What is rational function and irrational function?

The height of the tower where surveyor positions a transit that is 2 m tall  = 45.19 m

Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent are the six trigonometric functions. The trigonometric functions in mathematics are real functions that link the angle of a right-angled triangle to the ratios of the lengths of the two sides.

given:

transit height = 2 m

distance from the base of the tower = 35 m

angle of elevation = 51°

Utilize trigonometric functions to solve this problem. The tangent trigonometric function connects the opposite side and the neighboring side.

Opposite side / Adjacent side = tan Ф

solving opposite side = Opposite side = Adjacent side x tan Ф

Opposite side = 35 x tan 51°

Opposite side = 35 x  1.234

Opposite side = 43.19 m

calculating the tower height = 43.19 + 2 m

= 45.19 m

therefore the height of the tower = 45.19 m

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The polynomial 3xA² is a linear monomial having one term and a degree one.

A polynomial is an algebraic expression.

A polynomial of degree n in variable x can be written as,

a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² +...+ aₙ.

There are many types of polynomials according to the number of terms they have.

If a polynomial has one term it is called a monomial, Has two terms called a binomial, and having three makes it a trinomial.

Given, A polynomial 3xA².

Now, The variable is 'x' and raised to the power of 1 so it is linear and consists of only one term so it is a monomial as 3 and A are constants.

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The image of the point after the translation is (5,3)

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

Point =(4, 4)

This point can be represented as

(x, y) = (4, 4)

The translation equation is given as

(x,y)-->(x+1,y-1)

Substitute the known values in the above equation, so, we have the following representation

(x,y)-->(4+1,4-1)

Evaluate

(x,y)--> (5,3)

Hence, the image is (5,3)

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Similar to the plane reflection, the line reflection is the inversion through a straight line. In reality, it goes through the same transformation as a 180-degree rotation.

A reflection is referred to as a flip in geometry.

A reflection is the shape's mirror image.

A line, called the line of reflection, will allow an image to reflect through it.

Every point in a figure is said to reflect the other figure when they are all equally spaced apart from one another. The line reflection is the inversion through a straight line, much as the plane reflection.

Actually, it undergoes the same change as a 180-degree rotation.

For this reflection, the rule would be written as RX-axis (x,y)→(x,y).

Rule of Notation: This notation rule tells you that image A has been reflected across the y-axis and the x-coordinates have been multiplied by -1.

It has the form by-axis A→B = axis (x,y)→(x,y).

Therefore, similar to the plane reflection, the line reflection is the inversion through a straight line. In reality, it goes through the same transformation as a 180-degree rotation.

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The formula for finding the altitude of an equilateral triangle is ((sqrt (3))/2)a.

As we know, the sides of an equilateral triangle are equal. Let the side of the triangle be 'a'.

Dropping a perpendicular from the vertex of the triangle, the triangle gets divided into two right triangle where the base of the triangle is divided into two as well.

Therefore, by using the Pythagoras' theorem,

we can get the altitude (say 'h') of the triangle by:

h^2 = a ^ 2 - (a/2) ^ 2 = 3a ^ 2/4

h = sqrt (3a^2/4) = ((sqrt (3))/2)a.

The question is incomplete, the complete question is:

What is the formula for finding the altitude of an equilateral triangle?

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15  is the mean of 20 and 10 .

What are the mean, median, and example?

A data collection is ordered from least to largest, and the median is the midpoint number. A data set's mode is the number that appears the most frequently. The most frequent number, or the one that happens the most frequently, is known as the mode.

                     Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.

x = 20 + 10/2

x = 30/2

x = 15

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Answer: The sample space of rolling an eight-sided die would be (1, 2, 3, 4, 5, 6, 7, 8)

Step-by-step explanation: Sample space is defined as a set of all possible outcomes. In an eight-sided die, there are eight faces and hence, we have the probability of getting any number within 1 to 8.

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Answer:

D

Step-by-step explanation:

Therefore, In the figure, a∥b  angle m∠7 = 34° .

An angle is a figure in Euclidean geometry made up of two rays that share a vertex, or common terminus, and are referred to as the sides of the angle. Angles of two rays lie in the plane containing the rays. Angles can also result from the intersection of two planes. Dihedral angles are the name given to them.

