Express the rational function as a sum or difference of two simpler rational expressions. 2x4 2x 2x3 2 х — Additional Materials eBook Submit Answer Practice Another Version +0/10 points Previous Answers osCalc1 7.4.190. Express the rational function as a sum or difference of two simpler rational expressions. X x2 36 (х+6)(х- 6) 1 (x - 1)x2 sum or difference of two simpler rational expressions. (Note: x Express the rational function as a 1).) x+ 9x2 x3 1 (x-1)2+ -1 t Additional Materials eBook + -/10 points OSCalc1 7.4.195 1. Express the rational function as a sum or difference of two simpler rational expressions. 44x2 6x4x3 3x 79 (x1)(x2 4)2

The final answer is  g'(x) = 0 when x = 0 or x = 6.

As an illustration, the phrase x + y is one where x and y are terms with an addition operator in between. There are two sorts of expressions in mathematics: numerical expressions, which only contain numbers, and algebraic expressions, which also include variables.

Given g'(x) = [tex]24x^2 - 4x^3[/tex], we need to find the value(s) of x such that g'(x) = 0. To do this, we factor the expression as follows:

[tex]g'(x) = 24x^2 - 4x^3 = 4x^2(6 - x)[/tex]

Setting g'(x) = 0, we have:

[tex]4x^2(6 - x) = 0[/tex]

This equation is satisfied when either [tex]4x^2 = 0[/tex]or 6 - x = 0. Solving for x, we get:

[tex]4x^2 = 0[/tex] => x = 0

6 - x = 0 => x = 6

Therefore, g'(x) = 0 when x = 0 or x = 6.

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The value of the function (fg)(x) = = ∛5

A function can be described as an equation or expression that is used to show the relationship between two variables.

The two variables are;

From the information given, we have that;

f(x) = 5x

g(x) = x^1/3

To determine the composite function (fg)(x), substitute the value of(x) as the value of x in the function g(x), we have;

(fg)(x) = 5^1/3

This is written as;

(fg)(x) =(∛5)¹

(fg)(x) = = ∛5

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This condition ensures that the distribution is not too skewed and allows us to use the Z-score to calculate probabilities.

(a) When we have a large sample size, the sample mean, X, follows a normal distribution with mean μ and standard deviation σ/√n, where σ is the population standard deviation. However, since we usually do not know the population standard deviation, we estimate it using the sample standard deviation, s. In this case, we use the t-distribution with n-1 degrees of freedom to calculate the 2-score, which has a slightly wider distribution than the standard normal distribution. To adjust for this wider distribution, we divide by the standard error, which is s/√n, instead of the population standard deviation a.

(b) We do not need the Central Limit Theorem to find P(X <72) if our sample size is 9 because the distribution of X is already normal. This is because the sample size is not too small, and we know the population standard deviation, so we can use the Z-score to calculate the probability directly.

(c) In order to convert 2 to a chi-square random variable, we need the assumption that the population follows a normal distribution. Specifically, if we have a random sample of size n from a normal population, then the sample variance s2 follows a chi-square distribution with n-1 degrees of freedom.

(d) In the Central Limit Theorem for 1 Proportion, we need to check the success/failure condition to ensure that the sample proportion, p, follows a normal distribution. Specifically, if np ≥ 10 and n(1-p) ≥ 10, then the sample proportion follows a normal distribution with mean p and standard deviation √(p(1-p)/n). This condition ensures that the distribution is not too skewed and allows us to use the Z-score to calculate probabilities.

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The rectangle with the formula x(x+1)=60 using the quadratic formula has  length = 8.26 and width = 7.26.

Given that,

A rectangle has the formula,

x (x + 1) = 60

x² + x = 60

x² + x - 60 = 0

Using the quadratic formula,

x = -1 ± √1 -(4 × 1 × -60) / 2

  = (-1 ± √241) / 2

x = 7.26

Width = 7.26

Length = x + 1 = 8.26

Hence the required length and width are 8.26 and 7.26.

