should be remembered that the scale factor only affects how big a figure is, not how it looks.

Given:

A map of kano is drawn to a scale of 1:50000.

The area that airport covers = 8 cm²

Also, 1 ha = 10000 m²

if it's 1 cm, then real life would be 50,000 cm.

Then, 50,000 cm x 50,000 cm = 2,500,000,000 cm² in real life.

Now, the Area of Airport in real

= 8 x 2,500,000,000

= 20,000,000,000 cm²

= 2000000 m²

= 2000000 / 10000

= 200 Ha

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The number of 5/4 is 1130

how to find the number x?

writing this as an algebraic expression.

An algebraic expression in mathematics is an expression that is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms. Also, solve questions in Algebraic Expressions Worksheets

[tex]\frac{5}{8}x[/tex] = 565

[tex]\frac{5x}{8}[/tex] = 565

5x = 565 * 8

5x = 4520

x = [tex]\frac{4520}{5}[/tex]

x = 904

so the number is 904

What is 5/4 of 904 ?

[tex]\frac{5}{4}[/tex] * 904

= 1130

so the answer to this question is 1130

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The joint pmf of X and Y 5/8, the marginal pmfs pX and pY is 3/7 and the conditional pmfs pX|Y (x|0) and pX|Y (x|1) is 5/28 and 41/56.

The probability that the chosen combination has a certain property can be set up by dividing the number of combinations where the property holds by the total number of possible combinations.

The order of the chips chosen does not matter so outcomes can be considered as combinations. We must choose 3 of the 9 chips in the box, where X are chosen from the 3 red chips, and Y are chosen from the 2 white, and the remainder are chosen from the 4 black.

For x∈{0,1,2,3}and y∈{0,1,2}, the probability that x red chips and y white chips are chosen is

To determine the marginal distribution of X, we need not distinguish white chips from black. X chips are chosen from the 3 red chips, and the remainder are chosen from the 5 non-red chips. For x∈{0,1,2,3} the probability that x chips are chosen is,

P(x) = (³ₓ) (₃-ₓ⁶) / (⁹₃)

Similarly, in determining the marginal distribution of Y, we choose Y white chips from among the 3 available, then choose the remaining chips from the 5 non-white chips.

c) To find the conditional distribution,

P(X|Y=0)= 5/28

P(X|Y=1)= 41/56

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The transformation rule for triangle PQR into triangle P'Q'R' is given as follows:

(x,y) -> (y,x).

Which is a reflection over the line y = x.

The vertices of the original triangle are given as follows:

P(3,2), Q(5,6) and R (4,-1).

The vertices of the transformed triangle are given as follows:

P'(2,3), Q'(6,5) and R'(-1,4).​

The x-coordinate and the y-coordinate were exchanged, hence the rule is defined as follows:

(x,y) -> (y,x).

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