Here,

Given that a and b are parallel lines,

∠3 and ∠7  are corresponding angles and congruent

m∠3 = 34°

thus ,m∠7 =m∠3 = 34°

so, m∠7 = 34°

Therefore, In the figure, a∥b  m∠7 = 34° .

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We are asked to find the first derivative of,

[tex]\longrightarrow g(x) = \sqrt{x^2+2x}[/tex]

Here we can write the term [tex]x^2+2x[/tex] by adding and subtracting 1 as,

[tex]x^2+2x = x^2+2x+1-1[/tex]

[tex]x^2+2x = (x+1)^2-1\quad[\because\, x^2+2x+1=(x+1)^2][/tex]

Thus,

[tex]\longrightarrow g(x) = \sqrt{(x+1)^2-1}\quad\dots(1)[/tex]

Now take,

[tex]x+1=\sec\theta\quad\dots(2)[/tex]

[tex]x=\sec\theta-1[/tex]

[tex]dx=\sec\theta\tan\theta\, d\theta[/tex]

[tex]\dfrac{d\theta}{dx}=\dfrac{1}{\sec\theta\tan\theta}\quad\dots(3)[/tex]

Then (1) becomes,

[tex]\longrightarrow g(x) = \sqrt{sec^2\theta-1}[/tex]

We have,

[tex]\sec^2\theta-1=\tan^2\theta[/tex]

So we get,

[tex]\longrightarrow g(x) = \tan\theta[/tex]

Now,

[tex]\longrightarrow g'(x) = \dfrac{d}{dx}\,[\tan\theta][/tex]

By chain rule,

[tex]\longrightarrow g'(x) = \dfrac{d}{d\theta}\,[\tan\theta]\cdot\dfrac{d\theta}{dx}[/tex]

[tex]\longrightarrow g'(x) = \sec^2\theta\cdot\dfrac{1}{\sec\theta\tan\theta}\quad\quad\textrm{[From (3)]}[/tex]

[tex]\longrightarrow g'(x) = \sec\theta\cdot\dfrac{1}{\tan\theta}[/tex]

[tex]\longrightarrow g'(x)=\dfrac{1}{\cos\theta}\cdot\dfrac{\cos\theta}{\sin\theta}[/tex]

[tex]\longrightarrow g'(x)=\dfrac{1}{\sin\theta}\quad\dots(4)[/tex]

But we have,

[tex]\sin^2\theta+\cos^2\theta=1[/tex]

[tex]\sin\theta=\sqrt{1-\cos^2\theta}[/tex]

[tex]\sin\theta=\sqrt{1-\dfrac{1}{\sec^2\theta}}[/tex]

[tex]\sin\theta=\dfrac{\sqrt{\sec^2\theta-1}}{\sec\theta}[/tex]

[tex]\sin\theta=\dfrac{\sqrt{(x+1)^2-1}}{x+1}\quad\quad\textrm{[From (2)]}[/tex]

[tex]\sin\theta=\dfrac{\sqrt{x^2+2x}}{x+1}[/tex]

Hence (4) becomes,

[tex]\longrightarrow\underline{\underline{g'(x)=\dfrac{x+1}{\sqrt{x^2+2x}}}}[/tex]

This is the first derivative of the given function.

Step-by-step explanation:

To find the ratio between Rohit's savings and expenditure, we need to divide his savings by his expenditure:

Savings / Expenditure = (1500 - 1250) / 1250 = 250 / 1250 = 1/5

The ratio between Rohit's savings and expenditure is 1/5.

To find the ratio between Rohit's expenditure and income, we need to divide his expenditure by his income:

Expenditure / Income = 1250 / 1500 = 5/6

The ratio between Rohit's expenditure and income is 5/6.

To find the ratio between Rohit's income and savings, we need to divide his income by his savings:

Income / Savings = 1500 / (1500 - 1250) = 1500 / 250 = 6/1

The ratio between Rohit's income and savings is 6/1.

Answer:

m = 5

Step-by-step explanation:

The equation is y = mx + b

The m here is the slope; in this case, the slope is 5

Answer:

1. y=x    2. x=4   3. y=3x-10   4. y=-2/5x-1/5    5. y=-8x-13    6. y=-x+3

Step-by-step explanation:

The rewritten expression using the summation notation is ∑ k = 1 1! + 2! + 3! + ... + (n+1)!

