find the range of sina(3cos2a cos4a 3sin2a sin2a cos2a) ^ tan a(sec a -sina tana) where a is not an integer multiple of 7t/2. (source: hmmt)

An example of such a function can be f(x)  { 1  x∈Q

                                                                      -1   x∈ R\Q}

A function in mathematics is like a machine that takes input and gives output. There is always a limit for output also called the range or image of the function.

A function has two values output and input. The limit within which input is defined is called the domain or pre-image of the function.

A function is represented as f(x) i.e., f of x or function of x. The input and output have a relation defined for them

A function must follow two rules they are:

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The like terms are listed as follows:

-y, 7y and y/2.

Like terms are terms that share these two features:

If two terms are like terms, then they can be either added or subtracted.

Hence the first option is the only option in which all the three terms are like terms, as:

The remaining options have terms with different letters(second and third), or different exponents (fourth), hence they are not like terms.

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The set of items that are part of both sets F and G is indicated by the symbol F G. It is the group of equilateral triangles that are also isosceles triangles, to put it another way.

Due to its three congruent sides, every equilateral triangle is an isosceles triangle. F G = F hence equals "equilateral triangles." By eliminating the unnecessary component, we can make F G simpler.

Consequently, F G = "equilateral triangles"

When seen collectively, a set is a group of items. The elements or members of the set are the things that make it up. Any mathematical object, such as integers, functions, or other sets, may be one of the elements of a set.

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The numeric value of the expression for this problem is given as follows:

[tex]\sqrt{2} + 1[/tex]

The expression for this problem is defined as follows:

[tex]\frac{4 - 3\sqrt{2}}{(\sqrt[4]{2} - \sqrt[4]{8})^2}[/tex]

Applying the square of the subtraction in the denominator, we have that:

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = \sqrt[4]{4} - 2\sqrt[4]{2}\sqrt[4]{8} + \sqrt[4]{64}[/tex]

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = 4^{\frac{1}{4}} - 2\sqrt[4]{16} + \sqrt[4]{4^3}[/tex]

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = (2^2)^{\frac{1}{4}} - 2 \times 2 + 3[/tex]

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = \sqrt{2} - 1[/tex]

Hence the simplified expression is given as follows:

[tex]\frac{4 - 3\sqrt{2}}{\sqrt{2} - 1}[/tex]

Rationalizing the denominator, we have that:

[tex]\frac{4 - 3\sqrt{2}}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1}[/tex]

Applying the subtraction of perfect square, the denominator simplifies to one, hence the value of the expression is given by the numerator as follows:

[tex](4 - 3\sqrt{2})(\sqrt{2} + 1) = 4\sqrt{2} + 4 - 3 \times 2 - 3\sqrt{2}[/tex]

[tex][tex](4 - 3\sqrt{2})(\sqrt{2} + 1) = \sqrt{2} + 1[/tex]

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The probability that a student rides a bike to school, given that the student's school has a crossing guard is given by 0.25.

For two events A and B, we have:

Probability that event A or B occurs = Probability that event A occurs + Probability that event B occurs - Probability that both the event A and B occur simultaneously.

We know that;

For two events A and B:

The chain rule states that the probability that A and B both occur is given by:

⇒ P (A∩B) = P (A) P (B|A) = P (B) P (A|B)

You can remember this fact by using a trick that when A is on left, it is behind B in next term P(A) P(B|A)

And when B is on left, then it is behind A in next term P(B)P(A|B)

P(A|B) = Probability that A will happen given B has already happened.

P(B|A) = Probability that B will happen given A has already happened.

Using those above stated rules, we have

Let the event A be that a student rides bike to school.

Let the event B be that the student's school has a crossing guard

Then we are given that:

P(A) = P{rides bike to school}) = 0.14

P(B) = P{has crossing guard}) = 0.48

P{rides bike and crossing guard}) = 0.12

We have to find P(a student rides a bike to school, given that the student's school has a crossing guard) = P(A | B)

From the chain rule of probability, we have;

⇒ P (A ∩ B) = P (B) P (A |B)

⇒ 0.12 = 0.48 × P (A|B)

⇒ P (A|B) = 0.12 / 0.48

⇒ P (A|B) = 0.25

Thus, The probability that a student rides a bike to school, given that the student's school has a crossing guard is given by 0.25.