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The drawing of each solid in three dimension is sketched and attached

Perspective views in drawing are a way of representing a three-dimensional object or scene on a two-dimensional surface. By using perspective, the artist can create the illusion of depth and spatial relationships between objects in the scene.

There are several types of perspective views in drawing, including:

One-point perspective: This type of perspective is used when the subject is viewed straight-on, and all lines converge to a single point on the horizon line.

Two-point perspective: This type of perspective is used when the subject is viewed from an angle, and two vanishing points are used to create the illusion of depth.

Three-point perspective: This type of perspective is used when the subject is viewed from a very high or very low angle, and three vanishing points are used to create the illusion of depth.

Perspective views are an essential tool for artists in many fields, including architecture, product design, and illustration. Understanding perspective is crucial for creating realistic and compelling images that accurately represent three-dimensional space on a two-dimensional surface.

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For a company that wants to teach self improvement seminars is holding of its seminar. The company will be take 12 attendees to break even. The company's total expenses and revenues both are equal to $384.

We have a company that teaches self-improvement seminars is holding one of its seminars. Flat fee spent by company to rent a facility, P = $324

Additionally, Spent by company on books for every attendee who registers = $ 5

The fee pay by each attendee for attending the seminar = $32

We have to determine the number of attendees. Let 'x' represent the total number of attendees who are registered. According to the above scenario, the break-even equation is written as R (revenue) = E( expenses), 32x = 5 x + 324

Simplify it,

32x - 5x = 324

=> 27 x = 324

=> x = 324/27

=> x = 12

So, It will take 12 attendees to break even. Now, the company's total expenses

= 5x + 324

= 5×12 + 324

= $384

The net income or revenue will also be

= 32 ×12

= $384

Hence, required value is $384..

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The point with polar coordinates (-1, -π/6) is plotted as a point on the terminal arm of an angle of -π/6 in the polar coordinate system. The Cartesian coordinates of the point are then determined using the relationships:
x = r cosθ and y = r sinθ, where r is the radius and θ is the angle in radians.

To find the Cartesian coordinates of the point, we substitute the given values for r and θ in the above equations:

x = (-1) cos(-π/6) = (-1) × (√3/2) = -√3/2

y = (-1) sin(-π/6) = (-1) × (-1/2) = 1/2

Therefore, the Cartesian coordinates of the point are (-√3/2, 1/2).

In summary, the point with polar coordinates (-1, -π/6) is plotted as a point on the terminal arm of an angle of -π/6 in the polar coordinate system. Then, using the relationships between polar and Cartesian coordinates, the Cartesian coordinates of the point are determined to be (-√3/2, 1/2).

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Answer:

1. The base is a square.

2. Its dimensions are 3 by 3 (or 3 x 3).

3. Their faces are triangles. (Not too sure if this correct as the wording of the question is a little confusing.)

4. The base of each triangle measures 3 cm.

5. The height of each triangle measures 4 cm.

6. The polyhedron is a cube.

Step-by-step explanation:

1. The question says to draw a net for the square pyramid. From there we already have our answer for the base.

2. It is a general rule that all sides of a square are equal. Hence, if one side is measured to be 3 cm, so will the rest.

3. This one is clear by sight, you can clearly recognize that the rest of the shapes are triangles. If not, you can also deduce this from the instructions saying that is a square pyramid. According to byjus.com, "A pyramid has a polygonal base and flat triangular faces, which join at a common point called the apex."

4. The side of the square is 3 cm. However, the triangle's base must be equal to this in order to form the pyramid and connect with the other triangles and square. We can also see this fact from the diagram, as the base of the triangle and the side of the square are the same.

5. The diagram points out that the height of the triangle is 4 cm. This is the only measurement left in the diagram, so it is likely to be the correct answer. Otherwise, this would be very difficult to solve.

6. If you contruct the net together, you will find that it forms a cube. We can also notice that there are 6 squares shown, and the cube is a 6-sided polyhedron. I have experience with forming paper cubes from my previous math classes, so I can confidently confirm that is a cube. If you still have doubts though, you may also research the net of a cube.