The summation notation is a way to represent the sum of a series of terms. In the summation notation, the series of terms is represented by an expression that is followed by a summation symbol (∑). The summation symbol is usually followed by an index of the summation (k in this case), an equal sign (=), the starting value of the index, and a colon (:). The series of terms is then written below the summation symbol, with the index replacing the variable in each term.

To rewrite the given expression using the summation notation, we can use the following steps:

Identify the series of terms: 1 + 2 + 3 + ... + n.

Write the series of terms below the summation symbol: ∑ k = 1

Replace the variable in each term with the index: 1! + 2! + 3! + ... + (n+1)!

The rewritten expression using the summation notation is ∑ k = 1 1! + 2! + 3! + ... + (n+1)!

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Answer:

Answer is 4, but in the answer choices provided it is 4b.

Step-by-step explanation:

Use PEMDAS! (Order of PEMDAS: Parenthesis, exponent, Multiplication, division, addition, subraction)

1. Multiply 6b x 2= 12b.

2. Divide 12b by 3b = 4. or 4b

Answer:

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

Correct choice is D.

The correct option is A.Hence the given number follow the pattern of consective powers,so unit digit of given number will be 1.

The unit digit is a number placed in the one's position.It is placed right most digit of the given number.The value of every digit in the number system by specifying the position of the particular digit in the number such as ones,tens,hundreds,thousands and so on.

It is easy to identify the unit place of smaller number but when it comes to the raised to the power it becomes difficult to identify by simple observation.Then we use cyclicity chart.It is based on every digit in the number has its own repetitive pattern when raised to any specific power.

For example 2^5=32 ; Here 2 is the unit digit of given number.

Now we use cyclicity concept to  find and get

Unit digit of 27^1=7(7^1=7)

Unit digit of 27^2=9(7^2=49)

Unit digit of 27^3=3(7^3=343)

Unit digit of 27^4=1(7^4=2401)

Hence for every four consective ,the unit digit will follow pattern of 7,9,3,1.

SO Unit digit of given number is 1.

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If we can move one thing without affecting its size or shape such that it perfectly overlays the other picture, then two objects are congruent. These motions are what are known as congruence transformations.

While the image's size and shape are unaffected by rigid body motion, its location and orientation are. The three fundamental movements of a rigid body are translation, reflection, and rotation. Prior to movement, archetypes depict points or forms.

In order for two items to be congruent, one of them must be able to be moved over the other without affecting its size or form. Congruence changes are what we refer to as these motions. The transformation of an object into a congruent object is known as a congruence transformation.

∠A = ∠B

And, also

AB = BC

AD = BD

Therefore,

ΔABC ≅ ΔDEF

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Step-by-step explanation:

circumcentre:

In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle.

othocentre:

The orthocenter is the intersection point of three altitudes drawn from the vertices of a triangle to the opposite sides. A centroid is the intersection point of the lines drawn from the midpoints of each side of the triangle to the opposite vertex.

The roots of the given equation 2x2 + 4x = 0 are -1 and 0. It can be seen from the roots that the nature of the roots is real and equal.

The nature of the roots of a quadratic equation ax2 + bx + c = 0 with real coefficients a, b, and c can be determined by using the quadratic formula. The quadratic formula states that the two roots of the equation are given by:

x = (-b ± √(b2 - 4ac)) / 2a

For the equation 2x2 + 4x = 0, a = 2, b = 4, and c = 0. Substituting these values into the quadratic formula, we get the two roots of the equation as:

x = (-4 ± √(42 - 4(2)(0))) / 2(2)

x = (-4 ± 0) / 4

x = -1, 0

Therefore, the roots of the given equation 2x2 + 4x = 0 are -1 and 0. It can be seen from the roots that the nature of the roots is real and equal.

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Answer:

a) 47

b) 61

Step-by-step explanation: We can setup two equations from the information given above:

2c + 4b = 14 where c is the cost if one carousel ticket and b is the cost of four bumper car tickets.

6c + 9b = 33 where where c is the cost if one carousel ticket and b is the cost of four bumper car tickets like in the 1st equation.

If you divide the second equation by 3, you get 2c + 3b = 11, leaving 2c in both equations.

If you subtract both equations by the elimination method:

    2c + 4b = 14

-

    2c + 3b = 11

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

b = 3, which solves for the cost of one bumper car ticket

If you plug this value back into the first (or second) equation,

ex. 2c + 4(3) = 14 -> 2c = 2 -> c=1

You get c = 1, which solves for the cost of one carousel ticket.

Then you plug these values (c=1 and b=3) in a) and b):

a) 8*1 + 13*3 = 47

b) 10*1 + 17*3 = 61

A corporate bond has a face value of p dollars. The interest each year is 6% of the face value.  the total interest /, in dollars, as a function of the age f, in years, of the bond is

a)I(f) = 0.06 * f * p

b) P(f) = I(f) + p

Generally, (a) The total interest, in dollars, as a function of the age f, in years, of the bond can be expressed as:

I(f) = 0.06 * f * p

This represents the total interest paid over f years, which is calculated by multiplying the annual interest rate (0.06), the number of years (f), and the face value of the bond (p).

(b) The payout P, in dollars, as a function of f can be expressed as:

P(f) = I(f) + p

This represents the total payout over f years, which is calculated by adding the total interest paid (I(f)) and the face value of the bond (p).

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More than 2 in mathematical expression is written as >2.

The relationship between two quantities can be described using comparison terms. There are primarily three comparison terms: more than (>), less than (<), and equal to (=).

More than (>): When one quantity is greater than the other quantity, we use “more than”. For example, 5 > 3.

Less than (<): When one quantity is less than the other quantity, we use “less than”. For example, 8 < 10.

Equal to (=): When two quantities are the same, we use “equal to”. For example, 15 = 15.

More than 2 means, greater than but not including 2, which is written as >2

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The root of the graph of function f(x) = (x-5)³(x+2)² touch the x-axis at -2 and 5.

Given that,

The function is f(x)= (x-5)³(x+2)²

We have to find at which root does the graph function touch the x-axis.

We know that,

Mathematical calculus' core component is functions. The unique forms of relationships are the functions. When it comes to arithmetic, a function is represented as a rule that produces a different result for each input x.

Take the function

f(x) = (x-5)³(x+2)²

f(x) = 0 if a curve touches the x-axis.

⇒ (x - 5)³(x + 2)² = 0.

But if ab = 0

So, a=0 and b=0

⇒ (x - 5)³ = 0 and (x + 2)² = 0

⇒ (x - 5) = 0 and x + 2 = 0

⇒ x=5 and x=-2.

Therefore, The root of the graph of function f(x) = (x-5)³(x+2)² touch the x-axis at -2 and 5.

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The factors form of the given equation 3x²−6x is 3x(x−2)

Given that:

3x²−6x

To find : The factor form of 3x²−6x

The splitting or decomposition of an entity (such as a number, a matrix, or a polynomial) into the product of another entity, or factors, whose multiplication results in the original number, matrix, etc., is known as factorization or factoring in mathematics. You will mostly learn this idea in your lower secondary studies, which run from grades 6 to 8.

=3×x×x−2×3×x, factor each monomial

=3x(x−2), factor out common factor 3x

The factors form of 3x²−6x is 3x(x−2)

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The number less than 3,000.when you divide me by 32, my remainder is 30. when you divide me by 58, my remainder is 44 is one of this {798, 1726, 2654}.

There exists a, b∈N such as

N= 30+32a=44+58b

Therefore,

16a-29b= 22-15=7

Let us notice that 5×29-9×16=1

Therefore, 35×29-63×16=7

Hence, a= -63 is a non-fundamental solution to our equation.

We can show that the other solution are of the form a=-63+29k, with k∈Z

Therefore, the first positive solutions are

a= -63+3×29= 24

N= 798

a= -63+4×29= 53

N= 1726

a= -63+5×29=82

N= -63+6×29= 111

N= 3582

Therefore, the number is one of {798, 1726, 2654}.

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The graph of inequality x < −2 is as shown below.

In this question we need to graph x is less than or equal to negative 2.

i.e., we need to graph an inequality x < −2

First we draw a straight vertical line x = -2.

The graph of x = -2 is a straight line parallel to y-axis and intersects x-axis at (-2, 0)

We represent the graph of x = −2 by the dashed line.

The dashed line indicates that the line is the boundary of the inequality but not part of it. Shade in the left side of the line to indicate the inequality x<−2.

Therefore, the graph of x is less than or equal to negative 2 is as shown below.

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The probability of forming a bow-tie quadrilateral from four randomly chosen chords is approximately 6.25%.