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Answer: a) To calculate the total rate of return if Josh invests the same amount in each company, we need to add up the individual rates of return from each company and divide by the number of companies.

The total rate of return for the four companies is: (7.62% + -4.17% + -0.83% + 3.70%) / 4 = 2.52%

So the total rate of return if Josh invests the same amount in each company is 2.52%.

b) To calculate the total rate of return if Josh invests three times as much in Stiller Co. as in the other companies, we need to multiply the rate of return for Stiller Co. by 3 and add it to the rates of return for the other companies.

The total rate of return for the four companies is: (-4.17% x 3) + 7.62% + -0.83% + 3.70% = -7.32%

So the total rate of return if Josh invests three times as much in Stiller Co. as in the other companies is -7.32%.

It's important to note that the answer may change if the investment amount on each company is different.

Step-by-step explanation:

The alternative hypothesis for this test would be that the new dog food extends the average lifespan of dogs beyond 10 years.

Mathematically, the alternative hypothesis can be stated as:

Ha: μ > 10, where μ is the population mean of the lifespan of dogs on the new dog food.

In this scenario, the null hypothesis is that the average lifespan of dogs on the new dog food is equal to 10 years, which is the commonly accepted average lifespan of dogs. However, the purpose of the study is to determine if the new dog food can extend the lifespan of dogs beyond this average.

Thus, the alternative hypothesis is that the average lifespan of dogs on the new dog food is greater than 10 years. This is represented mathematically as Ha: μ > 10, where μ is the population mean of the lifespan of dogs on the new dog food.

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They marked it up to 2979 since 993*3(aka 200%). You then multiply 2979 by 1.14 to get the total of $3396.06

Answer:

$3396.06

Step-by-step explanation:

Markup is what a store adds to the price they paid in order to come up with a selling price.

store selling price = cost to store + markup

Store purchase price: $993

Store markup of 200%: 200% of $993 = 2 × $993 = $1986

Store selling price: $993 + $1986 = $2979

14% tax: 14% of $2979 = 0.14 × $2979 = $417.06

Total selling price including markup & sales tax: $2979 + $417.06 = $3396.06

Answer: $3396.06

If he continued at the same rate, he would travel  370 miles in 10 hours.

A proportion is described as an equation having two ratios that are set equal to each other.

It is the mathematical comparison of numbers or variables.

In mathematics, the different types of fractions are;

From the information given, we have that;

sauv drove 296 miles in 8 hours

Then,

If 296 miles  = 8 hours

x miles = 10 hours

cross multiply

8x = 2960

Make 'x' the subject of formula

x = 370 miles

Hence, the value is 370 miles

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According to the pie chart given -

a) The worker saved GH¢ 20.00.

b) 2/5 fraction of worker's salary was spent on food.

c) The worker spent GH¢ 120.00 on food.

What is a pie chart?

One sort of graph that illustrates the information in the circular graph is a pie chart. It is a sort of graphical representation of data where the slices of pie depict the relative sizes of the data. A list of numerical and categorical variables is necessary for a pie chart. Pie in this context refers to the entire thing, and slices to its component portions.

The rent charges is = GH¢ 80.00

The savings can be obtained by ratio method -

Savings is depicted by 24° and rent by 96° in the pie chart.

So, the ratio will be 24/96.

Then the savings value will be -

(24/96) × 80

(1/4) × 80

20

Therefore, the savings value is GH¢ 20.00.

The total angle of pie chart is 360°.

Savings is depicted by 24°, rent by 96°, entertainment by 48° and clothing by 48° in the pie chart.

The angle for food will be -

360° - (24° + 96° + 48° + 48°)

360° - 216°

144°

Now the fraction that is for food -

144/360

2/5

Therefore, food covers 2/5 part of pie chart.

The amount spent on food can be obtained by formula of angle sector -

(144/96) × 80

1.5 × 80

120

Therefore, money for food was GH¢ 120.00.

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Researchers might study the role of predators in natural selection by testing the hypothesis that animals with camouflage colorings have a greater chance of survival when they are exposed to predators.

They can do this by setting up an experiment in which they expose animals with different colorings, including camouflage, to predators and then measure the survival rate of the different colorings. This would test if animals with camouflage colorings have a higher chance of survival when they are exposed to predators.

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Answer:

24.5 days i believe

The statement 'A vector b is a linear combination of the columns of a matrix A is and only if the equation Ax = b has at least one solution.' is True.

We first thought of a matrix as a rectangular array of numbers. When the number of rows is m and columns is n, we say that the dimensions of the matrix are m×n.

A vector is most simply thought of as a matrix with a single column. There are two operations we can perform with vectors: scalar multiplication and vector addition. Both of these operations have geometric meaning.

Given a set of vectors and a set of scalars we call weights, we can create a linear combination using scalar multiplication and vector addition.

The product of two matrices can be seen as the result of taking linear combinations of their rows and columns. This way of interpreting matrix multiplication often helps to understand important results in matrix algebra.

The statement 'A vector b is a linear combination of the columns of a matrix A is and only if the equation Ax = b has at least one solution.' is True.

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Answer:x=13

Step-by-step explanation:

Answer: The simplified expression is 6x - 23

Step-by-step explanation: To simplify the expression (12x-13)-2(3x+5) we will follow the order of operations:

First, we simplify the inner parentheses: 2(3x+5) = 6x + 10

Next, we substitute this simplified version back into the original expression: (12x-13)-(6x+10)

Now we can simplify the expression by combining like terms: 12x-13-6x-10 = 6x - 23

So the simplified expression is 6x - 23

Answer:6x-23 this your answer to simplify the expression I hope this helps have a good day

Step-by-step explanation:

(12x-13)-2(3x+5)

Remove unnecessary parentheses

12x-13-2(3x+5)

then do this

distribute -2 through the parentheses

12x-13-6x-10

Collect like terms which something like this

6x-13-10

then last thing you do is

Calculate the difference which is like this

6x=23

To prove that triangles ABD and ΔABC are congruent by SSS, what needs to be true is: A. BC ≅ BD.

For two triangles to be congruent to each other by the SSS Congruence Theorem, the triangles must have three pairs of corresponding sides that are congruent to each other.

In the image given, we can deduce that the two triangles have the following properties:

AB ≅ AB (reflexive property)

AC ≅ AD (given)

This means that have two pairs of corresponding congruent sides. Therefore, in order to prove that ΔABD ≅ ΔABC by SSS, what needs to be true is: A. BC ≅ BD.

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The partials ∂y/∂x = 1 and ∂z/∂x = -1, so we get the Reciprocal Theorem.

Reciprocity Theorem: Let F(x, y, z) be a continuous function of x, y, and z, then we have

∂F/∂x = (∂F/∂y)x - (∂F/∂z)y

This is proven by writing the condition of F(x, y, z) = 0 into explicit forms x-x(y,z) or z-z(x,y). Then from the chain rule, we have

∂F/∂x = (∂F/∂y)(∂y/∂x) + (∂F/∂z)(∂z/∂x)

And since x = x(y,z) and z = z(x,y), the partials ∂y/∂x = -1 and ∂z/∂x = 1, so we get the Reciprocity Theorem.

Reciprocal Theorem: Let F(x, y, z) be a continuous function of x, y, and z, then we have

(∂F/∂y)x + (∂F/∂z)y = F(x, y, z)

This is proven by writing the condition of F(x, y, z) = 0 into explicit forms x-x(y,x) or z-z(x,y). Then from the chain rule, we have

∂F/∂x = (∂F/∂y)(∂y/∂x) + (∂F/∂z)(∂z/∂x)

And since x = x(y,x) and z = z(x,y), the partials ∂y/∂x = 1 and ∂z/∂x = -1, so we get the Reciprocal Theorem.

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Problem 22

The smaller inner circle has a radius of r = 5. That will be plugged into the circumference formula below.

C = 2pi*r

C = 2pi*5

C = 10pi

This is the exact distance around the circle in terms of pi. Think of it like the amount of fencing you need to enclose the circle.

If you want an approximate perimeter, then use something like pi = 3.14

So 10pi = 10*3.14 = 31.4 cm is the approximate perimeter aka circumference of the smaller circle.

-------------

The larger circle is the same idea. The radius is r = 5+5 = 10 cm

C = 2pi*r

C = 2pi*10

C = 20pi .... exact circumference

C = 20*3.14

C = 62.8 ... approximate circumference

Use more decimal digits of pi to get a more accurate value.

===================================================

Problem 23

The smaller circle has a diameter of 9 ft, which means the radius is r = 9/2 = 4.5 ft

C = 2pi*r

C = 2pi*4.5

C = 9pi .... exact circumference

C = 9*3.14

C = 28.26 .... approximate circumference

The units of the circumference are in feet

-------------

The larger circle has radius r = 4.5+2.5 = 7 ft

C = 2pi*r

C = 2pi*7

C = 14pi .... exact circumference

C = 14*3.14

C = 43.96 .... approximate circumference

===================================================

Problem 24

The larger circle has radius 22. This radius is the diameter of the smaller circle.

So the smaller circle has radius 22/2 = 11

C = 2pi*r

C = 2pi*11

C = 11pi .... exact circumference

C = 11*3.14

C = 34.54 .... approximate circumference

The circumference units are "meters".

-------------

Now onto the larger circle

C = 2pi*r

C = 2pi*22

C = 44pi .... exact circumference

C = 44*3.14

C = 138.16 .... approximate circumference

We come to the conclusion that if the scale factor from S to M is 3/2, the scale factor from M to S must be 2/4.

Assume we have a figure S. If we apply a stretch of scale factor K to our figure S, we may say that all of figure S's dimensions have been multiplied by K.

Following the stretch, we will have a bar of length M, such that:

M = S*K

If the scale factor is 3/2, the following scenario arises:

M = (3/2)*S

In the relationship mentioned above, S can be separated out as follows:

(2/3)*M = S

We now have an equation (similar to the first one).

As of right now, the scaling factor from M to S is 2/3, according to an equation (like the first one).

After that, we draw the conclusion that if the scale factor from S to M is 3/2, the scale factor from M to S must be 2/4.

The complete question is,

Scale factor is 3/2 for the range of S to M. How many are there in the scale from M to S?

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The mistake Derek did was d = 8.

The correct step is d = 9.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

2x + 4 = 55 is an equation.

We have,

8d = 72

Step 1:

Divide both sides with 8.

8d/8 = 72/8

[ 8 x 9 = 72 ]

So,

d = (8 x 9)/ 8

d = 9

Thus,

Derek's mistake is d = 8.

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Comparing rounding and compatible number, rounding gives a closer estimate than compatible number.

While compatible numbers are those that are simple to multiply mentally, rounding entails reducing one or both of the numbers to a close "friendly" number.

Estimating the result of 48 x 123 can be done using compatible integers and rounding.

Compatible numbers make it simpler to find an estimate calculation because they are close in value to the real number. A multiplication issue is presented here. By rounding the values to the closest hundreds place, you can obtain an approximation of the product. 400 and 100 are equivalent to 395 and 103.

1) Using compatible numbers :

48 = 50

123 = 100

50 × 100 = 5000

2) By rounding numbers :

48 = 50

123 = 120

50 × 120 = 6000

6000 is closer to the actual value of the product than 5000 ;

Therefore, rounding gives a closer estimate than compatible number.

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In general, the maximum order of the spectrum that can be observed with a diffraction grating is limited by the grating equation.

This equation states that mλ = d(sinθi - sinθr), where m is the order of the diffraction, λ is the wavelength of light, d is the spacing between the grating lines, θi is the angle of incidence, and θr is the angle of diffraction. Since d is fixed and θi is typically kept constant, the maximum order of the spectrum that can be observed is limited by the angle of diffraction (θr). This angle can be calculated using the equation θr = sin-1[(mλ/d)+sinθi]. Thus, the maximum order of the spectrum is limited by the value of θr, which is determined by the wavelength of light and the spacing between the grating lines.

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Check the picture below.

so following the inscribed quadrilateral conjecture, let's get those equations that add  up to 180°.

[tex](2y+74)+(7x+1)=180\implies 2y+7x+75=180\implies \underline{2y+7x=105} \\\\[-0.35em] ~\dotfill\\\\ (2y+100)+(6x-10)=180\implies 2y+6x+90=180\implies \underline{2y+6x=90} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{using the 1st equation}}{2y+7x=105}\implies 2y=105-7x \\\\\\ \stackrel{\textit{using the 2nd equation}}{2y+6x=90}\implies 2y=90-6x\implies \stackrel{\textit{substituting from above}}{105-7x=90-6x} \\\\\\ 105=90+x\implies \boxed{15=x}[/tex]

[tex]\stackrel{\textit{since we know that}}{2y+7x=105}\implies 2y+7(15)=105\implies 2y+105=105 \\\\\\ 2y=0\implies y=\cfrac{0}{2}\implies \boxed{y=0}[/tex]

The value of the standardized test statistic is D. 1.77.

A standardized test statistic is a measure of the difference between a sample statistic and the population parameter it estimates, expressed in units of the standard deviation.

The value of the standardized test statistic can be calculated using the formula:

(x - μ) / (s / √n)

where x is the sample mean, μ is the population mean, s is the population standard deviation, and n is the sample size.

Given the information provided in the question, the standardized test statistic is:

(20.3 - 20) / (1.2 / √50) = 1.7678

Rounding to two decimal places, the answer is D. 1.77.

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A transformation is applied to the input vector v by the function f. This transformation might include extending, rotating, or reflecting the vector in a certain way, depending on the precise shape of the function.

We may enter certain fixed vectors into the function, c, and watch as f transforms them to better understand its behavior (c). It may also be helpful to see the modified vectors as images or diagrams to have a better understanding of how the function behaves.

The term "vector" is used informally to describe constituents of particular vector spaces or some values that cannot be described by a single integer.

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Answer:

2

Step-by-step explanation:

The number of days a software engineer can go without wearing the same combination of three items will be 36.

A permutation is an act of correctly arranging things or pieces. Combinations are a method of taking stuff or pieces from an assortment of items or sets in which the order of the items is irrelevant.

A software engineer owns 3 pairs of pants, 4 shirts, and 3 pairs of shoes. Then the number of different ways is given as,

⇒ ³C₁ × ⁴C₁ × ³C₁

⇒ 3 x 4 x 3

⇒ 36

The number of days a software engineer can go without wearing the same combination of three items will be 36.

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The division of the value 8.8x10^9 and 2.2 x10^-3 in standard form is 4*10^12.

The concept that will be used here is division. One of the four fundamental arithmetic operations, or how numbers are combined to create new numbers, is division. The other operations are multiplication, addition, and subtraction.

Division in mathematics is the process of dividing an amount into equal parts. For instance, we may split a group of 20 people into four groups of 5, five groups of 4, and so on.

To simplify the values 8.8x10^9/2.2 x10^-3, the we can write as

8.8/2.2 *10^9*10^3

=4*10^12

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complete question:

simplify 8.8x109/2.2 x10-3

Answer:

(mn+5)(mn-5)

Step-by-step explanation:

This is a difference of squares as m^2n^2 = (mn)^2. So it cna be turned into the form a^2-b^2 = (a+b)(a-b) or in our case (mn+5)(mn-5).