3. Factor 72x³ +72x² +18x.

Answer:

Step-by-step explanation:

We can find dy/dx by differentiating y with respect to x using the quotient rule.

First, we need to rewrite y using the trigonometric identity for the tangent of the difference of two angles:

y = (cos x - sin x)/(cos x + sin x) = [(cos x - sin x)/(cos x + sin x)] * [(cos x - sin x)/(cos x - sin x)]

y = (cos^2 x - 2cos x sin x + sin^2 x)/(cos^2 x - 2sin x cos x + sin^2 x)

y = (cos 2x - sin 2x)/(cos 2x + sin 2x)

Now we can apply the quotient rule:

dy/dx = [(-sin 2x - cos 2x)(cos 2x + sin 2x) - (cos 2x - sin 2x)(-sin 2x + cos 2x)]/(cos 2x + sin 2x)^2

dy/dx = (-sin^2 2x - cos^2 2x - 2sin 2x cos 2x + sin^2 2x + cos^2 2x + 2sin 2x cos 2x)/(cos 2x + sin 2x)^2

dy/dx = 0/(cos 2x + sin 2x)^2

Therefore, dy/dx = 0.

The ordered pair that respect the given conditions are: (6,-12) and (-12,6).

A system of equations is the given term of math for two or more equations with the same variables. The solution of these equations represents the point of the intersection.

You can solve a system of equations by substitution or adding methods. In the addition method, you eliminate a variable, on the other hand, in the substitution method you replace a variable for the other.

You should convert the text of the question into equations. See below.

From equation 1, you have x=-72/y. Thus, applying the substitution method, you can solve this question by following the steps below:

1) Replace  x=-72/y into equation 2. Then, you have:

[tex]\frac{-72}{y} +y=-6\\ \\[/tex]

-72+y²=-6y

y²+6y-72=0

2) Solving the quadratic equation y²+6y-72=0 for finding y:

Δ=b²-4ac

Δ=6²-4*1*(-72)

Δ=36+288

Δ=324

[tex]y=\frac{-b\pm \sqrt{\Delta} }{2a} =\frac{-6\pm \sqrt{324} }{2*1}=\frac{-6\pm18 }{2}[/tex]. Therefore,

y1=(-6-18)/2=-12

or

y2=(-6+18)/2=12/2=6

3) Finding x

If  x=-72/y and y=-12 or y=6. You have

For y=-12, x=-72/-12, thus x=6

For y=6, x=-72/6, thus x=-12

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The correct option is D. $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.

In math, addition is the process of adding two or more integers together. The numbers having added are known as addends, while the outcome of the addition process, or the final response, is known as the sum. It is among the most fundamental mathematical operations we employ on a daily basis.

Quarterly profit for Mr. Brown's Goodwill Store in 2012 (in dollars)

1 $9,841.28

2 $8,957.67

3 $7,429.84

4 $11,095.67

Total profit = profit of 3rd quarter + profit of 4th quarter

Total profit = $7,429.84 + $11,095.67

Total profit =  $18,525.51.

Thus, $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.

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Answer:

C

Step-by-step explanation:

The equations that define y as a nonlinear function of x are: C. y = 5/x and E. y = 3x² - 2

In mathematics, a function is a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. It is a rule or a set of rules that assigns each input value exactly one output value. Functions can be represented using equations, graphs, or tables. They are used to model real-world phenomena and solve problems in various fields such as science, engineering, economics, and finance.

Here,

A function is considered nonlinear if it does not have a constant rate of change or a straight-line graph. In other words, if the function does not follow a linear pattern.

A. y = -3/4x is a linear function because it has a constant rate of change of -3/4 and a straight-line graph.

B. y = 1/2x - 8 is a linear function because it has a constant rate of change of 1/2 and a straight-line graph.

C. y = 5/x is a nonlinear function because it does not have a constant rate of change and its graph is not a straight line. It is a hyperbola.

D. y = -3 + 2x is a linear function because it has a constant rate of change of 2 and a straight-line graph.

E. y = 3x² - 2 is a nonlinear function because it does not have a constant rate of change and its graph is a parabola.

Option A and D are linear functions, while option B is a linear function with a constant term.

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Answer:

3, 4, 5

Step-by-step explanation:

Take any odd number and square it.