This is because there are exactly 28 possible quadrilaterals that can be formed from the seven given points, and only one of those 28 quadrilaterals is a bow-tie quadrilateral. This means that the probability of forming a bow-tie quadrilateral is 1/28, which is equal to 0.0357, or 3.57%.

To explain further, when we draw all of the chords from the seven given points, we have 21 distinct chords. Out of these, 10 chords form triangles, leaving us with 11 chords that form quadrilaterals.

Since four of these chords are randomly chosen, the number of possible combinations of quadrilaterals that can be formed is equal to 11C4, or 495. Out of these 495 combinations, only one is a bow-tie quadrilateral. This means that the probability of forming a bow-tie quadrilateral is 1/495, or 0.002041, or 0.2041%.

Since 0.2041% is too small to be expressed as a whole number, it is rounded up to 6.25.

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Answer:

-100

Step-by-step explanation:

To solve -2^6 - 6^2, we can evaluate each term separately.

2^6 = 64

6^2 = 36

-2^6 - 6^2 = -64 - 36 = -100

The area of the largest rectangle that can be inscribed in the ellipse is 2ab

The term ellipse in math is a circle that has been stretched in one direction to give it an oval shape

Here we have given the following equation x²/a² + y²/b² = 1

Now, we have to find the area of the largest rectangle that can be inscribed in the ellipse

Here we all know that the Area of rectangle = 4xy

That can be written as A = 4xy

Now, we have to differentiate the given expression then we get,

=> dA/dx = 4xdy/dx  + 4y ----------------------(1)

Here we have given an equation  x²/a² + y²/b² = 1

Now, we have to differentiate this one also, then we get,

=> 2x/a²  + 2y(dy/dx)/b² = 0

If we equate both of the equation then we get the value as.

=> dy/dx = - xb²/a²y

Now, substitute the values on equation (1), then we get,

=> dA/dx = 4x( - xb²/a²y)  + 4y

Equate the expression with 0, then we get the value of

=> a²y² = b²x²

=> x²/a² = y²/b²

When we simplify this one then we get the value of x as (a/√2) and  y as b/√2

Therefore, the Area is calculated as,

=> 4xy  =  4(a/√2)(b/√2)  = 2ab

So, the value 2ab is the area of the greatest rectangle that can be inscribed in an ellipse x²/a² + y²/b² = 1

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Answer: olynomial Long Division

Dividing :  x3 + x2 - 20x + 24

                             ("Dividend")

By         :    x - 3    ("Divisor")

dividend     x3  +  x2  -  20x  +  24

- divisor  * x2     x3  -  3x2        

remainder         4x2  -  20x  +  24

- divisor  * 4x1         4x2  -  12x    

remainder          -  8x  +  24

- divisor  * -8x0          -  8x  +  24

remainder                0

Quotient :  x2+4x-8  Remainder:  0

Factoring  x2+4x-8

The first term is,  x2  its coefficient is  1 .

The middle term is,  +4x  its coefficient is  4 .

The last term, "the constant", is  -8

Step-1 : Multiply the coefficient of the first term by the constant   1 • -8 = -8

Step-2 : Find two factors of  -8  whose sum equals the coefficient of the middle term, which is   4 .

     -8    +    1    =    -7

     -4    +    2    =    -2

     -2    +    4    =    2

     -1    +    8    =    7

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Find roots (zeroes) of :       F(x) = x3 + x2 - 23x + 24

    See theory in step 1.3

In this case, the Leading Coefficient is  1  and the Trailing Constant is  24.

The factor(s) are:

of the Leading Coefficient :  1

of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,8 ,12 ,24

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        47.00    

     -2       1        -2.00        66.00    

     -3       1        -3.00        75.00    

     -4       1        -4.00        68.00    

     -6       1        -6.00        -18.00    

     -8       1        -8.00        -240.00    

     -12       1       -12.00       -1284.00    

     -24       1       -24.00       -12672.00    

     1       1        1.00        3.00    

     2       1        2.00        -10.00    

     3       1        3.00        -9.00    

     4       1        4.00        12.00    

     6       1        6.00        138.00    

     8       1        8.00        416.00    

     12       1        12.00        1620.00    

     24       1        24.00       13872.00    

Polynomial Roots Calculator found no rational roots

Answer: x3 + x2 - 23x + 24

 ——————————————————

         x        

Step-by-step explanation: