How do you determine which table represents a function?.

We are aware that the numbers 4, 9, 25 and 49 are among those between 1 and 100 that have precisely three elements.

Given,

What number has three components?

In fact, the squares of primes are the only positive integers with precisely three factors. For instance, 1, 3, and 9 are the factors of 9, and 1, 7, and 49 are the factors of 49.

If a given number, such as x2, is perfect square, and its square root is prime, it must have precisely three different factors.

Here,

We are aware that the numbers 4, 9, 25 and 49 are among those between 1 and 100 that have precisely three elements.

1, 2, and 4 are the four factors. 1, 3, and 9 are the factors of 9.

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The vector-valued function at each given value of t  is :

a) i

b)  1/√2i+9/√2j

c)−cos(θ)i+9sin(θ)j

d)  −2sin(π/6+△t/2)sin(△t/2)i+18cos(π/6+△t/2)sin(△t/2)j

The given problem is related to trigonometric Ratios of Angle, where there are six trigonometric ratios sine, cosine, tangent, cotangent, cosecant, and secant. Since it is given that a  vector valued function:

r(t)= cos(t)i + 9 sin(t)j

r(t)=cos(t)i+9sin(t)j

so ,  r(0)=  cos(0)i+9sin(0)j = i ( sine cos0 =1 and sin0 =0)

r(π/4) =  cos(0)i+4sin(0)j = cos(π/4)i+9sin(π/4)j

           = 1/√2i+9/√2j

r(θ−π)= cos(θ−π)i+9sin(θ−π)j= −cos(θ)i+9sin(θ)j

(π/6+△t)−r(π/6)= cos(π/6+△t)i+9sin(π/6+△t)j−cos(π/6)i−9sin(π/6)j

=  cos(π/6+△t)i−cos(π/6)i+9sin(π/6+△t)j−9sin(π/6)j

= −2sin(π/6+△t/2)sin(△t/2)i+18cos(π/6+△t/2)sin(△t/2)j

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Evaluate (If possible) the vector-valued function at each given value of t. (If an answer does not exist, enter DNE.)

r(t)=cos(t)i+9sin(t)jr(t)

a)r(0)=?

b)r(π/4)=?

c)r(θ−π)=?

d)r(π/6+△t)−r(π/6)=?

Compound amount due in 8 year will be 495192.64

What do you mean by simple interest?

Simple interest is a quick and simple formula for figuring out how much interest will be charged on a loan. The daily interest rate, the principle, and the number of days between payments are multiplied to calculate simple interest.

Simple interest is a predetermined proportion of the principle amount borrowed or lent that is paid or received over a set period of time.

Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time.

Given principal, P = 2000000

Rate of interest, r  = 12%

Time period , n  = 8 years

Compound amount = [tex]P(1+\frac{r}{100})^n[/tex]

= [tex]200000(1+\frac{12}{100})^8[/tex]

= [tex]200000(1+0.12)^8[/tex]

= [tex]200000(1.12)^8[/tex]

= 495192.64

Therefore, compound amount due in 8 year will be 495192.64

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It is true that the expressions 5y + 4 − 3y = 2y + 4 are equivalent

From the question, we have the following expression that can be used in our computation:

5y+4−3y = 2y+4

This equation can be rewritten as follows

So, we have the following representation

5y + 4 − 3y = 2y + 4

Collect the like terms in the above equation

This gives

5y − 3y + 4 = 2y + 4

Evaluate the like terms in the above equation

This gives

2y + 4 = 2y + 4

Both sides of the equation are the same

Hence, the expressions are equivalent

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The pension is equal to $19,016

First,  we have to calculate the average of the salaries from the last three years:

avg salary = (38592 + 37590 + 37010)/3

= 113192/3

= 37730.66

Next, calculate the percentage that Mrs. Brown will get, multiplying 1.8% times the amount of years that she worked:

28 x 1.8 = 50.4

Finally, calculate what is 50.4% of the average income equal to:

50.4% of avg salary

50.4 % of 37730.66

(50.4/100) 37730.66

= 19016

Therefore, the pension is equal to $19,016

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Margin of error for the given standard deviation is  m = 5.497 , 5.703

What is standard deviation?

Standard Deviation is a measure that shows what quantity variation (such as unfold, dispersion, spread,) from the mean exists. the quality deviation indicates a “typical” deviation from the mean. it's a well-liked live of variability as a result of it returns to the first units of live of the info set.

