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The arc length of the sector is 10π inches, and the area of the sector is 60π square inches.

What is the arc length and area of the sector of a circle?

Arc length = [tex] \frac{ \theta}{ {360}^{o} } \times 2\pi \: r[/tex]

Area of sector = [tex] \frac{ \theta}{ {360}^{o} } \times \pi \: {r}^{2} [/tex]

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

Given that the angle is 150° and the radius is 12 inches, we can substitute these values into the formulas to get:

Arc length

[tex]= \frac{150}{360} \times 2π(12) \\ = \frac{5}{12} \times 2π(12) \\ = 10π \: inches[/tex]

Area of sector

[tex]= \frac{150}{360} \times π(12)^2 \\ = \frac{5}{12} \times π(144) \\ = 60π \: square \: inches[/tex]

Therefore, the arc length of the sector is 10π inches, and the area of the sector is 60π square inches.

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The solution to the system of equations is x = 5 and y = 3.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and operators (such as addition, subtraction, multiplication, and division) that are used to represent quantities and their relationships.

To solve the system using addition, we want to add the two equations together in a way that eliminates one of the variables. In this case, we can add the two equations directly because the x term will cancel out:

X - 4.7y = -9.1

+(-x + 5.8 = 12.4)

1.1y = 3.3

Now we can solve for y by dividing both sides by 1.1:

y = 3.3 ÷ 1.1

y = 3

Substituting y = 3 back into one of the original equations, we can solve for x:

x - 4.7y = -9.1

x - 4.7(3) = -9.1

x - 14.1 = -9.1

x = 5

Therefore, the solution to the system of equations is x = 5 and y = 3.

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The mean of the sampling distribution of sample means is approximately 5.2 ± 0.1339.

When a sampling distribution is created using samples of the ages at which 48 children begin reading, the mean of the sampling distribution of sample means can be calculated by dividing the population mean by the square root of the sample size. This is known as the standard error of the mean and is denoted as SE.

The formula for calculating the standard error of the mean is:

SE = σ / √n

where σ is the population standard deviation and n is the sample size.In this case, the sample size is 48 children. Since the population standard deviation is not given, it is assumed that the sample standard deviation is used as an estimate of the population standard deviation. The formula for calculating the sample standard deviation is:

S = √[(Σ(x - μ)²) / (n - 1)]

where S is the sample standard deviation, Σ is the sum of the squared deviations from the mean, x is the value of each observation, μ is the mean of the sample, and n is the sample size.Using the data given, the sum of the squared deviations from the mean is:

Σ(x - μ)² = (6 - 5.2)² + (8 - 5.2)² + (7 - 5.2)² + (9 - 5.2)² + (4 - 5.2)² + (7 - 5.2)² + (6 - 5.2)² + (5 - 5.2)² = 2.56 + 5.76 + 1.44 + 12.96 + 1.44 + 1.44 + 0.36 + 0.04 = 26.4

The sample standard deviation is:S = √[26.4 / (48 - 1)] = 0.9197Using the formula for standard error of the mean, the mean of the sampling distribution of sample means is:

SE = σ / √n= 0.9197 / √48≈ 0.1339

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With D as Center of dilation, the scale factor is √2.

In mathematics, a scaling factor is a ratio that describes how much a figure or object has been enlarged or reduced in size. A scaling factor is typically expressed as a fraction or decimal that represents the relative change in size between the original figure and the new, scaled figure.

Now,

To find the scale factor of dilation, we can use the distance formula to find the ratio of corresponding sides in the original and dilated triangles.

The distance between points A and B in the original triangle is:

[tex]\rm AB = \sqrt{(x_2 - x_1)\² + (y_2 - y_1)\²}[/tex]

[tex]= \sqrt{(0 - (-1))\² + (1 - (-1))\²} \\\\= \sqrt{1\² + 2\²}[/tex]

= √5

The distance between points A' and B' in the dilated triangle is:

[tex]\rm A'B' = \sqrt{(x_2' - x_1')\² + (y_2' - y_1')\²}[/tex]

[tex]= \sqrt{(-2 - (-1))\² + (-5 - (-2))\²}[/tex]

[tex]= \sqrt{1\² + 3\²}[/tex]

= √10

Therefore, the scale factor for the dilation is:

Scale factor = A'B' / AB = √10 / √5 = √2

Hence, the scale factor of dilation is √2.

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The solution to the system of equations is x = -1, y = 2.

A linear equation is a first-order (linear) term and a constant in an algebraic equation of the type y=mx+b, where m is the slope and b is the y-intercept. The previous equation, which has the variables y and x, is sometimes referred to as a "linear equation of two variables."

The adjective "linear" comes from the fact that the collection of solutions to such an equation forms a straight line in the plane.

These are the three types of linear equations:

By multiplying each equation by an appropriate constant, we may use the method of elimination to make the coefficients of x in both equations equal in size but opposite in sign. The following results from multiplying the first equation by 3 and the second equation by -2 in this situation:

a. 6x + 12y = 18

b. -6x - 10y = -14

To get rid of x, we can now combine these two equations:

a. 6x + 12y = 18

b. (-6x - 10y = -14)

2y = 4

By finding y, we obtain:

y = 2

We can find x by adding this value of y back into either of the initial equations:

2x + 4y = 6

2x + 4(2) = 6

2x + 8 = 6

2x = -2 x ⇒ -1

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the solution to the system of equations 2x+4y=6 and 3x+5y=7, after eliminating the x term, is x = -1 and y = 2.

The x term, we can multiply the first equation by -3 and the second equation by 2, so that the x term will have opposite coefficients and will cancel out when we add the two equations together.

[tex]-3(2x+4y=6) gives -6x - 12y = -18[/tex]

[tex]2(3x+5y=7) gives 6x + 10y = 14[/tex]

Adding these two equations gives:

[tex]-6x - 12y = -18[/tex]

[tex]+6x + 10y = 14[/tex]

[tex]-2y = -4[/tex]

Solving for y, we get:

[tex]y = 2[/tex]

Substituting this value of y back into either of the original equations, we can solve for x:

[tex]2x + 4y = 6[/tex]

[tex]2x + 8 = 6[/tex]

[tex]2x = -2[/tex]

[tex]x = -1[/tex]

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The greatest common factor of 36x²y and 54xy²z is 18xy

The greatest common factor (GCF) of a set of numbers or polynomial is the largest factor that all the numbers/polynomial share.

Hence, let's find the greatest common factor of 36x²y and 54xy²z as follows:

Hence, let's find the greatest common factor of a set of polynomials which is the largest positive integer/variables that divides evenly into all numbers with zero as the remainder.

36x²y and 54xy²z

Therefore,

GCF = 18xy

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Step-by-step explanation:

Acute angle means less than 90 degrees....so it is in the first Quadrant and sin and cos will be positive values

Trig identity

sin^2 + cos^2 = 1

sin^2 + .6^2 = 1    shows sin = .8

The tan  =   sin/ cos =  .8/.6 = 1.33

After answering the presented question, we can conclude that expression Therefore, the value of 5310² - 5330 × 5290 when p = 5330 and q = 10 is 400.

An expression in mathematics is a collection of representations, digits, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a combination of the two can be used as an expression. Mathematical operators include addition, subtraction, rapid spread, division, and exponentiation. Expressions are often used in arithmetic, mathematics, and form. They are employed in the representation of mathematical formulas, the solving of equations, and the simplification of mathematical relationships.

(i)

(p - 2q)² = p² - 4pq + 4q²

p(p - 4q) = p² - 4pq

(p - 2q)² - p(p - 4q) = (p² - 4pq + 4q²) - (p² - 4pq)

(p - 2q)² - p(p - 4q) = 4q²

(ii)

(5330 - 2×10)² - 5330(5330 - 4×10) = 4×10²

(5310)² - 5330 × 5290 = 400

Therefore, the value of 5310² - 5330 × 5290 when p = 5330 and q = 10 is 400.

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The estimated amount Franklin will pay in finance charges on his $1,520 purchase is $35.07.

What is annual percentage rate?

Annual Percentage Rate (APR) is the annual rate charged for borrowing or earned through an investment and is expressed as a percentage that represents the actual yearly cost of funds over the term of a loan or investment.

The finance charges on a credit card purchase can be calculated using the following formula:

finance charges = (APR/365) * balance * number of days

Where APR is the annual percentage rate, balance is the amount of the purchase, and the number of days is the time period for which the finance charges are being calculated.

Assuming that Franklin does not make any payments towards his credit card balance during the billing period, the number of days for which finance charges will be calculated will be equal to the billing cycle. Let's assume a billing cycle of 30 days for this example.

Using the formula above, we can calculate the finance charges as follows:

finance charges = (28.2%/365) * $1,520 * 30

= (0.00077397) * $1,520 * 30

= $35.07 (rounded to the nearest cent)

Therefore, the estimated amount Franklin will pay in finance charges on his $1,520 purchase is $35.07.

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Hence, a cylinder with height h and base area [tex]\pi r^{2}[/tex] may be proven to have the same volume as a triangular prism.

Two parallel circular bases joined by a curving surface form the three-dimensional geometric object known as a cylinder. Due of its parallel and congruent bases, it is a sort of prism.

When someone uses the word "cylinder,” they often mean a right circular cylinder with circles for bases and an axis that is perpendicular to the bases' planes.

A cylinder's volume may be calculated using the formula V = [tex]r^{2}[/tex]h, where r denotes the perimeter of the base and h the height of the cylinder. L = 2rh, where r is the radius of a base and h is the height of the cylinder, is the formula for a cylinder's lateral surface area.

Cavalieri's principle states that if two solid objects have the same height and every cross-section taken perpendicular to a common axis has the same area, then the two objects have the same volume.

Let's consider the given triangular prism with base area  [tex]\pi[/tex][tex]r^2[/tex] and height h. If we take a cross-section perpendicular to the base at a height y, we get a circle with radius r multiplied by a triangle with base 2r and height y. The area of this cross-section is:

[tex]A(y) = \pi r^2 + (2r)(y) / 2[/tex]

Simplifying, we get:

[tex]A(y) = \pi r^2 + ry[/tex]

Now, let's consider a cylinder with base area pi r² and height h. If we take a cross-section perpendicular to the height at a height y, we get a circle with radius r multiplied by a rectangle with base 2r and height h. The area of this cross-section is:

[tex]A'(y) = \pi r^ + (2r)(h) = \pi r^2 + 2rh[/tex]

By Cavalieri's principle, the triangular prism and the cylinder have the same volume if A(y) = A'(y) for all y from 0 to h. Let's check:

[tex]\pi r^² + ry = \pi r^² + 2rh[/tex]

[tex]ry = 2rh[/tex]

[tex]y = 2r[/tex]

So, we see that the areas are equal for all y between 0 and h, except at [tex]y = 2r.[/tex] However, this is just a single point and has no effect on the volume,

Therefore,  the cylinder with base area [tex]\pi[/tex][tex]r²[/tex] and height h has the same volume as the triangular prism, with base area [tex]\pi[/tex]r² and height h.

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The probability of selecting a blank marble on the first draw and an orange marble on the second draw is:

[tex](bo)/[(b+o)(b+o-1)][/tex]

Let B be the event that the first marble selected is blank and let O be the event that the second marble selected is orange. Since we are not replacing the marbles after each selection, the probability of the second marble being orange depends on the outcome of the first selection.

Let's assume that the bag contains b blank marbles and o orange marbles. The probability of selecting a blank marble on the first draw is b/(b+o), and the probability of selecting an orange marble on the second draw, given that the first marble was blank, is o/(b+o-1) (since there is one less marble in the bag after the first draw).

Therefore, the probability of selecting a blank marble on the first draw and an orange marble on the second draw is:

[tex]P(B\ and\ O )= P(B)*P(O|B)\\\\=b/(b+o)*o/(b+o-1)\\\\=(bo)/[(b+o)(b+o-1)][/tex]

Note that this assumes that there are no marbles that are both blank and orange. If there are such marbles in the bag, the calculation will be different.

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Dr. Yanira's main issue is that basing her request on a single observation of a private school with the highest per-capita income may not be representative of the broader population of schools in the county, and the observation could be biased and not provide a true representation of the overall situation.

The main issue with Dr. Yanira basing her request for budget, school supplies, and school lunches, among other things, on a single observation of a private school with the highest per-capita income in the area is that the results may not be representative of the broader population of schools in the county.

The sample of one school may not be statistically significant or diverse enough to draw valid conclusions about the entire population of schools. The observation could be biased and may not provide a true representation of the overall situation. Therefore, it is important to use a larger and more diverse sample to draw more reliable conclusions.

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a. The probability that the amount of collagen is greater than 60 grams per mililiter is 0.5398

b. The probability that the amount of collagen is less than 90 grams per mililiter is 0.9985.

c. Approximately 68% of the data falls within 1 standard deviation of the mean for a normally distributed dataset.

a. To find the probability that the amount of collagen is greater than 60 grams per milliliter, we'll use the z-score formula:
z = (X - μ) / σ
where X is the value we're interested in, μ is the mean, and σ is the standard deviation.
z = (60 - 61) / 9.8 ≈ -0.1
Now, we'll use a z-table or calculator to find the probability corresponding to this z-score.

A z-score of -0.1 corresponds to a probability of approximately 0.4602.

Since we're looking for the probability that the amount of collagen is greater than 60 grams per milliliter, we need to find the area to the right of this z-score:
P(X > 60) = 1 - P(X ≤ 60) = 1 - 0.4602 ≈ 0.5398
b. To find the probability that the amount of collagen is less than 90 grams per milliliter, we'll again use the z-score formula:
z = (90 - 61) / 9.8 ≈ 2.96
A z-score of 2.96 corresponds to a probability of approximately 0.9985.

Since we're looking for the probability that the amount of collagen is less than 90 grams per milliliter, we'll use this probability directly:
P(X < 90) ≈ 0.9985
c. To find the percentage of compounds formed from the extract of this plant that fall within 1 standard deviation of the mean, we'll use the empirical rule (68-95-99.7 rule), which states that approximately 68% of the data falls within 1 standard deviation of the mean for a normally distributed dataset.

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Answer:I think the answer should be 22 because you subtract 27 the largest number from the smallest number which is 5 which gives you 22

Step-by-step explanation:

1) 27-5=22

After factorization we get:

14. (m²+ 4)(m + 2)(m - 2).

15. (x + 2y + 2z)(x + 2y - 2z).

16. m²(m + 1)(m - 1) - 12.

17. (2x + 3)² - 3(2x + 3).

Factorization is the process of breaking down a mathematical expression, equation, or number into its constituent parts or factors. In algebra, factorization involves finding the factors of a polynomial expression, which are the individual terms that can be multiplied together to yield the original polynomial.

14. To factor m⁴ - 16, we can use the difference of squares formula, which states that a² - b² = (a + b)(a - b). In this case, we can rewrite the expression as (m²)² - 4², and use the formula:

m⁴ - 16 = (m² + 4)(m² - 4)

Now, we can use the difference of squares formula again to factor m^2 - 4:

m⁴ - 16 = (m²+ 4)(m + 2)(m - 2)

15. x² + 4xy + 4y² - 4z² = (x + 2y)² - 4z²

(x + 2y)² - 4z² = (x + 2y + 2z)(x + 2y - 2z)

Therefore, the factored form of x² + 4xy + 4y² - 4z² is (x + 2y + 2z)(x + 2y - 2z).

16. To factor m⁴ + m² - 12, we can start by factoring out m^2:

m⁴ + m² - 12 = m²(m² + 1) - 12

Now we have a difference of squares in the parentheses, which we can factor using the formula a² + b² = (a + b)(a - b):

m²(m² + 1) - 12 = m²(m + 1)(m - 1) - 12

Therefore, the factored form of m⁴ + m² - 12 is m²(m + 1)(m - 1) - 12.

17. To factor (2x + 3)² + 2(2x + 3) - 15, we can recognize that it is a quadratic expression in the form of ax² + bx + c. We can first simplify the expression by using the distributive property:

(2x + 3)² + 2(2x + 3) - 15 = (2x + 3)(2x + 3) + 4x + 6 - 15

= (2x + 3)² + 4x - 9

Now, we can factor out the common factor of (2x + 3)² :

(2x + 3)² + 4x - 9 = (2x + 3)²+ 4x - 12x - 9

= (2x + 3)² - 8x - 9

= (2x + 3)² - (3)(3 + 2x)

Finally, we can factor out the common factor of (3 + 2x) from the two terms:

(2x + 3)² - (3)(3 + 2x) = (2x + 3)² - (3 + 2x)(3)

= (2x + 3)² - 3(2x + 3).

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A function which could be used to determine the total amount of revenue the company earns from selling moving boxes include the following: B. r(x) = -49x² + 567x.

From the information provided, the amount of revenue with respect to price that's being generated in this scenario can be calculated by using the following function (equation):

R(x) = x × P(x)

Where:

Since it is a revenue function, we would simply substitute the value of the unit price and evaluate as follows:

Revenue, R(x) = x × P(x)

P(0) = 567

By substituting the given parameters into the formula, we have;

Revenue, R(x) = x × (567 - 49x)

Revenue, R(x) = -49x² + 567x

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The proportion of green lights is 0.4286.

(a) Yes, this situation is a Markov chain since the probability of transitioning to a new traffic light state only depends on the current state.

(b) Transition probability matrix:

G Y R

[tex]G 0.60 0.00 0.40[/tex]

[tex]Y 1.00 0.00 0.00[/tex]

[tex]R 0.30 0.00 0.50[/tex]

(c) There is only one class, as all states can be reached from any other state.

(d) After three more lights, the probability of being in each state is:

G Y R

G 0.3240 0.0000 0.6760

Y 0.5000 0.0000 0.5000

R 0.3150 0.0000 0.6850

(e) The proportion of green lights can be found by solving the steady-state equation. Assuming there are n traffic lights, the system of equations would be:

[tex]0.60g + 0.30r[/tex]= g

[tex]0.00g + 0.00r[/tex] = r

[tex]0.40g + 0.50r[/tex] = r

g + y + r = 1

Solving this system gives g =[tex]0.4286[/tex], y = [tex]0.1429[/tex], and r = [tex]0.4286.[/tex]

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The profit for the new container would be 3.47 each.

The gain from any business operation is referred to as profit. Every time a merchant sells a product, his goal is to make

a profit by getting something from the customer. In other words, if he sells the goods for more than the cost price, he

makes a profit; yet, if he needs to sell them for less, he loses money.

To calculate the profit for the new container, you need to follow these steps:

Determine the selling price. The marketing team suggests that the proposed container will sell for 3.97.

Determine the additional cost to make the new container. It will cost an additional 0.50 each to make.

Calculate the profit. Subtract the additional cost from the selling price to find the profit for the new container.

Profit = Selling Price - Additional Cost

Profit = 3.97 - 0.50

Profit = 3.47

So, the profit for the new container would be 3.47 each.

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Answer:

25 + 55 + 45 + 10 = 135 runners.

B is correct.

Answer:

The roots of p(z) are z = i, 2.91367 + 0.600499i, 1.22097 - 0.362212i, 0.306704 + 1.53771i.

Answer:

The answer is $45

Step-by-step explanation:

$60 is dicounted at 25% off

25% percent of 60 is 15

60 - 15 = 45

To multiply two numbers in scientific notation, we multiply their coefficients and add their exponents. So:

(0.4x10^-6)(0.7x10^-2) = (0.4)(0.7)x10^(-6-2) = 0.28x10^-8

Therefore, the answer is 0.28 x 10^−8 in scientific notation.

According to the given question quotient of a given equation [tex](x^{3} +x^{2} +x+2)/(x^{2} -1)[/tex] is [tex]x[/tex] [tex]+ (3x+2)/ (x^{2} -1)[/tex]

When the roots and solutions of two equations coincide, the two equations are compared.

The same quantity, symbol, or expression must be added to or subtracted from both of the equation's two sides in order to produce an equivalent equation.

By multiplying or dividing each side of such an equation by a nonzero number, we can also produce a comparable equation.

Given

[tex](x^{3} +x^{2} +x+2)/(x^{2} -1)[/tex] by using polynomial long devision

After dividing we get

[tex]x+ (3x+2)/ (x^{2} -1)[/tex]

Therefore the given question quotient of a given equation [tex](x^{3} +x^{2} +x+2)/(x^{2} -1)[/tex] is [tex]x[/tex] [tex]+ (3x+2)/ (x^{2} -1)[/tex]

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Answer:

Step-by-step explanation:

This can be solved using Pythagorean Theorem.

a² + b² = c²

8383²  +  b²   = 158158²

70274689 + b² = 25013952964

70274689 - 70274689 + b² = 25013952964 - 70274689

    b² = 24943678275

√b²  = √24943678275

b = 157935.7 cm

The solution to the operations of similar triangles are:

ΔABC is similar to ΔEDC

They are similar triangles by AAA Similarity rule

x = 24

Similar triangles are defined as triangles that have the same shape, but their sizes may vary. Thus, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

From the given image, we can see that:

AB is parallel to DE

Thus:

∠BAE ≅ ∠DCA because they are both alternate angles

Similarly:

∠ABD ≅ ∠EDB because they are both alternate angles

We can also say that:

∠ACB ≅ ∠DCE because they are both opposite angles

Thus: ΔABC is similar to ΔEDC

They are similar triangles by AAA Similarity rule

Using similar triangles ratio, we can say that:

x/15 =  16/10

x = (16 * 15)/10

x = 24

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The area of the parallelogram is 140 cm².

A parallelogram is a quadrilateral having two pairs of parallel sides (a four-sided polygon). This indicates that opposing sides of a parallelogram are parallel, and opposite angles have the same size.

Given :

Base of parallelogram = 10 cm

Height of parallelogram = 14 cm

Area of Parallelogram = B × H  

Area = 10 × 14

Area = 140 cm²

Area of parallelogram is 140 cm².

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The complete question is:

x=13, no idea what y is.