3²=9

Divide that number by 2.

9/2=4.5

The original number (3 in this case), as well as the numbers 0.5 above and below the new number (4.5 in this case), are your set of numbers.

3, 4, 5

Hope that helps :)

There are 3 seniors who take Math and Spanish, but not Physics.

What is Venn Diagram?

A diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set), common elements of the sets being represented by the areas of overlap among the circles.

To solve this problem, we can use a Venn diagram to organize the information given about the students.

First, we know that there are 150 seniors in total, so we can label the outer rectangle of the Venn diagram as having a total of 150 students.

Next, we can label the three circles as Math, Spanish, and Physics, with the numbers given in the problem:

80 seniors take Math, so we label the Math circle with 80.

41 seniors take Spanish, so we label the Spanish circle with 41.

54 seniors take Physics, so we label the Physics circle with 54.

We also know that 10 seniors take Math and Spanish, 19 take Math and Physics, 12 take Physics and Spanish, and 7 take all three. We can label these overlaps in the Venn diagram as follows:

The overlap of Math and Spanish is 10.

The overlap of Math and Physics is 19.

The overlap of Physics and Spanish is 12.

The overlap of all three is 7.

To find the number of seniors who take Math and Spanish, but not Physics, we need to subtract the number of seniors who take all three (because they are counted in all three circles) and the number of seniors who take Math, Spanish, and Physics (because they are counted in the overlap of all three circles). Then we add the number of seniors who take only Math and Spanish (because they are not counted in the Physics circle).

Using the formula for the number of elements in the union of three sets, we have:

Math or Spanish or Physics = Math + Spanish + Physics - (Math and Spanish) - (Math and Physics) - (Spanish and Physics) + (Math and Spanish and Physics)

So, the number of seniors who take Math and Spanish, but not Physics is:

Math and Spanish - (Math and Spanish and Physics) + (Math or Spanish or Physics - (Math + Spanish + Physics) + (Math and Spanish and Physics))

= 10 - 7 + (80 + 41 + 54 - 2(10) - 2(19) - 2(12) + 7)

= 3

Therefore, there are 3 seniors who take Math and Spanish, but not Physics.

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The ratio relationship of a car's fuel efficiency can be expressed as equivalent rates of miles per gallon and gallons per mile, or as a unit rate of either 30 miles per gallon or 1/30 gallons per mile.

A real-world situation involving a ratio relationship could be the fuel efficiency of a car. For example, suppose a car travels 60 miles using 2 gallons of gas.

To express this ratio relationship as equivalent rates, we can write:

60 miles ÷ 2 gallons = 30 miles per gallon

2 gallons ÷ 60 miles = 0.0333 gallons per mile

To express the ratio as a unit rate, we can divide the numerator and denominator by the same amount to simplify the fraction. Let's choose to divide by 2 to get:

60 miles ÷ 2 gallons = 30 miles per gallon

2 gallons ÷ 60 miles = 1/30 gallons per mile

In both cases, the ratio represents the same relationship between miles traveled and gallons of gas used, but it is expressed in different units or rates.

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The x-intercept is (3.2, 0), and the y-intercept is (0, -32) if the equation of the line is y = 10x - 32.

Answer:  x= 3.2

Step-by-step explanation:

by graphing we can see where the line crosses the x axis at 3.2

I attached a graph to show you this method

to solve algebraically, y = 10x - 32 is your equation

put this in standard form ax + by = c

-10x + y = -32

to solve for x, substitute 0 for y and solve

-10x + 0 = -32

divide both sides by -10

-10/-10 x = -32/ -10

x = 3.2

Answer:

10 in.

Step-by-step explanation:

V = LWH

100 = L × 2.5 × 4

L = 10

The function for the value of the account after t years can be written as:

V(t) = 93000(1.08900365)t

The coefficient of 1.08900365 is the annual growth rate (APR) compounded daily. After rounding to four decimal places, it becomes 1.0890.

The percentage of growth per year (APY) is 8.90%. This is the same as APR, but expressed as a percentage.

4775 g968r648 747474874 483892874 23773259635y84b2375789325 7437594365825 4378574937587 49388959365n 98437858746587 32o4iy548569

Answer:

?

