Consider a given function.

find a and b .

When 2, 325 is rounded to the nearest thousand, the result would be 2, 000.

To round a number to the nearest thousand, identify the place value of the rounding digit, which is the thousandth place (the third digit from the right).

Then, look at the digit to the right of the rounding digit. If this digit is 5 or greater, then round up the rounding digit by adding one to it. If the digit is less than 5, keep the rounding digit as it is.

The number 2, 325 would therefore be rounded to 2, 000 because the digit " 3 " is less than 5.

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The value of all trigonometric functions can be found with the help of a triangle,

from the diagram of the point and reference triangle needed to answer the problem.

Plot the point (2,-3) which is in quadrant 4

Draw a right triangle such that one leg is resting on the x axis and the other leg is perpendicular to the x axis. The hypotenuse goes from (0,0) to (2,-3).

The horizontal leg is 2 units long. So a = 2

The vertical leg is 3 units long. So b = 3

The hypotenuse is unknown but we can use the pythagorean theorem to find it. We'll call it c for now

Use the Pythagorean theorem to find c

   [tex]a^2+b^2=c^2\\\\2^2+3^2=c^2\\\\c=\sqrt{13}\\[/tex]

The hypotenuse is sqrt(13) which is shown in the attached image (Fi

Using figure 1, now refer to figure 2 to compute the ratios for sine, cosine, and tangent

Figure 3 shows how to compute the ratios for cosecant, sectant, and cotangent

With figure 1, the leg furthest from angle theta is the opposite side (in this case -3). The leg closest to the angle theta is 2 units long.

[tex]Sin\theta=\frac{3}{\sqrt{13}}\\\\cos\theta = 2/\sqrt{13}\\\\tan\theta=3/2\\\\same\ as,\\cot\theta=2/3\\sec\theta=\sqrt{13}/2\\cosec\theta=\sqrt{13}/3[/tex]

The complete question is -

A point on the terminal side of an angle in standard position is given. Find the exact value of each of the six trigonometric functions of θ.

Point given: (2,-3)

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The sample size required to estimate the fraction of tenth graders reading at or below the eighth grade level at the 95% confidence level with an error of at most 0.02 is 685.

Population is the term typically used to refer to the number of people in a single area.

To estimate the fraction of tenth graders reading at or below the eighth grade level at the 95% confidence level with an error of at most 0.02.

we can use the formula for sample size in a proportion estimation problem:

n = (Z^2 * p * (1 - p)) / E^2

Plugging in the values:

n = (1.96^2 * 0.17 * (1 - 0.17)) / (0.02^2)

n = 684.2

Since n must be a whole number, we round up to the next integer:

n = 685

So, the sample size required to estimate the fraction of tenth graders reading at or below the eighth grade level at the 95% confidence level with an error of at most 0.02 is 685.

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Yes, that is correct. A system of linear equations with fewer equations than unknowns is often referred to as an underdetermined system.

In this type of system, there are infinitely many solutions or no solutions, and it is not possible to determine a unique solution for the variables.

When a system of equations has more variables than the equation, therefore, it has fewer rules to curb the variables, giving the variables more freedom to take up values from their respective domain(Usually it is a set of real numbers), in such a condition the system of equations has infinite solutions. This kind of system is known as an underdetermined system.

Hence, the system of equations doesn't have a unique solution.

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QRST is a Rhombus, then:

Rhombus is a parallelogram with all congruent sides. Rhombus' diagonals are perpendicular and bisects its angles. Rhombus has 4 congruent right angles created from its perpendicular diagonals and form 2 pairs of two congruent isosceles triangle.

RT bisects of QRS because every diagonals on rhombus bisect their repsective angles.

QS is perpendicular to RT because both QS and RT are rhombus' diagonals and diagonals in rhombus are perpendicular with each other.

QS bisects RT because QS divide rhombus QRST into 2 congruent triangles, QRS and QTS. Since both triangles' height are half of RT, then QS bisects RT.

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Note that the rates that are unit rates are given as follows:

Note that a unit rate is a rate in which the second term equals 1. It represents a comparison of a quantity to 1.

Since:

Therefore, it is right or correct to state that 2/4,  9/1, and 10:1 are unit rates.

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The statement is false, as it is possible for three vectors to not span Rn but still lie on the same plane, such as the three vectors (1,1,1), (2,2,2), and (3,3,3), which all lie on a plane that passes through the origin.

(a) In order for three vectors un, U2, U3 to span Rn, all possible values of n must be 3 or greater. This is because three vectors can only span a 3-dimensional space or higher. Therefore, any n greater than or equal to 3 will work.

(b) Since {u, uy, u3} is a spanning set for Rn, then {u, u, + u2, u, -u3} is also a spanning set for Rn. This is because adding or subtracting any of the vectors in the original set will still form a set of three vectors that span Rn.

(c) This statement is always true. If three vectors ui, uy, u3 are linearly independent, then any linear combination of those three vectors (such as u,, u, + u2, u, -u3) will also be linearly independent.

(d) This statement is also always true. If three vectors ui, uy, u3 are linearly independent, then any linear combination of those three vectors (such as u,, u, + u2, u, -u3) will also be linearly independent.

(e) This statement is false. It is possible for three vectors ui, uy, u3 to not span Rn but still lie on the same plane. For example, the three vectors (1,1,1), (2,2,2), and (3,3,3) do not span R3, but they all lie on the same plane that passes through the origin (x+y+z=0).

The statement is false, as it is possible for three vectors to not span Rn but still lie on the same plane, such as the three vectors (1,1,1), (2,2,2), and (3,3,3), which all lie on a plane that passes through the origin.

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(i) the coordinates of the vertex S = Q + QS = (-1, -1) + (2, 2) = (1, 1).

(ii) the coordinates of the point of intersection of the diagonals PR and QQS is point (-1, 0).

(i) The coordinates of the vertex S can be found by shifting the vector PQ to QS. The vector PQ is given by Q - P = (-1 - (-3), -1 - (-1)) = (2, 0). The vector QS can be found by shifting PQ by 2 units vertically, so QS = (2, 2). To find the coordinates of S, we add the vector QS to the point Q: S = Q + QS = (-1, -1) + (2, 2) = (1, 1).

(ii) The point of intersection of the diagonals PR and QS can be found by finding the midpoint of each diagonal and checking if they are equal. The midpoint of PR is given by ( (1 + -3)/2 , (1 + -1)/2 ) = (-1, 0). The midpoint of QS is given by ( (1 + -1)/2 , (1 + 1)/2 ) = (0, 1). Since the midpoints are equal, the diagonals PR and QS intersect at the point (-1, 0).

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The estimated cost for constructing the house in New Orleans, Louisiana of the given square feet is option A. $113,750.

Multiplication of two numbers is defined as the addition of one of the number repeatedly until the times of the other number.

a × b means that a is added to itself b times or b is added to itself a times.

Given that, construction rate per square feet is $65.

We have to find the estimated cost for a two-story home of 950 ft2 on the first floor and 800 ft² on the second floor.

Total area of the plot = 950 ft² + 800 ft²

                                   = 1750 ft²

Estimated cost can be calculated by multiplying total area with the construction rate per square feet.

Estimated cost = 1750 ft² × $65

                         = $113,750

Hence the estimated cost is $113,750.

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The domain of the function is Domain: (-2, 2). The range is:

Range: (-2, 2). The given graph does not have exactly one output for one input value. Hence, the given is not a function.

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is typically represented as y = f. (x).

The domain of the function are the input values or the x-coordinate of the graph. From the given graph the domain is:

Domain: (-2, 2)

The range of the function are the output values, or the y-coordinate of the graph. From the given graph the range is:

Range: (-2, 2)

We know that a function is a relationship between inputs where each input is related to exactly one output.

The given graph does not have exactly one output for one input value.

Hence, the given is not a function.

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a)

The value of k is 0.61.

b)

The population in 2010 is 739013.

The population in 2015 is 15604434.

c)

In 5.2 years, the population will reach 2.5 million.

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

We have,

P = 1657.5 [tex]e^{kt}[/tex]

P is the population in t years.