Main body:

formula for confidence interval is :

m = x-bar ± z* σ/√n

given information =

n = 25

σ = 0.3

mean = 5.6 ft

C.I. = 90% = 0.9

value of 0.9 in z - table = 0.816

substituting value ,

m = 5.6 ±(0.816) * 0.3/√25

m = 5.6 ± 0.103

m = 5.497 , 5.703

Hence margin of error is m = 5.497 , 5.703

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The density of horses in royale stables would be 5.78 horses/sq.km  

How to find the population density?

To calculate the population density, you will divide the population by the size of the area. Thus, Population Density = Number of People/Land Area. The unit of land area should be square miles or square kilometers.

Given that there are 4564 horses in royale stables. and the stables have an area of 789 sq.km

So by using the formula for population density, we can find the population density of horses in royale stables as

Population density of horses in royale stables = number of horses in royale stables/ Area of Royale stables

                                                             = 4564 / 789

                                                             = 5.78 horses/sq.km        

Hence, the density of horses in royale stables would be 5.78 horses/sq.km  

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Using the exponential model F₀ = x, the annual rate of deforestation in El Salvador is 2.65%. .It is estimated that at the present rate of deforestation in El Salvador, in 20 years only 53%of the present forest will be remaining.

A number written as a superscript over another number is known as an exponent. In other words, it means that the base has been elevated to a particular level of power. Other names for the exponent are index and power. Mn indicates that m has been multiplied by itself n times if m is a positive number and n is its exponent.

Given,

It is estimated that at the present rate of deforestation in El Salvador, in 20 years only 53%of the present forest will be remaining.

Let the area present in forest be x,

F₀ = x,

Forest's final area:

F = 53/100 x

F = 0.53x

Time = 20 years

As we know,

Exponential model,

F = F₀ e^rt

0.53 =  e^r(20)

Equating en both sides,

en 0.53 = en e ^20r

en 0.53 = 20r

r = en 0.53/20

r = 0.0265

r - 2.65%

Using the exponential model F₀ = x, the annual rate of deforestation in El Salvador is 2.65%. .It is estimated that at the present rate of deforestation in El Salvador, in 20 years only 53%of the present forest will be remaining.

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The phenomena of hiding distribution characteristics in a system from applications and users is known as distribution transparency. Access transparency, location transparency are some examples.

Distributed databases have the attribute of distribution transparency, which keeps consumers from knowing the internal workings of the distribution.

Thus, access transparency, location transparency are some examples of the (distribution) transparency.

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The sum of first 20 multiples of 7 will be 1470.

Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

Given are first 20 multiples of 7.

The sequence of first 20 multiples of 7 will be -

7, 14, 21, 28, ..... , 140

Sum = (20/2)[2 x 7 + (19) x 7]

Sum = 10[14 + 133]

Sum = 10 x 147

Sum = 1470

Therefore, the sum of first 20 multiples of 7 will be 1470.

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[Question translation in english -

find the sum of the first 20 multiples of 7 with procedure.]

The first thing a sales associate should do when a customer has a problem with a product by Thanking  the customer for bringing the problem to their  attention

A sales associate may be a sort of sales professional who works in retail whose duties are geared towards interfacing directly with prospects as they enter a store and serving as a consultative resource who helps those customers discover the specific items that best suit their needs. when a customer approach a problem, understand the problem the customer has in a detailed way and ask them for the receipts, or warranties, credit card statements, canceled checks,  contracts, invoices, or other documents in order for payment issues.

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It will take 9 years for Kevin to be only 4 times as old as Daniel.

The number of years it will take for Kevin to be only 4 times as old as Daniel can be determined by using an algebraic expression

Consider x represents the number of years that will need to pass. To represent Kevin's age, the expression can be given as;

39 + x

To represent 4 times Daniel's age, the expression can be given as;

4 × (3 + x)

In other words, this equation can be represented as Kevin's age plus x years is equal to four times Daniel's age plus x years.

Since Daniel's age needs to be equal to 4 times Daniel's age, we set the expressions equal to each other:

39 + x = 4 ( 3 + x )

39 + x = 12 + 4x

39 - 12 = 4x - x

3x = 27

x = 27 ÷ 3

x = 9

Although a part of your question is incorrect, you might be referring to this question:

Kevin is 39 years old and Daniel is 3 years old how many years will it take until Kevin is only 4 times as old as daniel

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The mean of the 1234567 numbers is 4.

What is mean?

A amount that includes a price intermediate between those of the acute members of some set is known as mean.

Main body:

In this question we have to find the mean of the first 7 numbers. Whole numbers are all non-negative integers which start from 0. Now the mean of the first 7 numbers will be the mean of numbers from 1 to 7, as the count of numbers from 1-7 is 7. Use this concept to reach the answer.

Mean of numbers is equal to the sum of the numbers divided by the count of the numbers that is

Mean = sum of numbers/ total numbers          …………………….. (1)

To find the mean of the 1234567 , we will find the sum of numbers from 1 to 7 as 1-7 are the first 7  numbers.

So, the sum of the digits is 1+2+3+4+5+6+7 = 28

total numbers = 7

mean = 28/7 = 4

Hence mean of 1234567 is 4.

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The minimum size of the group so that it guarantees to contain at least 6 students from some states in America is  251  students .

In the question ,

it is given that ,

the number of states in America = 50 states

The maximum number of cases in which there are less than 6 students from any state is when there are 5 students from each state,

which is ,

total of 5x50 = 250 students.

By Pigeon Hole Principle ,

In this, if one more student is added, the state to which that student belongs will have a total of 6 students.

So, this is minimum number required to guarantee that there are at least 6 students from some state in America is 251.

5x50 + 1 = 251

Therefore , the minimum size of the group is 251 students .

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The differential equations that could be used to model this situation with respect to time t, where k is a positive constant is db/dt= kp(N-p)

A differential equation is a mathematical equation that contains one or more terms and the derivatives of one variable (the dependent variable) with respect to another variable.

A differential equation is an equation that connects one or more unknown functions and their derivatives. In most applications, functions represent physical quantities, derivatives represent their rates of change, and differential equations define a relationship between the two.

This was illustrated regarding the change with respect to the number of people.

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There are no potential roots for this polynomial.

An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). For example, a cubic function is of the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants and a ≠ 0. It has a degree of 3 and has 3 possible roots.

We have the following polynomial -

x⁴ – 2x³ + 5x² – 7x + 9.

Plot the graph of the function. The [x] - intercepts represent the possible roots of the given polynomial. It can be seen that it does not cut the [x] - axis, hence, there are no potential roots for this polynomial.

Therefore, there are no potential roots for this polynomial.

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The magnitude of 2(u-v +w) is 17.1.

The given vectors are u = 2i + 3j, v=-i, and w = 5i – 6j.

We need to find the value of 2(u-v+w).

How to find the magnitude of vector?

The formula to determine the magnitude of a vector (in two-dimensional space) v = (x, y) is: |v| =√(x²+ y²). This formula is derived from the Pythagorean theorem.

Now, 2(2i + 3j-(-i)+5i-6j)

=2(2i+3j+i+5i-6j)

=2(8i-3j)=16i-6j

The magnitude of 16i-6j=√16²+6²=√256+36=√292

Therefore, the magnitude of 2(u-v +w) is √292 which is 17.1.

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If the sample size is decreased for any given confidence level, the confidence interval for the population proportion will increase.

A larger sample size or lower variability results in a more narrow confidence interval with a smaller margin of error. A larger confidence interval with a larger margin of error will result from a smaller sample size or higher variability.

The width of the confidence interval grows as the confidence level grows. As the confidence level falls, so does the width of the confidence interval.

This means that as the sample size is reduced, the width of the confidence interval expands.

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An equation that has fractions with xs in the numerator, denominator, or both is referred to as rational. A rational equation looks like this: (4 / (x + 1)) - (3 / (x - 1)) = -2 / (x^2 - 1).

Consider the last time you used a fraction to solve an equation. 1/3 x = 8. The 3 in the denominator is a prisoner in our eyes, and we wish to free it. We multiply both sides of the equation by 3 to release the 3. Consider it as 3 notifying both parties of his departure. 3 (1/3 x) = 8 (3). (3).This procedure eliminated the fraction x = 24 and released our denominator. We add one more step to it when we need to solve rational equations. Sometimes, even though our answer to a rational equation appears to be sound, it actually contains a virus that prevents it from working. We refer to these as superfluous solutions. To answer a rational equation, follow these steps:

1. Determine the common denominator.

2. Divide all sums by the lowest common denominator.

3. Simplify.

4. Verify the solution(s) to make sure there aren't any unnecessary ones.

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The average age of all students in the college is a parameter. A parameter describes the whole populations.

Tha value of parameter or statistic can be average, percentage, and others.

A college reports that the average age of their students is 28 years old.

Is this a statistic of parameter?

The college would have access to data on all students. The data of all students in the college or the entire students is called population. So, this is a parameter.

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For the given function f(t) = (2t + 1) using definition of Laplace transform the required solution is L(f(t))s = [ ( 2/s²) + ( 1/s) ].

As given in the question,

Given function is equal to :

f(t) = 2t + 1

Simplify the given function using definition of Laplace transform we have,

L(f(t))s = [tex]\int\limits^\infty_0 {f(t)e^{-st} } \, dt[/tex]

          =  [tex]\int\limits^\infty_0[2t +1] e^{-st} dt[/tex]

          = [tex]2\int\limits^\infty_0 te^{-st} + \int\limits^\infty_0e^{-st} dt[/tex]

         = 2 L(t) + L(1)

L(1) = [tex]\int\limits^\infty_0e^{-st} dt[/tex]

     = (-1/s) ( 0 -1 )

     = 1/s , ( s >  0)

2L ( t ) = [tex]2\int\limits^\infty_0 te^{-st}[/tex]

        =  [tex]2[t\int\limits^\infty_0 e^{-st} - \int\limits^\infty_0 ({(d/dt)(t) \int\limits^\infty_0e^{-st} \, dt )dt][/tex]

        =  2/ s²

Now ,

L(f(t))s = 2 L(t) + L(1)

          = 2/ s² + 1/s

Therefore, the solution of the given function using Laplace transform the required solution is L(f(t))s = [ ( 2/s²) + ( 1/s) ].

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These terms represent the first four terms of the Taylor series expansion of the function f(x) around the point c. The higher order terms in the series involve higher order derivatives of f(x) at the point c.

The Taylor series is a mathematical concept that gives an approximate representation of a function as a sequence of terms determined from the function's derivatives at a particular location. It is named for mathematician Brook Taylor, who first proposed the idea in the 18th century. The Taylor series enlarges a non zero function into an infinite sum of terms, each of which is a power of the independent variable (often x) multiplied by a coefficient. The value of the function and its derivatives at the location where the expansion is centred determines each term's coefficient. The Taylor series expansion of the function f(x) at the point x=a, for example, is given by:

f(x) = f(a) + (x-a) (x-a)

(x-a)2f"(a)/2! + f'(a) + (x-a) (x-a)^3f'''(a)/3! + ...

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The maximum area of the rectangular enclosure around his model railroad is 72 ft².

For the stated question-

A = L×W

Perimeter P = L + barn + 2W

Barn = 0ft (for no fence)

L + 2W = 24ft

L = 24 - 2W

A = (450 - 2W)×W

A = 24W - 2W²

A = -2W² + 24W

We understand we get a parabola because our equation regarding area in measures of width has the formula ax² + bx + c (since C = 0).

Since the squared term's coefficient is negative (a = -2), we can infer that the parabola's opening is downward.

In the equation y = (x - h)² + k, where k is our greatest area while h is our maximizing number, the vertex will be at point (h, k), which in this case is the width value that provides us our maximum area.

Knowing that h = -b / 2a.

h = -24/ 2(-2) = 6

maximum width = 6 ft

Now,

A = -2(6)² + 24(6)

Maximum area A = 72 ft²

Thus, the maximum area of the rectangular enclosure around his model railroad is 72 ft².

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If a, b, and c be three distinct numbers such that (ab)/(a-b)=(bc)/(2b-2c)=(ca)/(3c-3a) then the value of 8a9b5c is 8 [-(b^2)c / (2b^2-bc)] × 9[-2ac^2 / (3c^2 - 5ac)] × 5[3(a^2)b / (4ab - a^2) ]

Let a, b and c be three distinct number

Given that

ab / a-b = bc / 2b - 2c = ca / 3c - 3a

Consider first two term

The equation will be

ab / a-b = bc / 2b - 2c

Cross multiply the terms

ab(2b - 2c) = bc(a - b)

2ab^2 - 2abc = abc - (b^2)c

2ab^2 - abc = -(b^2)c

a(2b^2-bc) = -(b^2)c

a = -(b^2)c / (2b^2-bc)

Similarly

bc / 2b-2c = ca / 3c-3a

bc(3c-3a) = ca(2b-2c)

3bc^2 - 3abc = 2abc - 2ac^2

3bc^2 - 5abc = -2ac^2

b(3c^2 - 5ac) = -2ac^2

b = -2ac^2 / (3c^2 - 5ac)

Then,

ab(3c-3a) = ca(a-b)

3abc - 3(a^2)b = (a^2c) - abc

4abc - 3(a^2)b = (a^2c)

4abc - (a^2)c = 3(a^2)b

c(4ab - a^2) = 3(a^2)b

c =  3(a^2)b / (4ab - a^2)

The value of 8a9b5c = 8 [-(b^2)c / (2b^2-bc)] × 9[-2ac^2 / (3c^2 - 5ac)] × 5[3(a^2)b / (4ab - a^2) ]

Hence, if a, b, and c be three distinct numbers such that (ab)/(a-b)=(bc)/(2b-2c)=(ca)/(3c-3a) then the value of 8a9b5c is 8 [-(b^2)c / (2b^2-bc)] × 9[-2ac^2 / (3c^2 - 5ac)] × 5[3(a^2)b / (4ab - a^2) ]

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A unit circle has  radius of 1 (r = 1).  For any point on the unit circle.

According to our question-

A unit circle has  radius of 1 (r = 1).  For any point on the unit circle, we have (by the Pythagorean Theorem):

x2 + y2 = 1

x2 + (-1/4)2 = 1

Hence, A unit circle has  radius of 1 (r = 1).  For any point on the unit circle

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For smallest value n=127, the decimal representation is possible.

When the first three digits after the decimal point are 0.251,

we consider the smallest value of n....

Assume the number is otherwise in the form of

[tex]\frac{m}{n} = 0.X251 \ldots,[/tex]

where X is a k-digit string and n is as small as possible.

Then

[tex]10^k \cdot \frac{m}{n} - X = \frac{10^k m - nX}{n} = 0.251 \ldots[/tex]

. Since  [tex]10^k m - nX[/tex] is an integer and [tex]\frac{10^k m - nX}{n}[/tex] is a fraction between 0 and 1, we can rewrite this as [tex]\frac{10^k m - nX}{n} = \frac{p}{q}[/tex], where [tex]q \le n[/tex]. Then the fraction [tex]$\frac pq = 0.251 \ldots$[/tex]   suffices.

Thus we have[tex]\frac{m'}{n} = 0.251\ldots[/tex], or

[tex]\frac{251}{1000} \le \frac{m'}{n} < \frac{252}{1000} \Longleftrightarrow 251n \le 1000m' < 252n \Longleftrightarrow n \le 250(4m'-n) < 2n.[/tex]

As [tex]$4m' > n$[/tex], we understand that the minimum value of [tex]$4m' - n$[/tex] is 1; hence we need [tex]$250 < 2n \Longrightarrow 125 < n$[/tex].

Since [tex]$4m' - n = 1$[/tex], we need [tex]$n + 1$[/tex] to be divisible by four, and this occurs for the first time when n = [tex]\boxed{ 127 }[/tex]

(note that if[tex]4m'-n > 1,[/tex] then [tex]$n > 250$[/tex]).

this gives [tex]$m' = 32$[/tex] and the fraction [tex]$\frac {32}{127}\approx 0.25196 \ldots$[/tex]).

Thus, the smallest value for n is 127.

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Your question is incomplete, please complete the question

The dilation of a line segment around a point can be accomplished by following a few simple steps as described in the detailed explanation.

The figure of the triangle required to dilate a line around a point is attached.

Here P is the point and AB is the line-segment.

The steps to draw a dilated line is as follows.

It appears that the points A', B', and C' are parallel. More points on l can be chosen and dilated around point P to reveal that the dilated points also appear to lie on the line passing through points A', B', and C'.

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Using Tap working problem solving,

when faucet and drain working together, the time to fill up the tub is 60 minutes.

We have the following information about question,

The time taken by faucet to fill up the tub

= 15 minutes

The time taken by drain to empty the tub

= 20 minutes

we have to calculate the time in which the tub is fully fill the tube with drain open.

Rate of fill up the tub by faucet = 1/15 i.e 1/15 th of tub in one minute .

Rate of empty the tub by drain = 1/20 i.e 1/20 th of tub in one minute.

In one minute both are working together and will fill the tub = 1/15 - 1/20 = 1/60 i.e 1/60 th of tub

Since, it takes one minute to fill 1/60th of the tub and it will take 60 minutes to fill the tub.

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If the two corresponding sides and the angle added to these sides of one triangle are the same as the corresponding sides and the included angle of the second triangle,then it is said that the two triangles are congruent.

If AB = XY, angle A = angle X, and angle C = XZ, then triangle ABC is comparable to triangle XYZ mathematically.

All of the sides and angles of one triangle will be equal to those of the other triangle if the SAS congruence theorem holds true for any two triangles.

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