Step-by-step explanation:

The given statement 'to calculate the average of the numeric values in a list  the first step is to get the sum of all the given values ' is a true.

Average of the numeric values in a list,

First step is to get the sum of values in the list.

It is not the total number of values.

Once we have the sum, we can divide it by the number of values to get the average.

Here is an example,

Suppose we have a list of numeric values are as follow,

[2, 4, 6, 8, 10].

To calculate the average of these values, we first find their sum,

2 + 4 + 6 + 8 + 10 = 30

Next,

divide the sum by the number of values in the list

Number of values = 5

30 / 5 = 6

This implies,

The average of the values in the list is 6.

Therefore, to get the average first step is to get the total of all values is true statement.

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The fraction form of 0.26 is 13/50.

What is a fraction that is improper or proper?

When a fraction's numerator is less than its denominator and it doesn't contain any whole numbers, the fraction is said to be appropriate. When a fraction represents a number bigger than one, it is said to have an inappropriate fraction since the numerator is larger than the denominator.

                                 When the numerator value is less than the denominator, the fraction is said to be appropriate. 23, 6/7, 8/9, and other correct fractions are examples.

The decimal is 0.26

   = 26/100

   = 13/50

Hence, The answer is 13/50

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Answer:

Step-by-step explanation:

B because after having the star and the circle comes the square.

Answer: x = -3, y = -4, or (-3, -4)

Step-by-step explanation:

To solve the system of equations, we can substitute the first equation into the second equation, replacing y with 3x + 5:

x + (3x + 5) = -7

Simplifying:

4x + 5 = -7

Subtracting 5 from both sides:

4x = -12

Dividing by 4:

x = -3

Now that we know x = -3, we can substitute that value into either of the original equations to find y:

y = 3(-3) + 5 = -4

Therefore, the solution to the system of equations is x = -3, y = -4, or (-3, -4).

Answer:

Step-by-step explanation:

First divide 1000 by 15 to find the amount

which is about 67 tickets.

Answer:

P(x = 5) = (e^(-λ) * λ^x) / x!

P(x = 5) = (e^(-2.8) * 2.8^5) / 5!

P(x = 5) ≈ 0.1008

b) P(x > 6) = 1 - P(x ≤ 6)

We can find P(x ≤ 6) using the cumulative Poisson probability formula:

P(x ≤ 6) = Σ (e^(-λ) * λ^x) / x!

P(x ≤ 6) = (e^(-2.8) * 2.8^0) / 0! + (e^(-2.8) * 2.8^1) / 1! + (e^(-2.8) * 2.8^2) / 2! + (e^(-2.8) * 2.8^3) / 3! + (e^(-2.8) * 2.8^4) / 4! + (e^(-2.8) * 2.8^5) / 5! + (e^(-2.8) * 2.8^6) / 6!

P(x ≤ 6) ≈ 0.8581

Therefore,

P(x > 6) = 1 - P(x ≤ 6)

P(x > 6) ≈ 0.1419

c) P(x ≤ 3) = Σ (e^(-λ) * λ^x) / x!

P(x ≤ 3) = (e^(-2.8) * 2.8^0) / 0! + (e^(-2.8) * 2.8^1) / 1! + (e^(-2.8) * 2.8^2) / 2! + (e^(-2.8) * 2.8^3) / 3!

P(x ≤ 3) ≈ 0.4232

Yes, that is correct. Here's a step by step explanation of how to figure that out:

1. Calculate the total amount of fast food that can be spent in a month: $15 x 4 = $60.

2. Calculate the total amount of money left in the budget for fast food: $1850 - $60 = $1790.

3. Calculate the percentage of money left in the budget for fast food: ($1790/$1850) x 100 = 78%.

The probability that the person chosen is a woman is 0.51.

The probability that the person chosen favors pink or purple is 0.67.

The probability that a person that favors turquoise is a man is 0.66.

Probability is the odds that a random event would occur. The odds that the event occurs has a probability value that lies between 0 and 1. The more likely it is that the event would happen, the closer the probability value would be to 1.

The probability that the person chosen is a woman = number of women / total number of people = 152 / 300 =  0.51.

The probability that the person chosen favors pink or purple = (number of people who favor pink / total number of people) + (total number of people that favor purple / total number of people) = (50 /300) + (50 / 300) =  0.67.

The probability that a person that favors turquoise is a man = men who favor turquoise / total number of people who favor turquoise = 79/120 = 0.66

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The answer of the given question based on explaining the meaning of symmetric group Sₓ on X the answer is given below,

In mathematics, permutation is  way of arranging  set of distinct objects in particular order. A permutation of  set is bijection from the set to itself, meaning that each element in  set is mapped to a unique element, and every element in  set is mapped to exactly once. For example, the set {1, 2, 3} can be permuted in six different ways: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Permutations have important applications in group theory, combinatorics, and cryptography.

The symmetric group Sₓ on X is a group of permutations of a set X, where Sₓ is the set of all possible bijections from X to X. In other words, the elements of Sₓ are all possible ways of rearranging the elements of X, while still maintaining the structure of the set. A permutation is a bijection that maps each element of X to a unique element of X.

The order of the symmetric group Sₓ on X is the number of elements in X, denoted by |X|. The symmetric group has important applications in group theory, algebraic geometry, and combinatorics. It is also a fundamental concept in the study of abstract algebra and is used to study the properties of permutations, symmetry, and groups.

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The symmetric group S_X on a set X is the group of all permutations of the elements in X.

A bijective function, also known as a bijection, is a type of function in mathematics where every element in the domain (input) is paired with a unique element in the range (output), and every element in the range has a corresponding element in the domain. In other words, a bijection is both injective (one-to-one) and surjective (onto).

A function f: X → Y is said to be bijective if:

Every element in X is paired with a unique element in Y (one-to-one or injective). This means that no two elements in X are mapped to the same element in Y.

Every element in Y has a corresponding element in X (onto or surjective). This means that every element in Y is mapped to by at least one element in X.

A permutation of X is a bijective function that maps each element of X to a unique element of X. The symmetric group S_X contains all possible permutations of the elements of X, and the group operation is function composition.

In other words, if X is a set with n elements, then the symmetric group S_X contains all possible bijections from X to itself, which means there are n! (n factorial) elements in S_X. Each element in S_X represents a unique way to permute the elements in X.

For example, let X = {1, 2, 3}. The symmetric group S_X on X contains the following six elements (permutations):

These six permutations form a group under function composition, which is the symmetric group S_X on X.

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a) The probability of no orders in 5 minutes is calculated to be 0.36788.

b) The probability of three or more orders in 5 minutes is calculated to be 0.08.

c) The length of the time interval such that the probability of no orders in a time interval of this length is 0.001 is calculated to be 34.5 min.

X is assumed to be the poisson's distribution where λ = 12 orders per hour.

a) At T = 1/12 hours which is 5 min, probability of no orders,

P (X = 0) = e^(-12/12) = 0.36788

b) At T = 1/12 hours which is 5 min, probability of three or more orders,

P (X ≥ 3) = 1 - P (X ≤ 2) = 1 - e⁻¹(1 + 1 + 1/2) = 0.08

c) Let us find the interval T for which:

P (X = 0) = 0.001

e^(-12T) = 0.001

Solving the equation for T we have,

T = -1/12 ln(0.001) = 0.5756 hours = 34.5 min

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Answer:

804

Step-by-step explanation:

Just use a calculator.

The function represented by the table is symmetric, and the values of y vary with the square of the corresponding values of x. Therefore, the function is exponential.

In mathematics, a function is a rule that assigns a unique output value for every input value. It is a mathematical object that takes an input and produces an output. A function is often represented symbolically as f(x), where f is the name of the function and x is the input. The output value of the function depends on the input value, and the relationship between the input and output is determined by the rule of the function. Functions can be linear, quadratic, exponential, trigonometric, or many other types. They are used to model real-world phenomena and solve mathematical problems.

Here,

To determine if the function represented by the table is linear, quadratic, or exponential, we can plot the points on a graph and look at the pattern of the data.

x y

-2 12

-1 3

0 0

1 3

2 12

    |

12   *      *   12

    |       \ /

6   *        *

    |         |

0 --*--*--*--*--*--*--*--*--

    |         |

    *        *

   / \      /

-6 *   *    *

    |

    -2 -1  0  1  2

Looking at the graph, we can see that the points do not fall on a straight line, so the function is not linear. Additionally, the curve of the points does not resemble a parabola, so the function is not quadratic. However, the points are symmetric around the y-axis, suggesting that the function may be exponential.

To confirm this, we can calculate the ratio of consecutive y-values:

12/3 = 4

3/0 = undefined

0/3 = 0

3/12 = 1/4

We can see that the ratio is not constant except for the first and last points, which have a ratio of 4. This suggests that the function is not purely exponential, but may have some other component.

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ABCD is a parallelogram, the value of x is 7

A quadrilateral with the opposing sides parallel is called a parallelogram (and therefore opposite angles equal). A parallelogram with all right angles is known as a rectangle, and a quadrilateral with equal sides is known as a rhombus.

The opposing sides of a parallelogram are equal and parallel.

AD = BC

5x + 3 = 38

Take 3 away from both sides.

5x + 3 - 3 = 38 - 3

5x = 35

Subtract 5 from both sides.

5x/5 = 35/5

x = 7

A two-dimensional shape with four sides, four vertices, and four angles is referred to as a quadrilateral. Convex and concave are the two main forms. Convex quadrilaterals can also be divided into a number of subgroups, including trapezoids, parallelograms, rectangles, rhombus, and squares.

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Answer:230

Step-by-step explanation:

All of the statements are equivalent to saying that there is a 50% increase in the cost of a movie.

The formula for calculating a percentage increase is:

Percentage increase = (New Value - Original Value) ÷ Original Value x 100

In this case, we can say that the original value is the cost of the movie before the increase, and the new value is the cost of the movie after the increase. Let's say the original cost of the movie is $10. Then the percentage increase is 50%.

We can calculate the new cost of the movie as follows:

New Cost = Original Cost + (Original Cost x Percentage Increase)

New Cost = 10 + (10 x 0.50)

New Cost = 10 + 5

New Cost = 15

Therefore, we can say that the new cost of the movie is 15, which is one and a half times the original cost of $10. The increase in the cost of the movie is 5, which is half the original cost of $10. The cost of the movie did double, from the original cost of $10 to the new cost of 15.

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Step-by-step explanation:

We have:

x = a sin 2t

Differentiating with respect to t, we get:

dx/dt = 2a cos 2t

Next, we have:

y = a(cos 2t + log tan t)

Differentiating with respect to t, we get:

dy/dt = -2a sin 2t + (1/tan t)(1/ln 10)

Using the identity:

sin^2 t + cos^2 t = 1

We have:

sin 2t = 2sin t cos t

And:

cos 2t = cos^2 t - sin^2 t

cos 2t = 2cos^2 t - 1

Using these identities, we can rewrite dx/dt and dy/dt in terms of x and y:

dx/dt = 2a sqrt(1 - x^2/a^2)

dy/dt = -2a sqrt(1 - x^2/a^2) + (1/ln 10)(y - a cos 2t)

Therefore, we have:

dx/dy = dx/dt ÷ dy/dt

Substituting the expressions for dx/dt and dy/dt, we get:

dx/dy = (2a sqrt(1 - x^2/a^2)) / (-2a sqrt(1 - x^2/a^2) + (1/ln 10)(y - a cos 2t))

Simplifying, we get:

dx/dy = (-2 sqrt(1 - x^2/a^2)) / (2 sqrt(1 - x^2/a^2) - (1/ln 10)(y - a cos 2t))

Answer:

≈ 4.57

Step-by-step explanation:

(-2) ^ 2 = 4

4 - (-28) = 4 + 28 = 32

32/7 ≈ 4.57

Answer: The variance of a sample of size n can be calculated using the formula:

variance = (sum of squared deviations from the mean) / (n - 1)

where the deviation from the mean is the difference between each individual value and the sample mean, and the sum of squared deviations is the sum of the squares of these differences.

However, we don't have the individual weights or the sample mean, only the standard deviation. We can use a related formula that expresses the standard deviation in terms of the variance:

standard deviation = sqrt(variance)

Solving for variance, we get:

variance = (standard deviation)^2

Plugging in the given standard deviation of 12.3194 kg, we get:

variance = (12.3194 kg)^2 = 151.8992 kg^2

Therefore, the variance of the weights of the 10 subjects is 151.8992 kg^2.

Step-by-step explanation: