help me please please please

The event (e ∩ f ∩ g)c is equal to the set {3, 8, 13, 18}.

To find the complement of the intersection of sets e, f, and g, denoted as (e ∩ f ∩ g)c, we first need to determine the intersection of sets e, f, and g.

The intersection of sets e, f, and g is the set of elements that are present in all three sets. In this case:

e ∩ f ∩ g = {23, 28}

To find the complement of this intersection, we need to consider all the elements that are not in the set {23, 28}.

Given that the original set s = {3, 8, 13, 18, 23, 28}, the complement of the intersection can be found by subtracting {23, 28} from set s:

(e ∩ f ∩ g)c = s - {23, 28}

Calculating this, we have:

(e ∩ f ∩ g)c = {3, 8, 13, 18}

Therefore, the event (e ∩ f ∩ g)c is equal to the set {3, 8, 13, 18}.

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The diameter of the tree trunk is 6.5 feet (to the nearest half-foot).

Given: Length of the rope wrapped around the tree trunk = 21 feet 8 inches.How the group of students can use this length to estimate the diameter of the tree trunk to the nearest half-foot is described below.Using this length, the students can estimate the diameter of the tree trunk by finding the circumference of the tree trunk. For this, they will use the formula of the circumference of a circle i.e.,Circumference of the circle = 2πr,where π (pi) = 22/7 (a mathematical constant) and r is the radius of the circle.In this question, we are given the length of the rope wrapped around the tree trunk. We know that when the rope is wrapped around the tree trunk, it will go around the circle formed by the tree trunk. So, the length of the rope will be equal to the circumference of the circle (formed by the tree trunk).

So, the formula can be modified asCircumference of the circle = Length of the rope around the tree trunkHence, from the given length of rope (21 feet 8 inches), we can calculate the circumference of the circle formed by the tree trunk as follows:21 feet and 8 inches = 21 + (8/12) feet= 21.67 feetCircumference of the circle = Length of the rope around the tree trunk= 21.67 feetTherefore,2πr = 21.67 feet⇒ r = (21.67 / 2π) feet= (21.67 / (2 x 22/7)) feet= (21.67 x 7 / 44) feet= 3.45 feetTherefore, the radius of the circle (formed by the tree trunk) is 3.45 feet. Now, we know that diameter is equal to two times the radius of the circle.Diameter of the circle = 2 x radius= 2 x 3.45 feet= 6.9 feet= 6.5 feet (nearest half-foot)Therefore, the diameter of the tree trunk is 6.5 feet (to the nearest half-foot).

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The formula for the indicated rate of change dc/dk is dc/dk = 32c.

To find the indicated rate of change, we need to calculate dc/dk, which represents the partial derivative of the function s(c, k) = c(32k) with respect to k while treating c as a constant.

To calculate dc/dk, we differentiate the function s(c, k) with respect to k while considering c as a constant:

dc/dk = d/dk (c * (32k))

Applying the product rule of differentiation, we have:

dc/dk = c * d/dk (32k) + (32k) * d/dk (c)

The derivative of 32k with respect to k is 32, as it is a constant multiple of k. The derivative of c with respect to k is zero since c is treated as a constant.

Therefore, dc/dk simplifies to:

dc/dk = c * 32 + 0

dc/dk = 32c

So, the formula for the indicated rate of change dc/dk is dc/dk = 32c.

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The sketched region enclosed by the given curves, y = 2/x, y = 8x, and y = x/8 is given below.

To sketch the region enclosed by the given curves, we'll first plot each curve separately and then identify the region between them. The curves are:

y = 2/x

y = 8x

y = x/8

Let's start by plotting these curves one by one:

y = 2/x:

Since x > 0, the curve y = 2/x is a hyperbola with the y-axis as an asymptote and passes through the point (1, 2).

y = 8x:

This is a straight line passing through the origin (0, 0) with a slope of 8. The line goes through the first quadrant.

y = x/8:

This is another straight line with a slope of 1/8. It passes through the origin (0, 0) and also goes through the first quadrant.

Therefore, the final graph is given below.

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The complete question:

Sketch the region enclosed by the given curves.

y = 2/x, y = 8x, y = x/8, x>0

Answer:

a = 3, -17

Step-by-step explanation:

a ² +  14a  - 51 = 0

1) put the a, not a ², in parenthesis.

2) half the coefficient (14) of a. that is 7. Put that into same parenthesis.

3) we have (a + 7)

4) square this and multiply out. (a + 7) ² = a ² + 7a + 7a +49 = a ² +14a + 49

5) this looks just like the original equation except for +49. What do we have to do to get back to original? 49 – (-51) = 49 + 51 = 100. We have to subtract 100

6) now we have (a + 7) ² – 100 =0

7) (a + 7) ² = 100

8) (a + 7) = ± √100

9) a = ± √100  - 7

a = ±10 - 7

= -17 and 3

Given: An angle of measure 115 degrees We know that: The supplement of an angle is equal to 180 degrees minus the angle, and the complement of an angle is equal to 90 degrees minus the angle

Now, we need to find the complement of the supplement of an angle measuring 115 degrees.So, let's first find the supplement of the given angle:

Supplement of 115 degrees = 180 - 115= 65 degrees

Now, we need to find the complement of the above angle which is:

Complement of 65 degrees = 90 - 65= 25 degrees Therefore, the complement of the supplement of an angle measuring 115º is 25 degrees.

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The equation of the line in the graph is

From the graph the line passed through points (4,-1) and (0,5)

using the slope-intercept form of a line, which is y = mx + b,

where

m is the slope and

b is the y-intercept.

the slope of the line

m = (5 - (-1)) / (0 - 4) = 6 / -4 = -3/2

form the points the y intercept is 5

Therefore, the equation of the line is y = -3/2 x + 5.

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The assembly time for a product is uniformly distributed between 6 to 10 minutes the standard deviation of assembly time in minutes is approximately 1.155.

To find the standard deviation of assembly time for a product that is uniformly distributed between 6 to 10 minutes, we can use the following formula for a uniform distribution:

Standard Deviation (σ) = √((b - a)² / 12)

Here, 'a' is the lower limit (6 minutes) and 'b' is the upper limit (10 minutes).

Step 1: Calculate (b - a)²
(10 - 6)² = 4² = 16

Step 2: Divide by 12
16 / 12 = 1.3333

Step 3: Find the square root
√1.3333 ≈ 1.155

So, the standard deviation of assembly time for a product in minutes is approximately 1.155.

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Main answer: Transformations of basic functions depend on the changes made to their variables.

Supporting answer :Functions can be transformed in different ways. The variable a modifies the vertical stretch or compression of a function. A negative value of a produces a reflection over the x-axis. The variable b is used to modify the horizontal stretch or compression of the function. A negative value of b produces a reflection over the y-axis. The variable h translates the graph to the left (h > 0) or to the right (h < 0). Lastly, the variable k translates the graph up (k > 0) or down (k < 0).

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To write and solve a story problem with 6 divided by 6, we need to come up with a situation in which 6 is divided equally among 6 parts. For example:

There are 6 pieces of candy to be divided equally among 6 children. Solution: To solve this problem, we can simply divide the total number of candies (6) by the number of children (6):6 ÷ 6 = 1Therefore, each child will receive 1 piece of candy. Another way to solve this problem is to use multiplication. Since division is the inverse of multiplication, we can think of this problem as:6 ÷ 6 = x can be rewritten as 6 = x × 6, where x is the number of candies each child receives. Then we can solve for x by dividing both sides by 6:x = 6 ÷ 6x = 1Therefore, each child will receive 1 piece of candy.

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The value of the definite integral ∫3−3(3x3−2x2 x 1) dx is 0.

The given question asks us to find the definite integral of a polynomial function of degree 3 over the interval [-3, 3]. When we integrate a polynomial function, we get a polynomial function of one degree higher. In this case, we get a degree 4 polynomial function, which we can evaluate at the upper and lower limits of the interval and take the difference to get the definite integral.

After simplifying the expression, we get the definite integral to be 0. This result suggests that the area under the curve of the given polynomial function over the interval [-3, 3] is zero. Definite integrals have many applications in calculus, physics, engineering, and economics, and understanding their properties is crucial in these fields.

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When government spending increases by $5 billion and the MPC = .8, in the first round of the spending multiplier process, spending increases by $20 billion.


The spending multiplier is the amount by which GDP will increase for each unit increase in government spending. It is calculated as 1/(1-MPC), where MPC is the marginal propensity to consume. In this case, MPC = .8, so the spending multiplier is 1/(1-.8) = 5.

Therefore, when government spending increases by $5 billion, the total increase in spending in the economy will be $5 billion multiplied by the spending multiplier of 5, which equals $25 billion. However, the initial increase in spending is only $5 billion, hence the increase in the first round of the spending multiplier process is $20 billion.

In summary, when government spending increases by $5 billion and the MPC = .8, the initial increase in spending is $5 billion, but the total increase in the first round of the spending multiplier process is $20 billion.

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The hydrated Calcium Chloride compound is CACl₂.0.2998H₂O

To determine the formula of the hydrated Calcium Chloride compound,  to calculate the number of water molecules present in the hydrated salt.

First to calculate the mass of water lost during the heating process. This can be done by subtracting the mass of the anhydrous salt from the mass of the hydrated salt.

Mass of water lost = Mass of hydrated salt - Mass of anhydrous salt

= 5.4769 g - 2.7745 g

= 2.7024 g

To convert the mass of water lost to moles. the molar mass of water, which is approximately 18.015 g/mol.

Number of moles of water lost = Mass of water lost / Molar mass of water

= 2.7024 g / 18.015 g/mol CACl₂x²

= 0.1499 mol

Calcium Chloride (CACl₂x²) has a molar mass of approximately 110.98 g/mol.

Since calcium chloride has a 1:2 ratio with water in the hydrated form,  the following equation:

0.1499 mol water / 1 mol CACl₂ = X mol water / 2 mol CACl₂x²

0.1499 mol water = X mol water / 2

X mol water ≈ 0.1499 mol water × 2

X mol water ≈ 0.2998 mol water

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The answer is:

(a) 47.

We can use the following formula to find the sample size needed:

n = [(t-value * standard deviation) / margin of error]²

where the margin of error is the maximum amount we allow the estimate to be off by, and the t-value is based on the desired level of confidence and the degrees of freedom (n-1).

In this case, we want the margin of error to be $100 and we want to have a 95% level of confidence. Using a z-value of 1.96 for a 95% confidence interval, we can find the corresponding t-value with 39 degrees of freedom (n-1) using a t-table or calculator.

t-value = 2.0227

Substituting the values into the formula, we get:

n = [(2.0227 * 350) / 100]²

n = 47.22

we get a required sample size of 47 (option a).

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The evaluated value of the given expression when j = 0.5 and k = 0.25 is 6.5.

The given expression is 7j+5−8k7j+5−8k7, j, plus, 5, minus, 8, k.

We need to evaluate the given expression when j=0.5j=0.5j, equals, 0, point, 5 and k=0.25k=0.25k, equals, 0, point, 25.

Now we substitute the values of j and k in the given expression.

7(0.5)+5−8(0.25)7(0.5)+5−8(0.25)7, times, 0, point, 5, plus, 5, minus, 8, times, 0, point, 25=3.5+5-2=6.5

The value of the expression when j=0.5j=0.5j, equals, 0, point, 5 and k=0.25k=0.25k, equals, 0, point, 25 is 6.5, which is the final answer.

Therefore, the evaluated value of the given expression when j = 0.5 and k = 0.25 is 6.5.

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The profit function is 9x - 7000 and the revenue function is 12x.

Given that the cost function for a company to produce a lunch box is c(x)= 3x+7000 where x is the number of lunch boxes and the company sells the lunch boxes for $12 each.

To write a profit function, the revenue function is required to calculate the profit earned by the company.

The revenue function is given as:

Revenue = Selling Price × Quantity Sold

Price is $12 for each lunch box, therefore

Revenue = $12 × Quantity sold

Quantity sold is represented as x, therefore,

Revenue = 12x

The profit function is given as:

Profit = Revenue - Cost

The cost function is given as c(x)= 3x+7000

Therefore,

Profit = 12x - (3x + 7000)

Profit = 9x - 7000

Hence, the profit function is 9x - 7000 and the revenue function is 12x.

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Sam the snail's rate is approximately 0.03 miles per hour.

To find Sam's rate in miles per hour, we need to convert his speed from feet per minute to miles per hour.

We know that 1 mile is equal to 5280 feet. First, we can convert Sam's speed from feet per minute to feet per hour by multiplying it by 60 since there are 60 minutes in an hour.

Therefore, Sam's speed in feet per hour is 2.64 ft/min * 60 min/hr = 158.4 ft/hr.

Next, we can convert Sam's speed from feet per hour to miles per hour. Since 1 mile is equal to 5280 feet, we can divide Sam's speed in feet per hour by 5280 to get his speed in miles per hour.

Therefore, Sam's speed in miles per hour is 158.4 ft/hr / 5280 ft/mi = 0.03 mi/hr.

Therefore, Sam the snail crawls at a rate of approximately 0.03 miles per hour.

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Answer:

Step-by-step explanation:

sqrt{-2x^{2}-2x+11 }=\sqrt{-x^{2} +3}

Square both sides:

-2x^2 - 2x + 11 = -x^2 + 3

0 = x^2 + 2x - 8

( x + 4)(x - 2) = 0

x = -4, 2.

As the original equation contains square roots some of these roots might be extraneous.

Checking:

x = -4

sqrt(-2(-4)^2 - 2(-4) + 11 =  sqrt(-13)

sqrt (-(-4)^2 + 3) = sqrt(-13)

x = 2:

sqrt(-2(4) - 2(2) + 11) = sqrt(-8 - 4 + 11) = sqrt(-1)

sqrt(-(2)^2 + 3) = sqrt(-1)

So both are roots

the series ∑(n=1 to infinity) [tex]n^{6}[/tex] / n! is convergent by using ratio test.

To apply the Ratio Test, we need to evaluate the limit of the ratio of consecutive terms, lim(n→∞) (a(n+1) / a(n)).

In this case, a(n) = [tex]n^{6}[/tex] / n! and a(n+1) =[tex](n+1)^{6}[/tex] / (n+1)!.

Taking the limit, we have:

lim(n→∞) [[tex](n+1)^{6}[/tex] / (n+1)!] / [[tex]n^{6}[/tex] / n!]

= lim(n→∞) [[tex](n+1)^{6}[/tex] / [tex]n^{6}[/tex]] * [n! / (n+1)!]

= lim(n→∞) [[tex](n+1)^{6}[/tex] / [tex]n^{6}[/tex]] * [1 / (n+1)]

= 1 * 0 = 0.

Since the limit of the ratio of consecutive terms is 0, which is less than 1, the series converges by the Ratio Test.

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A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It can be used to represent systems of linear equations, transformations in geometry, and a wide range of other mathematical concepts in a compact and organized form.

When you multiply a matrix A (n × n) by a vector x (in R^n), the resulting product Ax is a linear combination of the columns of matrix A.

Here's a step-by-step explanation:

1. Let A be an n × n matrix with columns C₁, C₂, ..., Cₙ, and x be a vector in R^n with elements [x₁, x₂, ..., xₙ]^T (transpose).

2. When you multiply the matrix A by the vector x, the resulting vector Ax can be represented as:

  Ax = A * x = [C₁ C₂ ... Cₙ] * [x₁, x₂, ..., xₙ]^T

3. The multiplication of A and x results in a new vector, where each element is formed by taking the dot product of the corresponding row of A with the vector x:

  Ax = [x₁*C₁ + x₂*C₂ + ... + xₙ*Cₙ]

4. In the resulting vector Ax, you can see that each column of matrix A is multiplied by its corresponding scalar from the vector x, forming a linear combination of the columns of matrix A.

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Option A is correct, 57.17% is the probability that he hits the target at least four out of next six attempts.

Let's calculate the probability of hitting the target exactly four times out of six attempts:

P(4 hits) = C(6, 4) × (0.65)⁴ ×  (1 - 0.65)⁶⁻⁴

The probability of hitting the target exactly five times out of six attempts:

P(5 hits) = C(6, 5) × (0.65)⁵ × (1 - 0.65)⁶⁻⁵

Now calculate the probability of hitting the target all six times:

P(6 hits) = (0.65)⁶

Now, we can find the probability that Tom hits the target at least four times by summing up the individual probabilities:

P(at least 4 hits) = P(4 hits) + P(5 hits) + P(6 hits)

P(at least 4 hits) = C(6, 4) × (0.65)⁴ ×  (1 - 0.65)⁶⁻⁴ + C(6, 5) × (0.65)⁵ × (1 - 0.65)⁶⁻⁵ +  (0.65)⁶

=57.17%

Hence,  57.17% is the probability that he hits the target at least four out of next six attempts.

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5.880 can be expressed as the ratio of integers 127/25.

To express 5.880 as a ratio of integers, we can write it as follows:

5.880 = 5 + 0.880

To convert the decimal part (0.880) into a fraction, we can write it as a repeating decimal by observing the repeating pattern:

0.880880880...

The repeating part is "880", which has three digits.

Now, we can express 5.880 as a ratio of integers:

5.880 = 5 + 0.880 = 5 + 880/1000

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 10:

5.880 = 5 + 880/1000 = 5 + (880 ÷ 10)/(1000 ÷ 10) = 5 + 88/100

Finally, we can simplify the fraction further:

5.880 = 5 + 88/100 = 5 + 22/25

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So there are Output pairs that cannot be obtained for any input pair.

(a) T(x, y) = (2x, y)

This function is one-to-one but not onto. It is one-to-one because different input pairs (x1, y1) and (x2, y2) will always result in different output pairs (2x1, y1) and (2x2, y2). However, it is not onto because for any y ≠ 0, there is no input pair (x, y) that maps to the output pair (0, y).

(b) T(x, y) = (x^4, y)

This function is onto but not one-to-one. It is onto because for any given output pair (a, b), we can find an input pair (x, y) such that T(x, y) = (a, b) by taking the fourth root of a for x and setting y to b. However, it is not one-to-one because different input pairs can result in the same output pair. For example, T(1, 2) = T(-1, 2) = (1, 2).

(c) T(x, y) = (sin(x), cos(y))

This function is neither one-to-one nor onto. It is not one-to-one because different input pairs can result in the same output pair due to the periodic nature of sine and cosine functions. For example, T(0, 0) = T(2π, 0) = (0, 1). It is also not onto because the range of the function is limited to the interval [-1, 1] for both x and y, so there are output pairs that cannot be obtained for any input pair.

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T(x, y) = (2x, y) is one-to-one and onto.

To show one-to-one, assume T(a, b) = T(c, d). Then we have (2a, b) = (2c, d), which implies a = c and b = d.

To show onto, we need to show that for any (x, y) in R2, there exists (a, b) in R2 such that T(a, b) = (x, y). If we take (a, b) = (x/2, y), then T(a, b) = (x, y).

(b) T(x, y) = (x^4, y) is one-to-one but not onto.

To show one-to-one, assume T(a, b) = T(c, d). Then we have (a^4, b) = (c^4, d), which implies a = c and b = d.

To show not onto, note that there is no (a, b) in R2 such that T(a, b) = (-1, 0), since x^4 is always non-negative.

(d) T(x, y) = (sin(x), cos(y)) is neither one-to-one nor onto.

To show not one-to-one, note that T(0, 0) = T(2π, 0), but (0, 0) ≠ (2π, 0).

To show not onto, note that there is no (x, y) in R2 such that T(x, y) = (0, 1), since sin(x) is always between -1 and 1.

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According to question  the value of ∫41(3f(x) 2x)dx is 73.

We know that the average value of the function f on the interval [1,4] is 8. This means that:

(1/3) * ∫1^4 f(x) dx = 8

Multiplying both sides by 3, we get:

∫1^4 f(x) dx = 24

Now, we need to find the value of ∫4^1 (3f(x) 2x) dx. We can simplify this expression as follows:

∫1^4 (3f(x) 2x) dx = 3 * ∫1^4 f(x) dx + 2 * ∫1^4 x dx

Using the average value of f, we can substitute the first integral with 24:

∫1^4 (3f(x) 2x) dx = 3 * 24 + 2 * ∫1^4 x dx

Evaluating the second integral, we get:

∫1^4 x dx = [x^2/2]1^4 = 8.5

Substituting this value back into the equation, we get:

∫1^4 (3f(x) 2x) dx = 3 * 24 + 2 * 8.5 = 73

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Answer:

a=4, b=11

Step-by-step explanation:

You have to complete the square.

x²-8x+5 = (x-4)²-16 +5 = (x-4)² - 11

The polar equation r = 3+6cosθ is symmetric to the polar axis with respect to the polar axis.

To test the polar equation r = 3 + 6 cos(θ) for symmetry, we will consider each type of symmetry one by one:

1. Polar axis symmetry: Replace θ with -θ and check if the equation remains the same.
r = 3 + 6 cos(-θ) = 3 + 6 cos(θ) (since cosine is an even function)
Since the equation remains the same, the curve is symmetric with respect to the polar axis.

2. Pole symmetry: Replace r with -r and check if the equation remains the same.
-r = 3 + 6 cos(θ)
This equation is not equivalent to the original equation, so the curve is not symmetric with respect to the pole.

3. Line θ = π/2 symmetry: Replace θ with (π - θ) and check if the equation remains the same.
r = 3 + 6 cos(π - θ) = 3 - 6 cos(θ) (since cos(π - θ) = -cos(θ))
This equation is not equivalent to the original equation, so the curve is not symmetric with respect to the line θ = π/2.

In conclusion, the polar equation r = 3 + 6 cos(θ) is symmetric with respect to the polar axis, but not with respect to the pole or the line θ = π/2.

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To determine the fraction that Lee writes on the first mark to the right of 17, we need to understand the numbering pattern and the position of the marks.

If Lee marks sixths on the number line, it means that the interval between each mark is 1/6.

Starting from 0, the first mark to the right of 17 would be located at 18.

To find the fraction written on this mark, we can calculate the difference between 18 and 17 and express it as a fraction of the interval between each mark (1/6).

18 - 17 = 1

Therefore, the fraction that Lee writes on the first mark to the right of 17 is 1/6.

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Therefore, the determinant of matrix B is 13. The determinant of A is the product of the pivots in the upper triangular matrix U is 6/5.

(a) Using the cofactor formula, we have:

|B| = 3 * |3 0 7|

- 2 * |2 0 1|

+ 0 * |-2 0 1|

= 3 * (3*1 - 0*5) - 2 * (2*1 - 0*(-2)) + 0 * (-2*0 - 0*1)

= 9 + 4 + 0

= 13

(b) To find the PA=LU factorization of matrix A, we perform Gaussian elimination with partial pivoting. The first step is to interchange the first and second rows to get a nonzero pivot in the (1,1) position:

| 3 1 2 | | 3 1 2 |

| 0 2 5 | -> | 0 -5 -1 |

| 3 5 5 | | 0 0 5 |

Next, we perform row operations to get zeros below the pivot in the second row:

| 3 1 2 | | 3 1 2 |

| 0 -5 -1 | -> | 0 -5 -1 |

| 0 4 3 | | 0 19 11 |

Finally, we divide the second row by -5 and subtract 3 times the second row from the third row to get zeros below the (3,2) position:

| 3 1 2 | | 3 1 2 |

| 0 1 1/5| -> | 0 1 1/5|

| 0 0 2/5| | 0 0 32/5|

Therefore, we have:

A = LU = | 3 1 2 | | 1 0 0 | | 3 1 2 |

| 0 1 1/5 | * | 0 1 0 | = | 0 1 1/5|

| 0 0 2/5 | | 0 0 32/5| | 0 0 2/5 |

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The smallest number of pigeons in the pigeonhole that contains the most number of pigeons is 2341.

To determine the smallest number of pigeons in the pigeonhole that contains the most number of pigeons, we can use the pigeonhole principle.

The pigeonhole principle states that if you distribute more than m objects into m pigeonholes, then at least one pigeonhole must contain more than one object.

In this case, we have 32761 pigeons and 14 pigeonholes. To minimize the number of pigeons in the pigeonhole that contains the most, we want to distribute the pigeons as evenly as possible.

Dividing 32761 by 14, we get:

32761 / 14 = 2340 remainder 1

This means we can evenly distribute 2340 pigeons to each of the 14 pigeonholes, leaving 1 pigeon remaining.

To minimize the number of pigeons in the pigeonhole that contains the most, we distribute the remaining 1 pigeon to one of the pigeonholes, resulting in the exact integer is 2341.

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The mechanistic basis for linear polystyrene having phenyl groups attached to alternate carbons of the polymer chain is due to the nature of the polymerization reaction, specifically free-radical polymerization.

1. Free-radical polymerization of styrene starts with the initiation step, where a free radical initiator generates a reactive radical site.
2. The reactive radical site reacts with the double bond of the styrene monomer, forming a new radical site on the styrene molecule.
3. This new radical site on the styrene molecule can now react with another styrene monomer, effectively joining them together.
4. As the radical site is always at the end of the growing polymer chain, the phenyl groups of each added styrene monomer will be attached to alternate carbons. This occurs because the reactive site is situated between the phenyl group and the double bond in the monomer, creating a zigzag pattern as the chain grows.

Conclusion:
The attachment of phenyl groups to alternate carbons of the polymer chain in linear polystyrene can be attributed to the free-radical polymerization mechanism. The reactive radical site, created during the polymerization, allows the phenyl groups to be connected in an alternating pattern along the chain.

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