How to solve triangles?.

There are four systems of equations in mathematics, namely linear equations, quadratic equations, polynomial equations, and exponential equations.

Each has its own unique properties that can be exploited for solving problems. Linear equations are the simplest system of equations and are the basic building blocks of more advanced systems of equations. Quadratic equations use the product of two terms to solve one equation. For example, 5x + 6y = 20 can be rewritten as 2x + 5y = 20. Polynomial equations use the power of a term squared to solve one equation. For example, x^2 + 7x + 12 = 55 can be written as x2 + 9x + 22 = 55. Exponential equations use the power to the base to solve one equation. For example, x^3 + 3x^2 - 10x - 16 = 0 can be written as x^3 + 3x^2 - 10x - 16 = 0e^-3 = e. These systems can be used to represent real world scenarios such as population growth,

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Answer:

[tex]a_4 = 154[/tex]

Step-by-step explanation:

In this problem, we are asked to find the fourth term ([tex]\square_4[/tex]) of a sequence [tex]a[/tex] given that [tex]a_1 = 5[/tex] and [tex]a_n = na_{n-1} + 2[/tex].

To solve for this, we can first find the second term in the sequence by inputting [tex]a_2[/tex] into the sequence definition.

[tex]a_2 = 2a_{2-1} + 2[/tex]

Then, we can replace [tex]a_{2-1}[/tex] with the value that we were given: [tex]a_1 = 5[/tex].

[tex]a_2 = 2(5) + 2[/tex]

[tex]a_2 = 10 + 2[/tex]

[tex]a_2 = 12[/tex]

Next, we can solve for the third term in the sequence.

[tex]a_3 = 3a_{3-1} + 2[/tex]

And we can replace [tex]a_{3-1}[/tex] with the value that we just solved for: [tex]a_2 = 12[/tex].

[tex]a_3 = 3(12) + 2[/tex]

[tex]a_3 = 36 + 2[/tex]

[tex]a_3 = 38[/tex]

Finally, we can solve for fourth term in the sequence.

[tex]a_4 = 4a_{4-1} + 2[/tex]

We can again replace [tex]a_{4-1}[/tex] with the value that we just solved for: [tex]a_3 = 38[/tex].

[tex]a_4 = 4(38) + 2[/tex]

[tex]a_4 = 152 + 2[/tex]

[tex]a_4 = 154[/tex]

Answer:

  a₄ = 154

Step-by-step explanation:

You want the value of a₄ given the recursive sequence definition ...

  [tex]\begin{cases}a_1=5\\a_{n}=n\cdot a_{n-1}+2\end{cases}[/tex]

The easiest way to find the first few terms of a recursively-defined sequence is to use the recursive definition directly:

  [tex]a_2=2\cdot a_1 +2 = 2(5) +2 = 12\\\\a_3=3\cdot a_2+2=3(12)+2 =38\\\\a_4=4\cdot a_3+2=4(38)+2=154[/tex]

The value of a₄ is 154.

__

Additional comment

The terms of this sequence get rapidly larger. The 10th term is 23,357,002. The general term can be defined in terms of the Euler gamma function and the incomplete gamma function:

  [tex]a_n=\Gamma(n+1)+2e\Gamma(n+1,1)[/tex]

The curve has a y-intercept of -4 and a slope of 2. We will need two points on the line y = 2x - 4 to plot the graph.

Y = 2x + b, where b is the y-intercept, is how the equation y = 2x - 4 can be expressed in slope-intercept form. We must perform the b-solve in order to determine the slope and y-intercept of the equation.

The line's crossing of the y-axis is known as the y-intercept. Setting x equal to 0 in the equation yields y = 2(0) - 4 and the y-intercept, which can be found. As a result, the y-intercept is determined to be y = -4.

The pace at which a line increases is referred to as its slope. Using the coefficient of x, which is 2, we can get the slope. Consequently, the slope of the equation is 2. Y = 2x – 4.

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Full Question = What are the slope and y − intercept of the equation y 2x − 4?

Answer:4 hours and 23 minutes

Step-by-step explanation:1:40-5:40 is 4 hours. Then add 20 minutes and you get to 6 o'clock. Then finally add 3 minutes and you get to 6:03. When you add 4 hr + 20 min + 3 min you get 4 hours and 23 minutes.

Answer:

Circumference = 56.55

Area=254.47

The value of x when solving for the alternate angles is; x = 20 degrees

Alternate angles are defined as angles that occur on the opposite sides of the transversal line and thus have the same size. There are two different types of alternate angles namely alternate interior angles and alternate exterior angles 3x + 50 degrees and 6x - 10 degrees are alternate angles and as such;

3x + 50 = 6x - 10

Rearrange to collect like terms to get;

6x - 3x = 50 + 10

3x = 60

x = 60/3

x = 20 degrees

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Answer:

To prove that cosA/sinA -sinA/cosA=2cos^2A-1/sinA cosA, we can start by multiplying both sides of the equation by sinA cosA:

cosA/sinA -sinA/cosAsinA cosA = (2cos^2A-1)/sinA cosAsinA cosA

Now we can simplify the left side of the equation by using the identity sin^2A + cos^2A = 1:

cosAcosA -sinAsinA = (2cos^2A-1)cosAsinA

Now we can simplify the right side of the equation by using the identity cos^2A = 1 - sin^2A:

cosAcosA -sinAsinA = (2(1 - sin^2A)-1)cosAsinA

Finally, we can simplify the right side of the equation by using the identity sinAcosA = (sinAcosA)/(sinA*cosA):

cosAcosA -sinAsinA = (2(1 - sin^2A)-1)(sinAcosA)/(sinA*cosA)

Since both sides of the equation are equal, we can conclude that the statement is true.

In the context of mathematics, a series has a stagnant difference between terms. Therefore , we can find out the sum of the terms in arithmetic series by multiplying the number of times the average of the very last and first terms.

Thus we can observe that series and knowing the sum of the terms of series is a very crucial task in mathematics. In this head, we will talk about some very known series and series formulas with instances.

Actually, a series in math is very easy the sum of the different numbers or elements of the series. For instance, to make a series from the channel of the first five positive integers 1, 2, 3, 4, 5 we will simply sum   them up. Therefore 1 + 2 + 3 + 4 + 5 is a series.

So, the series of a sequence is the sum of the sequence to some given number of terms, or occasionally til infinity. It is frequently written as S_n.

If the series is 2, 4, 6, 8, 10, … , then the sum of the first 3 terms:

S

3

=2+4+6

S3=12.

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Answer:

Step-by-step explanation:

In the context of mathematics, a series has a stagnant difference between terms. Therefore , we can find out the sum of the terms in arithmetic series by multiplying the number of times the average of the very last and first terms.

Thus we can observe that series and knowing the sum of the terms of series is a very crucial task in mathematics. In this head, we will talk about some very known series and series formulas with instances.

Actually, a series in math is very easy the sum of the different numbers or elements of the series. For instance, to make a series from the channel of the first five positive integers 1, 2, 3, 4, 5 we will simply sum   them up. Therefore 1 + 2 + 3 + 4 + 5 is a series.

So, the series of a sequence is the sum of the sequence to some given number of terms, or occasionally til infinity. It is frequently written as S_n.

If the series is 2, 4, 6, 8, 10, … , then the sum of the first 3 terms:

S

3

=2+4+6

S3=12.

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An increase of 1 on the Richter scale means that the scale has increased by approximately 31 units in terms of Richter scale.

A 1.0 increment in the Richter magnitude scale results in a 31-unit difference in the intensity of an earthquake. The Richter scale (ML), a numerical measure of an earthquake's size, was devised by American seismologists Charles F. Richter and Beno Gutenberg in 1935.

The magnitude of an earthquake is determined by taking the height of the largest seismic wave and dividing it by the logarithm of its amplitude. The original Richter scale has been replaced in modern science by newer, more accurate scales, yet it is still routinely misrepresented in news reports of earthquake intensity as the logarithmic scale on which earthquakes are measured.

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The number of elements in the set that represents the intersection of X and Y is given as follows:

Zero.

The number of elements in the intersection of X and Y is obtained according to the rule presented as follows:

n(X or Y) = n(X) + n(Y) - n(X and Y).

The parameters for this problem have the values given as follows:

Hence the number of elements in the intersection set is given as follows:

n(X or Y) = n(X) + n(Y) - n(X and Y).

40 = 15 + 20 - n(X and Y)

n(X and Y) = 0.

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Answer:

A - Sales associate

B - Summer Camp Counselor

Step-by-step explanation:

First, lets look at how much Joseph would averagely earn from each job.

Cashier in bookstore: in 6 weeks he will earn 900

Summer camp counselor: in 4 weeks 700

Sales associate: in 4 weeks he will earn 850

Overall, becoming a Sales associate will allow Joseph to earn his $850 quicker.

B - the Summer Camp Counselor job will not allow Joseph to make $850 because Summer Camp only lasts 4 weeks.

By solving a system of equations we will see that:

t = 33 and u = 308

The two triangles will be congruent if the correspondent sides have the same measure, then we can write two equations (or a system of equations) like the one below:

16t = 6u

20t -44 = 2u

So we have a little system of equations to solve.

We can isolate u on the second equation to get:

u = (20t - 44)/2

u = 10t - 22

Now we can replace that on the other equation to get:

16t = 6*( 10t - 22)

16t = 60t - 132

We can solve this for t to get:

132 = 60t - 16t

132 = 44*t

132/44 = t

33 = t

And the value of u is:

u = 10t - 22

u = 10*33 - 22 = 308

The values are t = 33 and u = 308

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Answer:

Mai needs 4/3 cups of strawberries in order to make the whole cake.

Step-by-step explanation:

1 cup of strawberries being equal to 3/4 of the cake is also the same as 1 cup of strawberries equaling 75% of the cake. So, what times 0.75 is equal to 1? That is 4/3.

The domain and range are described for a relation, and they are the sets of all the x-coordinates and all the y-coordinates of ordered pairs, respectively. For example, if the relation is R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then: Domain = {1, 2, 3, 4} and Range  = {2, 3}.

The domain and range of a function are the elements of a function. The domain is the set of all the input elements of a function, and the range is the possible outcome given by the function.

Domain→ Function →Range. If there exists a function f: α → β such that every element of α are mapped to elements in β, then α is the domain, and β is the co-domain.

The image of an element 'p' under a relation R is given by 'q', where (p,q) ∈ R. The function's range is the set of images. The domain and range of a function are denoted in general as follows:

Domain(f) = {x ∈ R} and range(f) = {f(x): x ∈ domain(f)}.

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The length of the original piece of fabric is 61 1/2 inches.

Let x be the length of the original piece of fabric.

If the first cut is 1/3 of the length of the original fabric, then we can express that as an algebraic expression 1/3x.

The next two cuts made from the remaining fabric is 10 inches and 5 inches.

10 + 5 = 15 inches

Therefore, the total length that was cut off was 1/3x + 15 inches = 35 1/2 inches.

With the algebraic equation written, solve for x.

1/3x + 15 = 35 1/2

To solve for x, isolate the terms containing the variable x on one sie and the constant terms on the other side.

1/3x = 35 1/2 - 15

1/3x = 20 1/2

Multiply both sides by 3 to get x.

1/3x = 20 1/2

x = 61 1/2 inches

Therefore, the original length of the fabric was 61 1/2 inches.

We can check our solution by adding the length of the cuts off to the original length:

1/3(61 1/2 inches) + 10 inches + 5 inches = 20 1/2 + 10 + 5 = 35 1/2

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The three fundamental trigonometric ratios, namely sine, cosine, and tangent, each yield the ratio of the lengths of the two sides of a right-angled triangle.

We are aware that the sides of a right-angled triangle are called the "legs" and the "hypotenuse," respectively. Thus, the longest side of a right-angled triangle is still referred to as the "hypotenuse" in trigonometry, while the other two legs are referred to as the "opposite side" and "adjacent side."

The opposite, adjacent, and hypotenuse sides of a right-angled triangle are used to define the sin, cos, and tan functions in trigonometry. These are the sin, cos, and tan formulae.

sinФ = OppositeHypotenuse

cos Ф = AdjacentHypotenuse

tan Ф = OppositeAdjacent

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The probability that fewer than 30% of the rolls result in a two is 0.450.

What is probability ?

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.

To find the probability that fewer than 30% of the rolls result in a two, we can use a normal distribution to model the number of twos rolled in 50 rolls.

Let's assume that the mean number of twos rolled in 50 rolls is μ = 50 * (2/6) = 16.67 (since the probability of rolling a two is 2/6). The standard deviation can be calculated as σ = sqrt(50 * (2/6) * (4/6)) = 2.94.

Now, we can use the normal distribution to find the probability that the number of twos rolled is less than 15 (30% of 50 rolls), which is P(X < 15) = P((X - μ) / σ < (15 - 16.67) / 2.94). We can use a standard normal table (z-table) to find this probability, which is approximately 0.450.

Therefore , the probability that fewer than 30% of the rolls result in a two is 0.450.

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There is a 0.450 percent probability that fewer than 30% of rolls will produce a two.

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true.

We can predict the frequency of twos rolled in 50 rolls using a normal distribution to determine the likelihood that less than 30% of the rolls produce two.

Given that the probability of rolling a two is 2/6, let's assume that the mean number of twos rolled in 50 rolls is = 50 * (2/6) = 16.67.

Calculating the standard deviation is as follows:

sqrt(50 * (2/6) * (4/6) = 2.94.

The chance that the number of twos rolled is less than 15 (30% of 50 rolls) can now be calculated using the normal distribution, and it is given by P(X 15) = P((X - ) / (15 - 16.67) / 2.94). It is possible to determine this probability, which is around 0.450, using a common normal table (z-table).

As a result, 0.450 represents the likelihood that fewer than 30% of rolls will produce a two.

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The exponential growth model is given by the equation y = 2000(1.06)ˣ

An equation is an expression showing the relationship between numbers and variables.

An exponential function is in the form:

y = abˣ

a is the initial value and b is the multiplication factor. if b > 1, it is an exponential growth whereas if b < 1, it is an exponential decay.

A Exponential growth model initial value 2,000 growth rate 6%, hence a = 2000, b = 100% + 6% = 1.06

The exponential model becomes y = 2000(1.06)ˣ

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Based on the table of values, the slope and y-intercept include the following:

In Mathematics, the y-intercept is also known as an initial value and the y-intercept of any graph such as a linear function, generally occur at the point where the value of "x" is equal to zero (x = 0). Therefore, the y-intercept is equal to -1/2.

From the information provided in the table above, the slope and data points on this line include the following:

In Mathematics and Geometry, the slope of any straight line can be calculated by using this expression;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Substituting the given points into the slope formula, we have;

Slope (m) = (1 - 0)/(1 + 1)

Slope (m) = 1/2

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To find the domain and range from given points on a graph: the set of all x-coordinates is the input and the set of all y-coordinates is the output.

We know that the domain of function is the set of all possible input values for which the function is well defined.

And the range of function is the set of all possible output values for given input.

To find the domain and range of function we can graphical methhod.

We know that the input values of function are shown on the x-axis and the output values of the function are shown on the y-axis.

This means, the domain of a graph shown on the x-axis. Whereas the range is shown on the y-axis.

To find the domain and range when points on a graph are given:

the set of all x-coordinates would be the domain and the set of all y-coodinates is the range of function.

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Only table {1} represents a exponential function.

       [tex]$e^{x} =\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+\cdots }[/tex]

Given are the function tables as shown in the question.

       {x} → 0  1  2  3  4

       {y} → 0  1  4  9  16

Therefore, only table {1} represents a exponential function.

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Answer: Domain and Range. The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the values that the function shoots out after we plug an x value in.

Step-by-step explanation:

The relation should be written as a set of ordered pairs as follows: A. . ordered pairs: {(0, 2), (2, 6), (4, 10)}.

In Mathematics, a function can be defined as a mathematical expression which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair.

This ultimately implies that, a function is typically used in mathematics for uniquely mapping an input variable (x-value or domain) to an output variable (y-value or range).

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph such as the following:

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The measure of angle PQR is given as follows:

To obtain the measure of angle PQR, first we verify if the triangle PQR is a right triangle or not.

To verify if the triangle is a right triangle we apply the Pythagorean Theorem, which states that the length of the hypotenuse squared is equals to the sum of the lengths of each side squared, in the case of a right triangle.

Hence, for this problem, considering the side lengths, we have that:

10² = 6² + 8²

100 = 36 + 64

100 = 100.

Which is true, hence this is in fact a right triangle and the measure of angle PQR is of 90º.

The problem asks for the measure of angle PQR.

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Answer:8u6

Step-by-step explanation:

8+9_rjs_+5

Answer:

  B(-π/3, 1/2), D(7π/6, -√3/2), E(11π/12, -1/2)

Step-by-step explanation:

You want the coordinates of various points marked on the graph of the sine and cosine functions.

The horizontal divisions on the graph are multiples of π/6. That is, there are 6 divisions between 0 and π. This means the x coordinates of interest are ...

  B = -π/3

  D = 7π/6

  E = 11π/6

Points B and D lie on the red curve, which the graph tells you is a plot of cos(x). Then the coordinates are ...

  B = (-π/3, cos(-π/3)) = (-π/3, 1/2)

  D = (7π/6, cos(7π/6) = (7π/6, -√3/2)

Point E lies on the blue curve, which is a plot of sin(x). Its coordinates are ...

  E = (11π/6, sin(11π/6)) = (11π/6, -1/2)

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Additional comment

The attachment shows the coordinates of points on the unit circle as (cos(θ), sin(θ)). On the given graph, θ is on the horizontal axis (x), and sine and cosine are on the vertical axis. To make use of the attachment, you find the angle, then look for the (cos, sin) coordinates at that angle.

Negative angles are clockwise from θ=0. To find the corresponding angle on the attachment, add 2π to any negative angle.

Answer:

y = −5x − 19

Step-by-step explanation:

Find the equation using slope-intercept form.

y = −5x − 19

The degree of 3x^1 is 1.

A degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 2x^3 + 4x^2 - 5x + 7, the degree is 3 (the highest power of x is x^3). A polynomial of degree 0 is a constant polynomial and a polynomial of degree 1 is a linear polynomial.

To find the degree of a polynomial, you can do the following:

→ Write the polynomial in standard form, with the terms arranged in descending order of exponents.

→ Take the exponent of the term with the highest exponent. This exponent is the degree of the polynomial.

3x^1 is a monomial with a coefficient of 3 and an exponent of 1. It is also called a linear term. The degree of a monomial is the exponent of the variable. So in this case, the degree of 3x^1 is 1.

Therefore, the degree of 3x^1 is 1.

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The solution of the given equations is x= 0.4 and y= 1.

The system of equations is y = -5x + 3 and y = 1. To find the solution, we can substitute the value of y from the second equation into the first equation.

To find the solution to a system of equations, we need to find the values of x and y that satisfy both equations simultaneously.

The system of equations is:

y = -5x + 3

y = 1

We can see that the second equation is already in the form y = a, which means that the y value is constant and equals to 1. Now we can substitute this value of y in the first equation:

1 = -5x +3

So the solution to the system of equations is x = 0.4, y = 1.

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