sequence and series

​

The value of the integrand is -ln | cos(x) | + C

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

Integral of tan(x) w.r.t x

The antiderivative (indefinite integral) of the tangent function, tan(x), with respect to x cannot be expressed in terms of elementary functions

However, it can be expressed in terms of special functions, such as the hyperbolic tangent and the dilogarithm.

One possible representation of the antiderivative of tan(x) is:

∫tan(x) dx = -ln | cos(x) | + C

where C is an arbitrary constant of integration.

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Answer:

D. 35°.

Step-by-step explanation:

1. additional building is performed and provided in the attachment;

2. m∠SUV=55/2x=27.5x;

3. m∠TUS=180-m∠SUV=180-27.5x;

4. m∠SUC=0.5(180°-95°)=42.5°, then m∠SUC=m∠CSU=42.5°;

5. m∠UST=90-m∠CSU=90-42.5=47.5°;

6. in ΔSTU: m∠UST+m∠TUS+m∠STU=180°, then

47.5°+180°-27.5x+10x+5°=180, ⇒ x=3;

7. finally, if x=3, then m∠STU=m∠STV=3*10+5=35°.

The value of x is 7.5. Each pair of alternate interior angles formed by a transversal cutting two parallel lines are equal. Each pair of interior angles on the same side of a transversal that cuts two parallel lines are supplementary, or they add up to 180 degrees.

Straight, equally spaced lines that never cross each other and are on the same plane are called parallel lines. Any line that crosses two straight lines at two different points is said to be a transversal. Different angles are created when any two parallel lines cross a transversal. Between these pairs of angles, there is a relationship.

Since,

The lines are bisected by parallel lines.

hence,

[tex]\frac{4}{6}=\frac{5}{x} \\\\x=\frac{30}{4} \\\\x=7.5[/tex]

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According to the problem Lily's net monthly income is $4,375.

Net monthly income is the total amount of money an individual earns each month after deducting any taxes, insurance, or other expenses from the total gross income. It is the amount of money a person has available to pay bills, save, or use for other purposes. Net monthly income is often referred to as take-home income, since it is the money an individual actually has after taxes and other deductions.

1: Lily's net monthly income is $4,375.

2: Lily's down payment is $46,000 (20% of $230,000).

3: Yes, a $1,200 monthly mortgage payment is within Lily's Spending Guideline for housing. The guideline recommends spending no more than 30% of net income on housing. In this case, 30% of $4,375 is $1,312.50, so $1,200 is within the guideline.

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Using Mathematical operations, Her monthly income is 4375. As 1200 < 4375, she can manage monthly mortgage payment within the Spending Guideline for housing.

Arithmetic operations are the foundation of mathematics. Addition, subtraction, multiplication, and division are the fundamental operations that can be carried out on relations.

We hardly never go a day without using them. Every day, we perform mathematical operations to determine total revenue and expenses for businesses, make monthly or annual budgets, calculate distances, and more.

For instance, we use them to figure out how many questions there were altogether on our assignment, how much time and money we spent, how many chocolates we consumed, how many points we obtained overall for all of our subjects, etc.

Given,

Lily makes a net annual income of $52,500,

Her monthly income will be = 4375

20% down payment on a $230,000 house = 0.20 × 230000 = 46,000

Her down payment = 46,000

her mortgage will be $1,200 per month.

As 1200 < 4375, she can manage monthly mortgage payment within

the Spending Guideline for housing.

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Tax on hotel room that costs $120 = $11.7

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation.

Given,

Function for tax of hotel room

T(x)=0.06x+4.5

where x is the cost in dollars of the room

T(x) = the tax in dollars.

Cost of the room = $120

Tax of the room

= 0.06(120) + 4.5

= 7.2 + 4.5

= $11.7        

Hence, $11.7 is the tax on room that costs $120.

