Decide whether each equation is true for all, one, or no values of x.

By using trigonometry, it can be calculated that

The value of sina is [tex]\pm \frac{1}{\sqrt{3}}[/tex]

What is Trigonometry?

Trigonometry shows the relationship between sides and angles of a right angled triangle.

There are six trigonometrical functions. They are

[tex]sin \theta, \ cos\theta, \ tan \theta, \ cot\theta, \ sec\theta, \ cosec\theta[/tex]

[tex]sin\theta = \frac{perpendicular}{hypotenuse}\\\\cos\theta = \frac{base}{hypotenuse}\\\\tan\theta = \frac{perpendicular}{base}\\\\cot\theta = \frac{base}{perpendicular}\\\\sec\theta = \frac{hypotenuse}{base}\\\\cosec\theta = \frac{hypotenuse}{perpendicular}[/tex]

Trigonometry is a very important tool in mathematics.

Here, the trigonometric equation is

sec 2a = 3

cos 2a = [tex]\frac{1}{3}[/tex]

we know,

[tex]1 - cos 2a = 2 sin^2a\\\\1 - \frac{1}{3} = 2sin^2a\\\\2 sin^2a = \frac{2}{3}\\\\sin^2a = \frac{2}{3} \times \frac{1}{2}\\\\sin^2a = \frac{1}{3}\\\\ sina = \pm \frac{1}{\sqrt{3}}[/tex]

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The statement that is true is All rational numbers can be written as the quotient of Integers

Any number that can be written as a fraction and has an integer as both the numerator and the denominator is referred to as rational number.

In a rational number, the denominator cannot be zero.

Integer refers to whole numbers. They are generally of two types the positive integers and the negative integers

Quotients refers to the end product of division example 2 ÷ 1 = 2/1

2/1 is the quotient, both 2 and 1 are integers

All decimals are rational - A counter example to this is √2

It has decimal and not a rational number because it cannot be represented as a fraction

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The statement that is true is All rational numbers can be written as the quotient of Integers

What are rational numbers ?

Any number that can be written as a fraction and has an integer as both the numerator and the denominator is referred to as rational number.

In a rational number, the denominator cannot be zero.

Integer refers to whole numbers. They are generally of two types the positive integers and the negative integers

Quotients refers to the end product of division example 2 ÷ 1 = 2/1

2/1 is the quotient, both 2 and 1 are integers

All decimals are rational - A counter example to this is √2

It has decimal and not a rational number because it cannot be represented as a fraction

The rate of change in the account for a week is $157.5.

Annalise has some money in her bank account. She withdraws $22.50 per day from her account for a week. We need to represent the change in the account for the week. The number of days in a week is seven. So, it means that she withdraws money from her bank account daily for seven days. The total amount of money withdrawn from the bank account is the product of the withdrawal amount per day and the total number of days in a week.

A = $22.5*7 = $157.5

Let the initial amount of money in the bank be "$x". The change in the bank balance is $157.5. The current balance is $(x - 157.5).

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The equation of the parabola from the vertex is f(x) = 7(x - 3)² + 5

The equation of the function is given as

f(x) = 7x²

Also, we have

Vertex = (3, 5)

The form of the equation is given as

f(x) = a(x - h)² + k

Where

Vertex = (h, k)

This means that

Vertex = (h, k) = (3, 5)

Substitute (h, k) = (3, 5) in the equation f(x) = a(x - h)² + k

So, we have

f(x) = a(x - 3)² + 5

This also means that

f(x) = 7(x - 3)² + 5

The 7 is gotten from f(x) = 7x²

Hence, the equation of the parabola is f(x) = 7(x - 3)² + 5

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a) Amount of sugar in the tank initially is 0.

b) The amount of sugar in the tank after t minutes is 13.6(1 - [tex]e^{\frac{4t}{1360} }[/tex]).

c) The concentration of sugar in the solution in the tank after 60 minutes is 3.672.

a) Assign the sugar content of the tank for time t to the symbol A(t). Starting with only clean water, the tank

A(0) = 0

b) Sugar flows in at a rate of

(0.02 kg/L) × (2 L/min) = 0.04 kg/min

and flows out at a rate of

(A(t) ÷ 1360 kg/L) × (2 L/min) = (2A(t) ÷ 1360) kg/min

so that the net rate of change of A(t) is governed by the ODE,

dA(t) ÷ dt = (4 ÷ 100) - (2A(t) ÷ 1360)

A'(t) + (2A(t) ÷ 1360) = 2 ÷ 100

Multiply both sides by the integrating factor [tex]e^{\frac{4t}{1360} }[/tex] to condense the left side into the derivative of a product:

[tex]e^{\frac{4t}{1360} }[/tex] A'(t) + (2 [tex]e^{\frac{4t}{1360} }[/tex] A(t) ÷ 1360) = 4 [tex]e^{\frac{4t}{1360} }[/tex] ÷ 100

( [tex]e^{\frac{4t}{1360} }[/tex] A(t))' = 4 [tex]e^{\frac{4t}{1360} }[/tex] ÷ 100

Integrate both sides:

[tex]e^{\frac{4t}{1360} }[/tex] A(t) = ∫ 4 [tex]e^{\frac{4t}{1360} }[/tex] ÷ 100

[tex]e^{\frac{4t}{1360} }[/tex] A(t) = 13.6 [tex]e^{\frac{4t}{1360} }[/tex] + C

Solve A(t);

A(t) = 13.6 + C [tex]e^{\frac{4t}{1360} }[/tex]

A(0) = 0, substitute the value

0 = 13.6 + C [tex]e^{\frac{4t}{1360} }[/tex]

C = -13.6

so that the amount of sugar at any time t is,

A(t) =  13.6 - 13.6[tex]e^{\frac{4t}{1360} }[/tex]

A(t) = 13.6(1 - [tex]e^{\frac{4t}{1360} }[/tex])

c) As t → 60, the exponential term converges to 0 and is left with

[tex]\lim_{t \to \(60}[/tex] A(t) = 13.6 × (1 - 0.73)

[tex]\lim_{t \to \(60}[/tex] A(t) = 3.672 kgs of sugar.

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Answer:

x=6

Step-by-step explanation:

-2x-6=-18

     +6. +6

-2x=-12

/-2. /-2

x=6

Hopes this helps please mark brainliest

Answer:

[tex]y=2x+6[/tex]

Step-by-step explanation:

Slope-intercept form is [tex]y=mx+b[/tex].

Therefore, rearranging the given formula:

[tex]10x-5y=-30[/tex]

[tex]-5y=-10x-30[/tex]

[tex]\frac{-5y}{-5}=\frac{-10x}{-5}-\frac{30}{-5}[/tex]

[tex]y=2x-(-6)[/tex]

[tex]y=2x+6[/tex]

Answer: y=2x+6

Step-by-step explanation:

A slope-intercept form is y=mx+b

Hence,

[tex]10x-5y=-30\\\\10x-5y+5y=-30+5y\\\\10x=-30+5y\\\\10x+30=-30+5y+30\\\\10x+30=5y[/tex]

Divide both parts of the equation by 5:

2x+6=y

Thus, y=2x+6

The Kilometer that Alan was left to walk is given as c six and twenty five thirty thirds kilometers

In order to solve for the solution, we would have to write out the expressions in mathematical forms first

eight and two thirds kilometers = [tex]8\frac{2}{3}[/tex]

This is is the time that Alan is said to travel for him to get to school

The expression one and ten elevenths is written mathematically as [tex]1\frac{10}{11}[/tex]

This is the kilometer that he spent running.

We are to find the time that was used to walk to school.

To get this we have to subtract the kilometer that he ran from the total kilometer that he traveled.

[tex]8\frac{2}{3} - 1\frac{10}{11}[/tex]

we have to express as a mixed fraction then we would subtract

26/3 - 21/11

= 286 - 63 / 33

= 223/33

= 6.75

= 6 25/33

Hence the solution is C.

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the population would be 7,518,400

469,900 • 2^20/5

* 20/5 is part of the exponent

Answer:

7518400

Step-by-step explanation:

Doubling time model:

         [tex]P= P_o 2^{\frac{t}{D}}[/tex]

Here, P is the initial value, D is the time taken to double the quantity, t is the given period of time.

 [tex]P_o = 469,900\\\\\\D = 5 \ years\\t = 20 \ years[/tex]

                    [tex]P = 469,900 * 2^\frac{20}{5}\\\\P = 469,900 * 2^4[/tex]

                         = 469,900 * 2 * 2 * 2 * 2

                         = 7518400

0.0456  is the probability that a sample mean is out of control.

What is probability in math?

UCL = 520

LCL = 480

Mean (X ) = 500

Standard Deviation of sample (Sn) = 11.55

Z (for UCL) = (UCL - Mean)/Sn

Z (for UCL) = (520-500)/10

Z (for UCL) = 2

Z (for LCL) = (LCL - Mean)/Sn

Z (for LCL) = (480 - 500)/10

Z (for LCL) = -2

*Use Z table to find confidence level between Z value of -2 and 2.

Confidence level = 0.4772 + 0.4772

Confidence level = 0.9544

Risk (α ) = 1 - confidence level

Risk (α) = 1 - 0.9544

Risk (α) = 0.0456

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Answer: final answer = 3/8

Step-by-step explanation:
it says 1/8 + 1/4. before you can add them you have to make sure that both fractions have the same denominator first you have to make both of the bottom denominators the same so its 1/8+2/8.
1/8+2/8=3/8

The advantages of bankruptcy include the following: e. All of these choices are correct.

In Financial accounting, bankruptcy can be defined as a legal process that is initiated when an individual or business organization (debtors) is relieved of the responsibility to pay their debts, honor its financial obligations to creditors, or protect them while they are trying to repay their debt.

Generally speaking, there are several advantages of bankruptcy and these include the following:

In conclusion, we can reasonably infer and logically deduce that bankruptcy is generally considered as both a friend and foe (enemy).

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The number of early-admission tickets that it sells is 30.

From the information, 30% of the tickets sold at a school carnival were early-admission tickets and the school sold 100 tickets in all.

Therefore, the number of early admission will be:

= Percentage for early admission × Total number of tickets

= 30% × 100

= 0.3 × 100

= 30

Therefore, there are 30 early admission.

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The rate of change of the surface area of the balloon is 32cm²/sec

We know that volume of the sphere is,

V = 4/3πr³

Given,

dV / dt = +200

r = 5cm

d(SA) / dt = ?

dV / dt = 4πr² dr/dt

80 = 4π (25) dr/dt

dr / dt = 0.8 / π

SA = 4πr²

d(SA) / dt = 8πr dr / dt

d(SA) / dt = 8π × 5 × 0.8 / π

d(SA) / dt = 32cm²/sec

Hence the rate of change of the surface area is 32cm²/sec

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Considering the triangle with exterior angles and interior angles as described. The statements that are true include

The problem is solved considering this 3 theorems about triangles

exterior angles theorem: m∠2 + m∠3 = m∠6

If a triangle side is created, the outside angle that results is equal to the sum of the two opposite internal angles.

Linear pair theorem: m∠5 + m∠6 =180°

Linear pairs are particular instances of supplementary angles that only involve two neighboring angles.

Sum of angles in a triangle is equal to 180 degrees:

m∠2 + m∠3 + m∠5 = 180°

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The number of real roots found are-

1. (x-4)(x+3)^2(x-1)^3=0 ; 6 real roots.

2. X^2(x^3-1)=0 ; 3 real roots.

3. X(x+3)(x-6)^2=0 ; 4 real roots.

4. 3x(x^3-1)^2=0 ; 3 real roots.

5. (x^3-8)(x^4+1)=0​ ; 1 real roots.

Consider the given cases-

1. (x-4)(x+3)^2(x-1)^3=0,

Put each equals zero.

x – 4 = 0 or (x + 3)^2 = 0 or (x – 1)^3 = 0,

Thus,

x = 4

x = –3 twice

x = 1 thrice

All are real numbers, thus given polynomial has 6 real roots.

2. x^2(x^3-1)=0

Put each equals zero.

x² = 0

x³ – 1 = x³ – 1³ = (x – 1)(x² + x + 1) = 0

The quadratic expressions has 2 imaginary roots.

Use the discriminant formula to find-

D = b² – 4ac = 1² – 4(1)(1) = 1 – 4 = –3 < 0.

Thus, roots that are not real.

So, the real roots of this polynomial : 0(twice) and 1.

Hence, thus given polynomial has 3 real roots.

3. x(x+3)(x-6)^2=0

Put each equals zero.

x = 0

x = 3

x = 6 twice

Hence, thus given polynomial has 4 real roots.

4. 3x(x^3-1)^2=0

Put each equals zero.

3x = 0 and  x = 0

(x^3 – 1)² = [(x – 1)(x² + x + 1)]² = (x – 1)²(x² + x + 1)² = 0,

Here, only two roots are real

The quadratic expression do have unreal roots, thus 1 twice are the real roots.

Hence, thus given polynomial has 3 real roots..

5. (x^3-8)(x^4+1)=0

Put each equals zero.

x³ – 8 = x³ – 2³ = (x – 2)(x² + 2x + 4) = 0

There is only one real root i.e, 2, the quadratic expression do have imaginary roots

Check using the discriminant formula

D = b² – 4ac = 2² – 4(1)(4) = 4 – 16 = – 12 < 0.

x⁴ + 1 = 0 has imaginary roots.

Hence, thus given polynomial has 1 real roots..

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The given equation  m<2 = 28⁰ so here in this case option A is correct .

Simplification:

       <1 = <3

<1 = 5x + 8              ………………equation 1.

<3 = 12x – 20         ………………equation 2.

By angle property <1 = <3  [ given ]

and

<1 = <2   [ vertically opposite angles ]

We have some options so we can use it to get answer more quickly:

Put <1=28 and <3=28

And solve it for X.

28 = 5x + 8                     Eq1

20=5x => x=4

28 = 12x - 20                 Eq2

28 + 20 = 12x

48 = 12x

4 = x

X = 4

As <1 = <2 ( vertically opposite angles )

From the first equation  <1 = <2  = 5x + 8 ......( equation 3)

Putting the value of x = 4 in (equation 3)

<2 = 5* 4+ 8

<2=28

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The dependent variable depends on a particular situation. The independent variable does depends on the result of an experiment. The controlled variable is a variable whose value is constant.

What is dependent variable?

The variable being measured or tested in an experiment is known as the dependent variable. The results of the participants' tests, for instance, since that is what is being measured, would be the dependent variable in a study looking at how tutoring affects test scores.

What is independent variable?

This isn't the same as an experiment's independent variable, which is a variable that can stand on its own. Tutoring would be the independent variable in the aforementioned case. The dependent variable (test results), however, may alter depending on the independent variable (tutoring).

What is controlled variable?

Anything kept constant or constrained in a research study is referred to as a control variable. Despite not being relevant to the study's objectives, this variable is controlled because it might have an impact on the results.

Numbers called parameters are used to define the characteristics of entire populations. Statistics are numerical representations of sample characteristics. One population parameter is the average income in the United States.

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The correct inequality symbol that makes the statement true is:

1000 < (6²)³.

The inequality symbols, and their meaning, are presented as follows:

In the context of this problem, the numbers being compared are presented as follows:

1000 and (6²)³.

The power of power property of a number with multiple exponents means that the base is kept while the exponents are multiplied.

The number in which this property is applied in this problem is (6²)³, in which:

Hence the equivalent number is given as follows:

(6²)³ = 6^6 = 6 x 6 x 6 x 6 x 6 x 6 = 46,656

Which is a greater number than 1,000, meaning that 1,000 is less than the number, and the inequality is:

1000 < (6²)³.

The problem is given by the image at the end of the answer.

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0.05093 is the probability that exactly 13 students will take up application development in the coming year .

Describe probability using an example?

Average rate (number) of students who chose to study Application Development (λ)= 18

Poisson random variable(number of students taking up Application Development in the coming year)(x)=13

Poisson distribution = P(X = x) = e^−λ (λ)^x/ x!

Here, probability of 13 students choosing application development next year= P(X=13)

               =   ^(−18 ) 18^13/  13!

P(X=13) = 0.0509286= 0.05093

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The top of the 20-foot ladder used for painting will be  19 feet up from the base of the house

information given in the question

Brian borrowed a 20-foot extension ladder: hypotenuse = 20-foot ladder

he sets the base of the ladder six feet from the house

adjacent = 6 feet from the house

opposite =  how far up will the top of the ladder reach = ?

The problem is solved using the Pythagoras theorem is applicable to right angle triangle.  the formula of the theorem is

hypotenuse² = adjacent² + opposite²

substituting the values into the equation

let x be thow far up will the top of the ladder reach

20² = 6² + x²

400 = 36 + x²

400 - 36 = x²

x² = 364

x = √364

x = 19.08

The ladder Brain borrowed will go 19 feet high

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complete question

Brian borrowed a 20-foot extension ladder to use to paint his house. If he sets the base of the ladder six feet from the house how far up will the top of the ladder reach?

The correct option is  a.) a questionnaire, used as observing the relationship between independent variable as well as a dependent variable is the goal of a quantitative research technique known as employed as.

Quantitative methods place an emphasis on accurate measurements or the statistical, quantitative, or numerical evaluation of information gathered through surveys, polls, and other types of research, as well as the manipulation of statistical data that has already been obtained using computing methods.

Quantitative research involves gathering numerical information and using it to understand a specific event or generalise it across groups of individuals.

Its primary attributes are:

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Answer:

see explanation

Step-by-step explanation:

(a)

AB = 4 + x

(b)

the area (A) of rectangle ABCD is calculated as

A = length × breadth

   = AB × AD

   = (4 + x)(6 + x) ← expand using FOIL

   = 24 + 10x + x²

given A = 28 , then

x² + 10x + 24 = 28 ( subtract 24 from both sides )

x² + 10x = 4 ← in the form x² + ax = b

with a = 10 and b = 4

A statement that two numbers or expressions are equal is an equation.

An equation is a mathematical expression that states that two expressions are equal. An equation's solution is the value that, when substituted for the variable, makes the equation true. To accomplish our goal, we employ two equality principles: the addition principle and the multiplication principle.

An equation is the statement that illustrates the variables given. In this case, two or more components are taken into consideration to describe the scenario. It is vital to note that an equation is a mathematical statement which is made up of two expressions that are connected by an equal sign.

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With the help of the given trendline equation, the length of the flight will be 3,613.34 miles.

So, the length of the flight will be:

Now, calculate for D as follows:

We can write as D = 3,613.34

Therefore, with the help of the given trendline equation, the length of the flight will be 3,613.34 miles.

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x = -1 and y = -13, (-1,-13) is the solution to the system of equation y = 3x - 10 and y = -2x - 15.

Given the system of equation in the question;

To find the solution to the system of equation, replace the occurrence of y in the second equation with y = 3x - 10  and solve for x.

y = -2x - 15

3x - 10 = -2x - 15

Add 10 to both sides

3x - 10 + 10 = -2x - 15 + 10

3x = -2x - 15 + 10

Add 2x to both sides

3x + 2x = -2x + 2x - 15 + 10

3x + 2x = - 15 + 10

Add like terms

5x = -5

x = -5/5

x = -1

Now, to solve for y, replace all occurrence of x with -1 in the first equation and solve for y.

y = 3x - 10

y = 3( -1 ) - 10

y = -3 - 10

y = -13

Therefore, the solution to the system of equation is (-1,-13).

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The value of m is 36 degrees and the value of n is 27 degrees

Consider the triangle on right side

Two sides of the triangle is equal, therefore it is an isosceles triangle.

One angle of the triangle = 126 degrees

Other two angles of the triangle is equal

The sum of interior angles of the triangle is 180 degrees

Then, the equation will be

126 + n + n = 180

2n + 126 = 180

2n = 180 - 126

2n = 54

n = 54/2

n = 27 degrees

Consider the full triangle

The equation will be

(m +27) + 90 +27 = 180

m + 27 + 90 + 27 = 180

m + 144 = 180

m = 180 - 144

m = 36  degrees

Hence, the value of m is 36 degrees and the value of n is 27 degrees

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