Hopefully this helped you out with your problem! I've answered this based from my own knowledge so please let me know if I misunderstood anything in the questions.

With probability we can predict that, Bennet will select a red marble about 60 times in 200 draws, which corresponds to option C.

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain.

The probability of selecting a red marble on any given draw is the number of red marbles divided by the total number of marbles in the bag:

P(Red) = 15 / (8 + 7 + 15 + 20) = 15 / 50 = 0.3

Since Bennet is replacing the marble after each selection, each draw is independent and has the same probability of selecting a red marble. Therefore, the number of red marbles selected in 200 draws follows a binomial distribution with n = 200 (the number of trials) and p = 0.3 (the probability of success on each trial).

The expected value of the number of red marbles selected can be found using the formula:

E(X) = np

where X is the number of red marbles selected, n is the number of trials, and p is the probability of success on each trial.

Substituting the values we get:

E(X) = 200 * 0.3 = 60

Therefore, we can predict that Bennet will select a red marble about 60 times in 200 draws, which corresponds to option C.

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The standard deviation by taking the square root of the variance:
2 x 0.21 + 3 x 0.37 + 4 x 0.22 + 5 x 0.10 + 6 x 0.07 + 7 x 0.03 = 3.42
So the mean number of defects in the batch is 3.42.

Next, we can calculate the variance
(2-3.42)^2 x 0.21 + (3-3.42)^2 x 0.37 + (4-3.42)^2 x 0.22 + (5-3.42)^2 x 0.10 + (6-3.42)^2 x 0.07 + (7-3.42)^2 x 0.03 = 1.8074 ⇒ √1.8074 = 1.34
Therefore, the closest answer is 1.28.

To find the standard deviation of this variable, we first need to calculate the mean (expected value) and then use the formula for standard deviation.

Mean (Expected value) = Σ (Defects × Probability)
= (2 × 0.21) + (3 × 0.37) + (4 × 0.22) + (5 × 0.10) + (6 × 0.07) + (7 × 0.03)
= 0.42 + 1.11 + 0.88 + 0.50 + 0.42 + 0.21
= 3.54

Next, we find the variance:
Variance = Σ[((Defects - Mean)² × Probability)]
= ( (2-3.54)² × 0.21) + ( (3-3.54)² × 0.37) + ( (4-3.54)² × 0.22) + ( (5-3.54)² × 0.10) + ( (6-3.54)² × 0.07) + ( (7-3.54)² × 0.03)
= 0.477 + 0.273 + 0.094 + 0.211 + 0.168 + 0.103
= 1.326

Now we find the standard deviation by taking the square root of the variance:
Standard Deviation = √(Variance)
= √(1.326)
≈ 1.28
Therefore, the closest answer is 1.28. However, please note that this answer may vary slightly depending on rounding or the level of precision required.

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The volume of the solid formed by rotating the region inside the first quadrant enclosed by y = a°, y = 25 about the r-axis is [tex]V = \pi(25^3 - (a^\circ)^3)[/tex].

To find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = a°, y = 25 about the r-axis, we can use the method of cylindrical shells.

First, we need to determine the limits of integration for y.

The region is enclosed by y = a° and y = 25, so the limits are a° and 25.

Next, we need to determine the radius of each cylindrical shell. Since we are rotating about the r-axis, the radius is simply the y-value.

So, the radius is r = y.

Finally, we need to determine the height of each cylindrical shell.

The height is the circumference of the shell, which is 2πr.

So, the height is h = 2πy.

The volume of each cylindrical shell is then given by V = 2πy * (y - a°)

To find the total volume, we integrate this expression with respect to y from a° to 25:
[tex]V = \int_{a^\circ}^{25} 2\pi (y - a^\circ) dy[/tex]

Evaluating this integral, we get:
[tex]V = \pi(25^3 - (a^\circ)^3)[/tex]

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The vertex form of the quadratic equation is:

y = (x - 3)² - 16

To complete squares we need to use the perfect square trinomial:

(a + b)² = a² + 2ab +b²

The given quadratic can be rewritten as:

y = x² - 2*3*x - 7

We can add and subtract 3²  = 9 to get:

y = x² - 2*3*x + 9 - 9- 7

y = (x - 3)² - 16

That is the quadratic equation in vertex form, wehre the vertex is (3, -16)

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Answer:

16pi/9 in

Step-by-step explanation:

length of arc = (angle/360) x (2πr)

where angle is the central angle of the arc in degrees, r is the radius of the circle, and π is the mathematical constant pi (approximately equal to 3.14159).

In this case, the radius is given as 8 inches and the central angle is 40 degrees. Substituting these values into the formula, we get:

length of arc = (40/360) x (2π x 8)

length of arc = (1/9) x (16π)

length of arc = 16π/9

So the length of the arc is 16π/9 inches. Rounded to the nearest hundredth, this is approximately 5.60 inches. Therefore, the answer is 16 pi 1/9 inches, when expressed in mixed number form.

The integer has 6 positive divisors and the smallest non-negative integer with the given factorization is 12.

We can use the fact that the number of divisors of an integer is equal to the product of one more than the exponent of each prime factor in its prime factorization.

In this case, the prime factorization of our integer is p^2*q, so the exponent of p is 2 and the exponent of q is 1. Therefore, the number of positive divisors is (2+1)*(1+1) = 3*2 = 6. So the integer has 6 positive divisors.

For the second question, we're asked to find the smallest nonnegative integer with the given factorization. Since p and q are unique primes, the smallest possible values for them are 2 and 3 (in some order). If we let p = 2 and q = 3, then the factorization becomes 2^2 * 3 = 12.

So the smallest non-negative integer with the given factorization is 12.

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The total wage bill for hiring 3 students to deliver the catalogues and leaflets will be 3 x £51.30 = £153.90.

Number is an abstract concept used to count, measure and identify a quantity. It is a fundamental part of mathematics, and is also used in many other fields, such as physics and computing. Numbers can represent both discrete, such as the number of people in a room, and continuous, such as the temperature outside. Numbers are also used to represent abstract ideas, such as the amount of money in a bank account.

In order to calculate the minimum number of students needed to deliver the catalogues and leaflets, we need to calculate how many catalogues and leaflets can be delivered in 8 hours by one student.

16 catalogues and 90 leaflets can be delivered in 1 hour by one student. Therefore, 128 catalogues and 720 leaflets can be delivered in 8 hours by one student.

Therefore, the total number of catalogues and leaflets that can be delivered in 8 hours by one student is 848 (128 + 720).

Since the total number of catalogues and leaflets to be delivered is 384 + 1890 = 2274, the minimum number of students required to deliver all the catalogues and leaflets is 3 (2274 / 848 = 2.67).

Therefore, the total wage bill for hiring 3 students to deliver the catalogues and leaflets will be 3 x £51.30 = £153.90.

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The eigenvalues of the 5 x 5 matrix are 0 (multiplicity: 3), 5 (multiplicity: 1), and 9 (multiplicity: 1).

Eigenvectors and eigenvalues have a large number of applications. The word eigenvalue means characteristic value. Therefore, the eigenvalues of the 5 x 5 matrix are represented by the roots of the characteristic polynomial. There is a term called multiplicity of an eigenvalue which is the number of times that a root is repeated.

Now, by algebraic means, we will determine the roots of the characteristic polynomial as follows:

[tex]p = λ^5 - 14λ^4 + 45λ^3\\p = λ^3( λ^2-14λ + 45)\\p = λ^3( λ^2 - 9λ - 5λ + 45)\\p = λ^3. (λ-5). (λ-9)[/tex]

The eigenvalues of the 5 x 5 matrix are 0 (multiplicity: 3), 5 (multiplicity: 1), 9 (multiplicity: 1)

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When there is a problem with Solver being able to find a solution, many times it is an indication of mistakes in the formulation of the problem. This means that the problem may not be correctly defined, or the constraints may not be properly specified.

However, it is also possible that the problem is a nonlinear programming problem, which can be more difficult for Solver to solve. In either case, it is important to carefully review the problem formulation and constraints to ensure that they are correct and accurately represent the problem at hand. It is also important to note that there may be some problems that cannot be solved using Solver or any other optimization tool, due to their inherent complexity or other factors.

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The probability that the proportion of red beads you select from bin b is higher than the proportion of red beads you select from the bin a is 0.005, assuming the proportions of red beads in each bin are 0.3 and 0.6 respectively.

To calculate the probability that the proportion of red beads you select from bin b is higher than the proportion of red beads you select from bin a, you would need to know the actual proportions of red beads in each bin. Let's say in bin a has 100 beads, of which 30 are red, while bin b has 200 beads, of which 60 are red.

To calculate the probability, you would need to use the formula for the probability of the difference between two binomial proportions being greater than zero:

P(p(b) - p(a) > 0) = 1 - P(p(b) - p(a) <= 0)

where p(b) is the proportion of red beads in bin b, and p(a) is the proportion of red beads in the bin a.

Using the binomial distribution, we can calculate the probability of selecting a certain number of red beads from each bin, and then use those probabilities to calculate the probability of the difference between the proportions being greater than zero.

For example, if we randomly select 20 beads from each bin, the probability of selecting 10 or more red beads from bin b is:

P(X >= 10) = 1 - P(X < 10)

where X is the number of red beads selected from bin b.

Using the binomial distribution with n=20 and p=0.3, we get:

P(X >= 10) = 1 - binom.cdf(9, 20, 0.3) = 0.236

Similarly, the probability of selecting 10 or more red beads from bin a is:

P(Y >= 10) = 1 - P(Y < 10)

where Y is the number of red beads selected from bin a.

Using the binomial distribution with n=20 and p=0.6, we get:

P(Y >= 10) = 1 - binom.cdf(9, 20, 0.6) = 0.979

Now, we can calculate the probability that the proportion of red beads selected from bin b is higher than the proportion selected from the bin a:

P(p(b) - p(a) > 0) = P(X >= 10) * (1 - P(Y >= 10))

= 0.236 * (1 - 0.979)

= 0.005

So the probability that the proportion of red beads you select from bin b is higher than the proportion of red beads you select from the bin a is 0.005, assuming the proportions of red beads in each bin are 0.3 and 0.6 respectively.

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The expression t^2+6t-16 is negative in the interval 2≤t≤4 when 0≤t≤6.

To determine the interval where the expression t^2+6t-16 is negative, we need to solve the inequality t^2+6t-16<0. We can do this by factoring the quadratic expression or using the quadratic formula, but it's quicker to notice that the expression can be written as (t+4)(t-2)<0.

This means that the expression is negative when t is between -4 and 2, or when t is between 2 and 4, because the product of two factors is negative when one factor is negative and the other is positive. However, we are only interested in the interval between 0 and 6, so we need to check which of these subintervals satisfy that condition.

The subinterval between -4 and 2 is entirely outside the interval between 0 and 6, so we can ignore it. The subinterval between 2 and 4 is partially inside the interval between 0 and 6, but only the part between 2 and 4 is relevant. Therefore, the expression is negative when 2≤t≤4.

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Therefore, the order-reversed integral is:

∫ba∫dcsin(x^2) dydx, where a= 0, b= 3, c= √(3y), and d= √(9y).

We have:

∫30∫3ysin(x^2) dxdy

To reverse the order of integration, we need to express the limits of integration as inequalities of x and y:

3y ≤ x^2 ≤ 9y

√(3y) ≤ x ≤ √(9y)

0 ≤ y ≤ 1

So, we have:

∫30∫√(9y)√(3y)sin(x^2) dxdy

Integrating with respect to x first, we get:

∫√(9y)√(3y) [-cos(x^2)/2] |_0^(√(3y)) dy

= ∫30 [-cos(3y)/2 + cos(y)/2] dy

= [-sin(3y)/6 + sin(y)/2] |_0^3

= (-sin(9)/6 + sin(3)/2) - (0 - 0)

= (-sin(9)/6 + sin(3)/2)

Therefore, the order-reversed integral is:

∫ba∫dcsin(x^2) dydx, where a= 0, b= 3, c= √(3y), and d= √(9y).

Note: We can also check the answer by evaluating the original integral and comparing it with the answer obtained by reversing the order of integration.

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[tex]log1/9^{(6)}1/2[/tex] can be expressed in terms of p as -6p.

Given [tex]log6^3=p[/tex]. Express[tex]log1/9^{(6)}1/2[/tex] in terms of p

We can use the logarithmic identity that states:

[tex]loga(b^c) = c \times loga(b)[/tex]

Using this identity, we can rewrite log6^3 as:

[tex]log6^3 = 3 \times log6[/tex]

Since [tex]log1/9^{(6)}1/2[/tex] can be rewritten as [tex]log(1/9^{(1/2))}^{6}[/tex], we can apply the same identity to obtain:

[tex]log(1/9^{(1/2)})^6 = 6\times log(1/9^(1/2))[/tex]

Now we need to express [tex]log(1/9^{(1/2)})[/tex] in terms of p.

[tex]Since 1/9^{(1/2)} = 1/(3^2)^{(1/2)} = 1/3[/tex], we can rewrite [tex]log(1/9^{(1/2)})[/tex] as:

[tex]log(1/9^{(1/2)}) = log(1/3) = -log(3)[/tex]

Therefore, we can express[tex]log(1/9^{(1/2)})^6[/tex]in terms of p as:

[tex]log(1/9^{(1/2)})^6 = 6log(1/9^{(1/2)}) = 6(-log(3)) = -6 \times log(3)[/tex]

Finally, substituting the value of p we obtained earlier, we have:

[tex]log(1/9^{(1/2)})^6 = -6log(3) = -6p[/tex]

Therefore, [tex]log1/9^{(6)}1/2[/tex] can be expressed in terms of p as -6p.

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The horizontal translations of exponential functions only affect the range of the function. The domain remains the same, and the asymptote is unaffected.

The domain of the function still persists as the same – all real numbers. When it comes to range, its variation may occur due to a horizontal shift. If C > 0, then the range will offset upwards by the value of C units, whereas if C < 0, then the range shifts downwards with the magnitude of the shift having no influence on the contour of the function.

Furthermore, exponential functions have a fixed horizontal asymptote at y =0, that is not affected by any horizontal translations and which overtly remains equal to its original function.

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After the second outer loop iteration, the list is [6,1,7,8,9].

To determine the list after the second outer loop iteration, let's assume we're working with a simple bubble sort algorithm. Here are the steps:
1. First outer loop iteration:
  - Compare 6 and 9; no swap.
  - Compare 9 and 8; swap to get [6,8,9,1,7].
  - Compare 9 and 1; swap to get [6,8,1,9,7].
  - Compare 9 and 7; swap to get [6,8,1,7,9].
2. Second outer loop iteration:
  - Compare 6 and 8; no swap.
  - Compare 8 and 1; swap to get [6,1,8,7,9].
  - Compare 8 and 7; swap to get [6,1,7,8,9].

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a)The price in dollars he paid for each bowl is given by: 32/x.

b) his proposed selling price for each bowl was $ (64+3x)/2x

c) the sum of money received is 5 × (64 + 3x)/x

d)The sum of money received for them is 2x - 20

e) this equation is 13x² - 655x + 1280 = 0

f) the number of bowls bought by the stallholder is 40.

In mathematics, a fraction is a number that represents a part of a whole or a ratio of two quantities. A fraction is written in the form of a numerator and a denominator separated by a line, where the numerator represents the number of parts being considered and the denominator represents the total number of equal parts in the whole.

(a) The price in dollars he paid for each bowl is given by: 32/x.

(b) The proposed selling price for each bowl at a profit of $1.50 is the cost price plus the profit, which is:

32/x + 1.50

= (32 + 1.50x)/x

Multiplying both numerator and denominator by 2, we get:

= (64 + 3x)/2x

(c) For 10 bowls sold at this price, the sum of money received is:

10 × (64 + 3x)/2x

= 5 × (64 + 3x)/x

(d) The remaining bowls were sold at $2 each, so the sum of money received for them is:

(x - 10) × 2

= 2x - 20

(e) The total amount of money received by the stallholder is the sum of money received for the 10 bowls and the remaining bowls, which is given by:

5 × (64 + 3x)/x + 2x - 20

Simplifying this expression, we get:

(13x² - 572x + 1280)/x

Since the total amount received is $83, we have the equation:

(13x^2 - 572x + 1280)/x = 83

Multiplying both sides by x, we get:

13x²- 572x + 1280 = 83x

Simplifying this equation, we get:

13x² - 655x + 1280 = 0

(f) Solving the quadratic equation using the quadratic formula, we get:

x = (655 ± √(655² - 4 × 13 × 1280)) / (2 × 13)

x ≈ 40.31 or x ≈ 12.38

Since the number of bowls must be a whole number, the solution is x = 40. Substituting x = 40 into the expression for the selling price, we get:

(64 + 3x)/2x = (64 + 3(40))/2(40) = 2.15

Therefore, the number of bowls bought by the stallholder is 40.

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To solve a congruence like 5x ≡ 11 (mod 37), we need to find the modular inverse of 5 (mod 37) which is 15, then x ≡ 11(15) ≡ 27 (mod 37). Similarly, to solve 11y ≡ 5 (mod 37), we find the modular inverse of 11 (mod 37) which is 20, then y ≡ 5(20) ≡ 24 (mod 37).

The first step is to find the value of xy (mod 37). To do this, we simply multiply x and y and take the result modulo 37. For example, xy ≡ 27(24) ≡ 648 ≡ 11 (mod 37).

To solve a congruence like 5x ≡ 11 (mod 37), we need to find a number y such that 5y ≡ 1 (mod 37), since then we can multiply both sides of the congruence by y to get x ≡ 11y (mod 37). To find y, we use the extended Euclidean algorithm, which involves finding the greatest common divisor of 5 and 37 and expressing it as a linear combination of 5 and 37. In this case, we find that y = 15 is the modular inverse of 5 (mod 37), since 5(15) ≡ 1 (mod 37). Therefore, x ≡ 11(15) ≡ 27 (mod 37).

Similarly, to solve the congruence 11y ≡ 5 (mod 37), we need to find a number x such that 11x ≡ 1 (mod 37), since then we can multiply both sides of the congruence by x to get y ≡ 5x (mod 37). Using the extended Euclidean algorithm, we can find that x = 20 is the modular inverse of 11 (mod 37), since 11(20) ≡ 1 (mod 37). Therefore, y ≡ 5(20) ≡ 24 (mod 37).

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The area of q is 20.

The area of a parallelogram determined by two vectors u and v is given by the magnitude of the cross product of u and v: |u x v|.

So, the area of the parallelogram p is:

| [4;1] x [3;-1] | = |(4)(-1) - (1)(3)| = |-7| = 7

To find the area of q, we apply the transformation T to each of the vertices of p and then compute the area of the resulting parallelogram.

First, we find the images of the vertices of p under T:

T([4;1]) = [3 1;1 2][4;1] = [16;6]
T([3;-1]) = [3 1;1 2][3;-1] = [6;1]

The sides of the parallelogram q are determined by the vectors T([4;1]) - T([3;-1]) = [10;5] and T([3;-1]) - [0;0] = [6;1].

The area of q is the magnitude of the cross product of these vectors:

| [10;5] x [6;1] | = |(10)(1) - (5)(6)| = |-20| = 20

Therefore, the area of q is 20.

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