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Answer:

X=-2-y

2(-2-y)+2y=8

-4-2y+2y=8

Step-by-step explanation:

-4-8=0

=-12

Answer:

x=-4.5 y=13.5

Step-by-step explanation:

plug in y=-3x to x+y=9

x+(-3x)=9

distribute

x-3x=9

combine like terms

-2x=9

isolate the variable

-2x/-2=9/-2

x=-4.5

plug in x=4.5 into y=-3x

y=-3(-4.5)

simplify

y=-13.5

done :D

According to the problem,Therefore, x = 12714.29 and y = 1285.71. Mitchell invested $12714.29 at 6% and $1285.71 at 7%.

Investment is the process of using money to purchase assets in order to generate income or increase in value. Investing involves the purchase of assets such as stocks, bonds, mutual funds, real estate, commodities, or other investments in order to achieve financial goals. It is a long-term strategy that involves setting aside funds for the future, with the expectation of obtaining a return on the investment. The ultimate goal of investing is to maximize the return on an investment while minimizing the risk of loss.

Let x be the amount invested at 6%, and y be the amount invested at 7%.

We know that x + y = 14,000, as this is the total amount of the advance.

We also know that the total interest each year is $930. We can calculate the interest individually using the formula I = Prt, where P is the principal (the amount invested), r is the interest rate and t is the time (1 year in this case).

For the 6% investment: I = (x)(0.06)(1) = 0.06x

For the 7% investment: I = (y)(0.07)(1) = 0.07y

We now have two equations:

0.06x + 0.07y = 930

x + y = 14,000

We can use substitution to solve for x and y.

Substitute x + y = 14,000 into equation 0.06x + 0.07y = 930 to get 0.06(14,000) + 0.07y = 930.

Simplifying, we get 840 + 0.07y = 930.

Subtract 840 from both sides to get 0.07y = 90.

Divide both sides by 0.07 to get y = 1285.71.

We now know that y = 1285.71.

Substitute y = 1285.71 into x + y = 14,000 to get x = 14,000 - 1285.71 = 12714.29.

Therefore, x = 12714.29 and y = 1285.71. Mitchell invested $12714.29 at 6% and $1285.71 at 7%.

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The evaluation of (f - g)(x) of the functions gives -x² - 4x + 5

A function is an expression that shows the relationship between the independent variable and the dependent variable.  A function is usually denoted by letters such as f, g, etc.

We have the functions, f(x) = 5x + 5 and g(x) = x² + 9x.

In order to find (f - g)(x), we will subtract g(x) = x² + 9x from f(x) = 5x + 5. That is:

(f - g)(x) = (5x + 5) - (x² + 9x)

(f - g)(x) = 5x + 5 - x² - 9x

(f - g)(x) = -x² - 4x + 5

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Answer:

4.6%

Step-by-step explanation:

Correct me if I'm wrong

219.66-210=9.66

210x4.6%=9.66

Answer: 4.6%

Step-by-step explanation:

I just took the test :)

The mean commute distance (in miles) for the students in the sample is given as follows:

7.9 miles.

The mean of a data-set is calculated as the sum of all observations in the data-set divided by the number of observations.

The histogram gives the number of observations in each range.

Hence, to obtain the sum of the observations, we consider the midpoint of each range, as follows:

S = 20 x 2 + 20 x 6 + 14 x 10 + 7 x 14 + 4 x 18 + 3 x 22

S = 536.

The number of people is given as follows:

20 + 20 + 14 + 7 + 4 + 3 = 68.

Hence the mean is of:

536/68 = 7.9 miles.

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Answer:

Step-by-step explanation:

The two correct representations of the inequality -3(2x - 5) < 5(2 - x) are:

-6x + 15 < 10 - 5x

6x - 15 > 5x - 10

The equation of the line passing through the points (-2,1) and (2,-11) the previous is -3 is -3x - 4.

It is also commonly represented by the variable "m" in the equation of a line: y = mx + b. A positive slope indicates that the line rises as x increases, a negative slope indicates that the line falls as x increases, and a slope of zero indicates that the line is horizontal.

The equation of the line with slope -3 and passing through the point (-2,1) can be found using point-slope form of a line:

y - 1 = -3 (x + 2)

y = -3x - 5 + 1

y = -3x - 4

The equation of the line passing through the points (-2,1) and (2,-11) is y = -3x - 4.

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The equation that represents f(x) is f(x) = x²-2x-8.

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We can use a system of equations to solve x + 8 = 12 and the both equations are true, we can conclude that the number is 4.

What is system of equations ?

A system of equations is a collection of two or more equations that are solved simultaneously to find the values of the variables that satisfy all of the equations. Each equation in the system represents a constraint on the variables, and the solution to the system is the set of values for the variables that satisfy all of the constraints.

We can use a system of equations to solve this problem. If the number in question is called "x", we can write:

x + 8 = 12

y

3 * x = 12

Now we can solve for "x". First, we will isolate "x" from the first equation:

x + 8 = 12

x = 12 - 8

x = 4

Now we can check if this result also satisfies the second equation:

3 * x = 12

3 * 4 = 12

Since both equations are true, we can conclude that the number is 4.

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The gravitational force exerted by the spheres on a particle of mass m located at (a)r=a,(b)r=b, and (c)r=c are

    [tex]F=\frac{Gm_1m_a}{R_1^2}\\\\F=\frac{Gm_1m}{b^2}}\\\\F=m(\frac{Gm_1}{c^2} +\frac{Gm_2}{c^2} )[/tex]

Total gravitational Field at r=a

Total gravitational Field at r=a=Field due to solid sphere+ field due to spherical shell

we know the gravitational field inside the spherical shell is zero.

and gravitational field due to the solid sphere inside = [tex]\frac{GMr}{R^2}[/tex]

so here

Total gravitational Field=Field due to solid sphere+ field due to spherical shell

[tex]G=\frac{Gm_1a}{R_1^2} +0\\\\G=\frac{Gm_1a}{R_1^2}[/tex]

Force exerted on particle F=mG

[tex]F=\frac{Gm_1m_a}{R_1^2}[/tex]

when r=b:

we know the gravitational field inside the spherical shell is zero.

and gravitational field due to a solid sphere outside at a distance:

[tex]r=\frac{GM}{r^2}[/tex]

so here

Total gravitational Field=Field due to solid sphere+ field due to spherical shell

[tex]G=\frac{Gm_1}{b^2}}+0\\\\G=\frac{Gm_1}{b^2}}[/tex]

Force exerted on particle F=mG

[tex]F=\frac{Gm_1m}{b^2}}[/tex]

gravitational field due to a hollow sphere outside at a distance:

[tex]r=\frac{GM}{r^2}[/tex]

and gravitational field due to a solid sphere outside at distance:

​[tex]r=\frac{GM}{r^2}[/tex]

so here

Total gravitational Field=Field due to solid sphere+ field due to spherical shell

[tex]G=\frac{Gm_1}{c^2} +\frac{Gm_2}{c^2}[/tex]

Force exerted on particle F=mG

[tex]F=m(\frac{Gm_1}{c^2} +\frac{Gm_2}{c^2} )[/tex]

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When 6,132 is divided by the exponent (10 to the 3rd power) it yields the value 6.132.

What is an exponent?

The way of representing huge numbers in terms of powers is known as an exponent. Exponent, then, is the number of times a number has been multiplied by itself.

The integer value is given as 6132.

The exponent value is given as 10³.

Use the arithmetic operation of division -

6132 / 10³

Write the numerator of the expression in the exponent form -

(6.132 × 10³) / 10³

Then use the exponent rule - a^x / a^y = a^(x-y).

6.132 × 10^(3-3)

6.132 × 10^0

Anything raised to the power 0 is 1.

6.132 × 1

6.132

Therefore, the value is obtained as 6.132.

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The correct notation for the sample size in this research study is "n = 100."

A sample is a portion of the population that is selected to represent the entire population.

In this study, the sample size of 100 individuals is a representation of the entire population of Americans, and the researchers are using this sample to gather information about the effect of caffeine on memory.

The notation "n" and "N" are commonly used in statistics to refer to sample size and population size, respectively. In this research study, the sample size is 100 individuals, so the correct notation is "n = 100."

In this case, the sample size of 100 individuals is considered a small to medium sample size.

Therefore, option (a) is correct.

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The graph that shows the relationship would be C. graph of a line passing through the points negative 2 comma 4 and 0 comma 0

To find the graph, first find the slope of the line using the points from the table, ( - 1, 2 ) and ( 1, - 2).

The slope is:

= ( Y2 - Y1) / ( X 2 - X 1 )

= ( - 2 - 2 ) / ( 1 - ( - 1 ))

= - 2

This means that if a point on the graph has the x value of - 2, the y value would be:

= x - value x slope

= - 2 x - 2

= 4

The point would therefore be ( -2, 2 ) which makes option C correct.

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The two types of tickets cost $36 and $21.

A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.

Variables are the name given to these symbols because they lack set values.

In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.

Given:

Let x = cost of a crab meal

and, y = cost of a vegetarian meal

So, the system of equations

36x + 16y = 1632.......(1)

21x + 28y = 1344 ...........(2)

Solving equation (1) and (2) we get

36x + 16 ( 1344 - 21x) / 28 = 1632

36x + 768 - 12x = 1632

24x = 864

x= 36

and, y = 1344 - 21(36) / 28

y = 21

Hence, tickets cost are $36 and $21.

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Answer:

year 1998

Step-by-step explanation:

The Probability that an incoming lot contain 36% of imperfect cloths will be acceptable is 0.00957

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. The probability of an event is a number between 0 and 1,

Given that Cappelli Inc. designs and manufactures women's apparel using material from various mills.

Their acceptance sampling plan states that 10 two inch squares of the incoming material must be carefully checked. If 3 or less squares reveal imperfections, the lot is accepted

Here no of defects in the 10 checked is Binomial because each trial is independent and there are only two outcomes.

X = no of defects is Bin (10, 0.36)

The probability  that an incoming lot from Blufton Mills that contains 36 percent imperfect cloth will be accepted

P (X ≤ 3) = 0.00957.

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The solution of the given differential initial value problem is

y(x) = 1/25 e⁻⁵ˣ(6e⁵ˣ - 5x - 6).

Given is the initial value problem -

y''' + 10y'' + 25y' = 0

We can write the -

[tex]$25 \frac{d}{d x} y{\left(x \right)} + 10 \frac{d^{2}}{d x^{2}} y{\left(x \right)} + \frac{d^{3}}{d x^{3}} y{\left(x \right)} = 0$[/tex]

The given differential equation can be solved and written as -

y(x) = 1/25 e⁻⁵ˣ(6e⁵ˣ - 5x - 6)

Therefore, the solution of the given differential initial value problem is

y(x) = 1/25 e⁻⁵ˣ(6e⁵ˣ - 5x - 6).

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(c) The argument is valid, and we can conclude that Penelope missed class because she got a detention.

(e) The argument is valid, and we can conclude that Penelope did not miss class because she got an A and did not get a detention.

(c) To prove this argument's validity, we need to define the predicates and express the hypotheses and conclusion using them:

Let "M(x)" be the predicate "x missed class", and "D(x)" be the predicate "x got a detention".

Hypotheses: M(Penelope), D(Penelope)

Conclusion: M(Penelope)

Using modus ponens, which states that if P implies Q and P is true, then Q must be true, we can conclude that M(Penelope) is true:

From M(Penelope) and "Every student who missed class got a detention", we have D(Penelope)

From D(Penelope), we have M(Penelope)

So, the argument is valid, and we can conclude that Penelope missed class because she got a detention.

(e) To prove this argument's validity, we need to define the predicates and express the hypotheses and conclusion using them:

Let "M(x)" be the predicate "x missed class", "D(x)" be the predicate "x got a detention", and "A(x)" be the predicate "x got an A".

Hypotheses: A(Penelope), ~D(Penelope)

Conclusion: ~M(Penelope)

Using modus tollens, which states that if P implies Q and Q is false, then P must be false,

we can conclude that M(Penelope) is false:

From A(Penelope) and "Every student who missed class or got a detention did not get an A",

we have ~M(Penelope) & ~D(Penelope)

From ~D(Penelope), we have ~M(Penelope)

So, the argument is valid, and we can conclude that Penelope did not miss class because she got an A and did not get a detention.

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Note that since the diagonals of rectangle QRST intersect at P and m∠PTS = 34°, thus,  m∠SRT is 56°.

A rectangle is a quadrilateral with four right angles in Euclidean plane geometry. Rectangles have the following fundamental properties: A rectangle is a quadrilateral.

With regard to the above problem:

Since ∠QRS = 90° (based on the qualities of a rectangle) ; and

∠QRT ≅ ∠RTS = 34° (based on the rule of alternate angles,

Thus, m∠SRT = 90° - 34°

∠SRT = 56°

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Full Question:
The diagonals of the rectangle QRST intersect at P. Given that

m∠PTS= 34° and QS=10, find m∠SRT.

See attached image.

The minimum quantity that the factory must sell in order to obtain benefits is x is 3.

Assuming that the unit price and unit cost are known, in addition to the fixed costs, the following formulas are proposed to determine the minimum quantity number of units that the factory must sell in order to obtain benefits: Input, cost, total cost, benefit, and utility. This example illustrates these formulas.

Unities sold = x =?

Unitary selling price = [tex]S.(2x-2)[/tex]

Unit cost is [tex]\frac{S}{(x+4)}[/tex]

Fixed costs = [tex]\frac{S}{160}[/tex]

Assumption: I(x) = p*x= x*

[tex](2x-2)[/tex]

Total cost: C(x)=[tex]x*(x+4)+160[/tex]

U(x) = [tex]x*(2x-2)-x*(x+4)+160[/tex]

U(x) = [tex]2x^{2} -2x-x(4)+160[/tex]

U(x)= [tex]x^{2} -6x+160[/tex]

derived from and equal to zero:

U'(x)= [tex]2x-6=0[/tex]

x = 3​.

Therefore, the minimum quantity that the factory must sell in order to obtain benefits is x is 3.

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Answer:

= 11.496503

Step-by-step explanation:

Answer:

[tex]x=\frac{y-3}{2\\}[/tex]

Step-by-step explanation:

Swap sides so that all variable terms are on the left-hand side. Subtract 3 from both sides. Divide both sides by 2. Dividing by 2 undoes the multiplication by 2.

By rewriting the 3 given systems, we will see that:

1) Independent and consistent.

2) Independent and inconsistent.

3) Dependent.

Let's analyze each system.

Let's look at the first one, we can rewrite it as:

y = 2x + 3

y = -x - 3

So, the slopes are different, meaning that we have one solution, so this system is consistent and independent.

Now let's look at the second system, we can rewrite it as:

y = 3x - 2

y = 3x + 2

Here both lines have the same slope, so the lines never do intersect, meaning that the system is inconsistent and independent.

For the final system, we have:

y = (-1/2)*x + 2

y = (-1/2)*x + 2

So, we have two equal lines, so this system is dependent.

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Based on the information, it can be inferred that after one hour Jake and Lisa will have filled 3 socks each.

To find the time necessary for both of them to have filled the same number of socks, we must take into account the speed at which each one fills the socks.

Additionally, when Jake arrives at work, Lisa has already filled two stockings. According to the above, after one hour Lisa will complete all 3 socks and Jake will have filled the first 3 socks for him.

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