Now,

In 2000, t = 0,

a)

In 2005, t = 5 and P = 1940,000

So,

P = 1657.5 [tex]e^{kt}[/tex]

1,940,000 = 1657.5 [tex]e^{5k}[/tex]

1940000/1657.5 = [tex]e^{5k}[/tex]

1170.44 = [tex]e^{5k}[/tex]

Taking logs on both sides.

log 1170.44 = 5k

3.07 = 5k

k = 0.61

b)

In 2010, t = 10

So,

kt = 0.61 x 10 = 6.1

P = 1657.5 [tex]e^{6.1}[/tex]

P = 1657.5 x 445.86

P = 739012.95

P = 739013

In 2015, t = 15

So,

P = 1657.5 [tex]e^{9.15}[/tex]

P = 1657.5 x 9414.44

P = 15604434

c)

P = 2.5 million = 2,500,000

Now,

P = 1657.5 [tex]e^{6.1}[/tex]

Solve for t.

2,500,000 = 1657.5 [tex]e^{0.61t}[/tex]

2500000/1657.5 = [tex]e^{0.61t}[/tex]

1508 = [tex]e^{0.61t}[/tex]

Taking logs on both sides.

log 1508 = 0.61t

3.18 = 0.61t

t = 3.18/0.61

t = 5.21

t = 5.2

Thus,

a)

k = 0.61

b)

P = 739013 in 2010

P = 15604434 in 2015

c)

In 5.2 years, the population will reach 2.5 million.

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Answer:

Step-by-step explanation:

3 (w - 2) ≥ 10

3w - 6 ≥ 10

3w ≥ 16

w ≥ 16/3

2528.5 centimeters is approximately 0.0253 kilometers.

To convert centimeters to kilometers, we first need to convert centimeters to meters, then meters to kilometers.

1 kilometer = 1000 meters

1 meter = 100 centimeters

So, to convert 2528.5 centimeters to meters, we divide by 100:

2528.5 cm / 100 = 25.285 meters

And to convert meters to kilometers, we divide by 1000:

25.285 m / 1000 = 0.025285 km

Therefore, 2528.5 centimeters is approximately 0.0253 kilometers.

The answer can be rounded to 4 significant figures, which is the number of significant figures in the original measurement of 2528.5 centimeters:

0.0253 km (rounded to 4 significant figures)

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The multiplication of the expression given is 60x² - 28x - 8.

Expressions are mathematical statements which consist of two or more terms and terms are connected to each other using mathematical operators like addition, multiplication, subtraction and so on.

The given expression is (5x + 1) (12x - 8).

We have to multiply the expressions and simplify.

For any numbers or algebraic expressions, a, b, c and d,

(a + b) (c + d) = ac + ad + bc + bd

(5x + 1) (12x - 8) = (5x × 12x) + (1 × 12x) + (5x × -8) + (1 × -8)

                         = 60x² + 12x - 40x - 8

                         = 60x² - 28x - 8

Hence the multiplication of the given expression gives 60x² - 28x - 8.

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Answer:

Option A) The volume is multiplied by 1/8.

Step-by-step explanation:

For the first volume problem involving the rectangular pyramid, the normal volume formula is V = 1/3 (l)(w)(h).  Since all 3 dimensions are being multiplied by 1/2, this involves multiplying the original formula by (1/2)(1/2)(1/2) which, put together, is multiplying by 1/8, so the volume will be 1/8 of what it was with the original dimensions.

The volume of the rectangular pyramid is multiplied by ( 1/8 )

The volume of the rectangle is given by the product of the length of the rectangle and the width of the rectangle and the height of the rectangle

Volume of Rectangle = Length x Width x Height

Volume of Rectangle = Area of Rectangle x Height

Given data ,

Let the length of the rectangle be L

Let the width of the rectangle be W

Let the height of the rectangle be H

Now , the volume of rectangle be V = L x W x H

And , when the length , width and height are multiplied by ( 1/2 ) , we get

L' = L/2

W' = W/2

H' = H/2

So , the volume V' = ( L' x W' x H' )

The volume of rectangular pyramid V' = ( 1/8 ) V

Hence , the volume is multiplied by ( 1/8 )

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The numerical value of x in the similar pair of triangle is 10.

A ratio is simply the relation between two amounts showing how many times a value is contained within another value.

From the diagram,

Side A of the small triangle = 10

Side B of the small triangle = x + 2

Side A of the Big triangle = 10 + x

Side B of the big triangle = x + 14

Hence;

Ratio of Side A to Side B of the small triangle = 10 / x + 2

Ratio of Side A to Side B of the Big triangle = 10 + x / x + 14

Since the triangle area similar, equate the two ratio and solve for x.

10 / ( x + 2 ) = ( 10 + x ) / ( x + 14 )

Cross multiply

10( x + 14 ) = ( 10 + x )( x + 2 )

Expand using FOIL method

10x + 140 = 10x + 20 + x² + 2x

10x + 140 = x² + 12x + 20

x² + 12x + 20 = 10x + 140

x² + 12x + 20 - 10x - 140 = 0

x² + 2x - 120 = 0

Factor the left side of the equation

( x - 10 )( x + 12 ) = 0

Hence;

x - 10 = 0 and x + 12 = 0

x = 10 and x = -12

Since the dimension of the triangle cannot be negative,

x = 10

Therefore, the value of x is 10.

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The probability of finding a particle in a one-dimensional box of length a between 0 and a/2 is[tex](2/a)^3/3[/tex].

The probability of finding a particle in a one-dimensional box of length a between 0 and a/2 can be calculated using the equation [tex]P = (1/h)^3 * (∫0^a/2Ψ^2dx)[/tex]. The equation is derived from the equation of probability density of finding a particle in a given state which is given as [tex]P = (1/h)^3 * ∫Ψ^2dx[/tex]. In this equation, h is the Planck's constant, and Ψ is the wave function of the particle.

For a particle in a one-dimensional box of length a, the wavefunction can be written as [tex]Ψ(x) = (2/a)^1/2 * sin (nπx/a)[/tex]. Substituting this in the equation for probability density, the probability of finding a particle between 0 and a/2 is [tex]P = (1/h)^3 * (2/a)^3 * (1/3 - 0)[/tex]. Therefore, the probability of finding a particle in a one-dimensional box of length a between 0 and a/2 is [tex](2/a)^3/3[/tex].

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The solution is (FoF) (0) = -8, when f(x) = 3x-2.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

f(x) = 3x-2

i.e. fof(x)

 =3(3x-2)-2

=9x-8

so. fof(0)

   =9*0-8

   = -8

Hence, The solution is (FoF) (0) = -8, when f(x) = 3x-2.

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The solutions to the equations of the triangle is given by x = 86 and the value of y = 56

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the triangle be represented as ABC

Now , the equation is

3x - 5y = -22   be equation (1)

Now , the measure of ∠A = ( y - 18 )°

The measure of ∠B = y°

The measure of ∠C = x°

Now , for a triangle

∠A + ∠B + ∠C = 180°

Substituting the values in the equation , we get

y - 18 + y + x = 180

On simplifying the equation , we get

2y + x = 198   be equation (2)

Multiply equation (2) by -3 , we get

-6y - 3x = -594   be equation (3)

Adding equation (3) and equation (1) , we get

-5y - 6y = -594 - 22

-11y = -616

Divide by -11 on both sides of the equation , we get

y = 56

Substitute the value of y in equation (2) , we get

2 ( 56 ) + x = 198

x + 112 = 198

Subtracting 112 on both sides of the equation , we get

x = 86

Hence , the solution to the equation is ( 86 , 56 )

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The statement 8x 8y 8z; (z < 0 ^ x < y) ) yz > xz is true, since the statement x < y and z < 0 both hold true, and thus yz > xz must also be true. The negation of this statement would be 8x 8y 8z; (z ≥ 0 ∨ x ≥ y) ) yz ≤ xz.

The first-order language of (directed) graphs is L = {r}, where r is a binary relation symbol. The only terms are the variables x , y, z. Atomic formulas have the form (x, y, z). These formulas represent relationships between the terms, such as x being adjacent to y and z.

These relationships can be used to model different types of graphs. For example, to model a directed graph, the binary relation x->y can be used to represent a directed edge from x to y. These formulas can be combined and manipulated to express more complex formulas, such as the perpendicularity relation between two lines or the perpendicular bisector theorem. These formulas allow for solving problems related to geometry and algebra in graphs.

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The speed of Usain Bolt is given by the equation S = 23.25 miles per hour

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as S

Now , the value of S is

The speed at which Usain Bolt won the gold medal = 34.1 feet per second

So , the speed at which Usain Bolt won the gold medal in miles per hour is calculated by dividing the value by 1.467

Substituting the values in the equation , we get

The speed of Usain Bolt S = ( speed of Usain / 1.467 ) miles per hour

On simplifying the equation , we get

The speed of Usain Bolt S = 34.1 / 1.467

The speed of Usain Bolt S = 23.25 miles per hour

Therefore , the value of S is 23.25 mph

Hence , the equation is S = 23.25 miles per hour

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Answer:

last option

number of students in class plus new students

Answer:

yes

Step-by-step explanation:

The largest force that can be supported before any member ruptures is 8, the maximum force that bar BC can support.

To solve this problem, we need to find the maximum force (stress) that each bar can withstand before rupturing.

This can be found using the stress-strain diagram.

For bar AB with a cross sectional area of 2.3 in^2, the maximum stress that it can withstand is given by the maximum slope of the stress-strain diagram, which is approximately 1.

For bar BC with a cross sectional area of 4 in^2, the maximum stress that it can withstand is given by the maximum slope of the stress-strain diagram, which is approximately 2.

Since the force (stress) is equal to the stress times the cross sectional area, the largest force that bar AB can support is given by 1 * 2.3 = 2.3 and the largest force that bar BC can support is given by 2 * 4 = 8.

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The number of longboards that will be defective based on the defective longboards in the month can be predicted to be 2, 500 longboards.

The number of longboards that would be defective for the other skateboarding company is 2, 446 longboards.

To find the defective longboards, first find the percentage of the longboards that were produced in the month. Then multiply this result with the number of longboards to be produced :

= 204 / 10, 200 x 125, 000

= 2 % x 125, 000

= 2, 500 longboards

The same procedure can be used for the other skateboarding company to find the defective products to be:

= 225 / 11, 500 x 125, 000

= 2, 446 longboards

The product for the second company therefore has fewer defects.

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Because in the first expression 4 was factored out to make it 4(2y-6). Before 4 was factored out the expressions were the same.

The correct options are a) ∠ R corresponds to ∠ P'QQ'  and d) side RQ corresponds to QQ'

Dilation means changing the size of an object without changing its shape. The size of the object may be increased or decreased based on the scale factor.

Given that, a triangle PQR is being dilated to form a new triangle P'Q'Q following a rule: (x, y) → (2x, 2y)

This show that, the scale factor is 2 and the dilation is an enlargement,

We know that, dilation transformation makes similar figures, and in similar figures, the corresponding angles and ratio of corresponding sides are equal,

Here, the corresponding angles are:

∠ P = ∠ P'

∠ PQR = ∠ P'Q'Q

∠ R = ∠ P'QQ'

And, the corresponding side:

P'Q'/PQ = P'Q/PR = Q'Q/QR

Therefore, true statements are;

a) ∠ R corresponds to ∠ P'QQ'  

d) side RQ corresponds to QQ'

Hence, the correct options are a) ∠ R corresponds to ∠ P'QQ'  and d) side RQ corresponds to QQ'

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The Taylor expansion of cos(x) around x=0 is [tex]cos(x) = 1 - (x^2/2!) + (x^4/4!) - (x^6/6!) +[/tex]......

We would use this expansion to calculate the cosine of a given angle with an error tolerance of 0.1%.

To do this, we must first write a function taylorcosine() which takes two arguments: the angle (in radians) and the relative percentage error. The function should then return two values: the Taylor approximation for the cosine of the angle, and the number of terms required in the Taylor series to reach this level of error.

For example, if we want to find the approximate value of cos(3π) with an error tolerance of 0.1%, we would call the function as follows:

[tex]val, nsteps = taylorcosine(3π, 1e-3)[/tex]

To find the percentage relative error, we must multiply the relative error by 100.

We can also use the bisection method to determine the roots of a function. To do this, we must write a function myBisection() which takes

three arguments: the function handle, an array containing the lower and upper limits of the starting interval, and the maximum allowable relative error. The return value of the function is the estimate of the root.

For example, if we want to find the root of f(x) = x^2 - 3x + 2, with an initial bracket of [1.3, 0] and an error tolerance of 1%, we would call the function as follows:

root = [tex]myBisection(f, [1.3, 0], 1%)[/tex]

The error is defined as the absolute value of the ratio of the difference between the old and the new estimates of the root and the new estimate of the root.

Error = [tex]|x_new/x_new - x_old|/x_new[/tex]

By writing the taylorcosine() and myBisection() functions, we can calculate the cosine of an angle with a given error tolerance, and determine the roots of a function with a given error tolerance, respectively.

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