If θ = 31π/4 , the reference angle is

On solving the question we have that the answer of the equation will be [tex]Answer: "f(x) = 1 + x +\frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ..."[/tex]

A math equation is a mechanism for connecting two statements and indicating equivalence with the equals sign (=). To explain the connection between the two sentences put on each side of a letter, a statistical method can be employed. The software and the logo are usually interchangeable. 2x - 4 equals 2, for example. An equation is a logical expression that asserts the equality of some mathematical expressions in algebra. In the equation 3x + 5 = 14, for example, the equal sign separates the numbers 3x + 5 and 14.

[tex]"f(x) = {e^x} \Rightarrow f'(x) = {e^x} \Rightarrow f''(x) = {e^x} \Rightarrow f'''(x) = {e^x}"[/tex]

Then

[tex]"f(x) = f(0) + \frac{{f'(0)}}{{1!}}x + \frac{{f''(0)}}{{2!}}{x^2} + \frac{{f'''(0)}}{{3!}}{x^3} + ... = {e^0} + \frac{{{e^0}}}{1}x + \frac{{{e^0}}}{2}{x^2} + \frac{{{e^0}}}{6}{x^3} + ... = 1 + x + \frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ..."[/tex]

[tex]Answer: "f(x) = 1 + x + \frac{1}{2}{x^2} + \frac{1}{6}{x^3} + ..."[/tex]

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Answer:

therefore, the expression for John's age in terms of y is 4y - 16.

Step-by-step explanation:

Answer:

John's age = 4y - 16

Step-by-step explanation:

We can start by using the information given to write expressions for the ages of Libby and John in terms of Alfred's age y.

We know that Libby is 4 years younger than Alfred, so her age can be written as:

Libby's age = Alfred's age - 4

We also know that John is 4 times as old as Libby, so his age can be written as:

John's age = 4 × Libby's age

Substituting the expression for Libby's age from the first equation into the second equation, we get:

John's age = 4 × (Alfred's age - 4)

Simplifying this expression, we get:

John's age = 4 × Alfred's age - 16

Therefore, an expression in terms of y for John's age is:

John's age = 4y - 16

Answer:

To write the time taken to fill a tank to the nearest 5 minutes, we first need to convert 2 hours and 43 minutes to minutes. 2 hours = 2 x 60 = 120 minutes So, the total time taken in minutes is: 120 + 43 = 163 minutes To round this to the nearest 5 minutes, we need to consider the last digit of the number of minutes, which is 3. Since 3 is less than 5, we round down to the nearest multiple of 5, which is 160. Therefore, the time taken to fill the tank to the nearest 5 minutes is: 2 hours and 43 minutes = 160 minutes = 2 hours and 40 minutes

To write the given time to the nearest 5 minutes, round down the minutes value if the last digit is less than 5, and keep the hours value as it is.

To write the given time to the nearest 5 minutes, we need to round it to the closest multiple of 5. The given time is 2 hours and 43 minutes. To round to the nearest 5 minutes, we look at the last digit of the number of minutes, which is 3. Since 3 is less than 5, we round down the minutes to 0 and keep the hours as it is. Therefore, the time to the nearest 5 minutes is 2 hours and 40 minutes.

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a. The sketch is attached

b. The vertical height is 18.13 feet

c. The given angle 65° is the angle of elevation.

d. The foot of the ladder is 8.45 feet away from the wall.

7. The angle of elevation is congruent to angle of depression because they are on the alternate sides of two parallel lines.

8.

a. Diagram of triangle CDE is attached

b. The formula that can be used to solve triangle CDE is the Law of Sines

c.

∠ e = 11

c = 75.47

d = 8.77

b. To find the vertical height of the ladder that is touching the wall, we use the trigonometric function

sin 65 = vertical height / height of ladder

sin 65 = vertical height / 20

vertical height = sin 65 * 20

vertical height = 18.13 feet

c. The given angle 65° is the angle of elevation.

d. To find how far the foot of the ladder is from the wall, we can use the trigonometric function cosine.

cos 65° = x / 20

x = 20 cos 65°

x ≈ 8.45 feet

7. The statement is true. The angle of elevation is the angle between the horizontal line and the line of sight from the observer to the object above the horizontal line.

The angle of depression is the angle between the horizontal line and the line of sight from the observer to the object below the horizontal line.

If the two objects are on parallel lines (e.g., the ground and a bridge above it), then the angle of elevation from one object is congruent to the angle of depression from the other object.

8. a. Diagram of triangle CDE is attached

b. The formula that can be used to solve triangle CDE is the Law of Sines

∠ e = 180 - 52 - 17 = 111

sin E / 28 = sin C / c = sin D / d

c = 28 sin 52 / sin 111 = 23.63

d = 28 sin 17 / sin 111 = 8.77

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a) The constant of proportionality is solved to be 9

b) The equation of the relationship is y = 9x

a. To find the constant of proportionality, we can use the formula:

K = y/x

where

k is the constant of proportionality,

y is the dependent variable (output), and

x is the independent variable (input).

In this case, The ordered pair (3, 27) will be used them to solve for k.

where

x = 3

y = 27

k = y/x = 27/3

k = 9

b. The equation for this relationship is written using the formula

K = y/x

rewriting in  terms of y

y = kx

y = 9x

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answer

306.87 degrees and 666.87 degrees.

steps

4tanθ+5=0.

Subtract 5 from both sides to get 4tanθ=-5.

Divide both sides by 4 to obtain tanθ=-5/4.

θ = arctan(-5/4)

Using a calculator, we find that θ is approximately -53.13 degrees or 306.87 degrees.

Since we need to specify the interval [0,360), we add 360 degrees to the negative solution:

θ = -53.13 + 360 = 306.87 degrees

Therefore, the solutions to the equation 4tanθ+5=0 on the interval [0,360) are approximately 306.87 degrees and 666.87 degrees.

Equation with tangent function.

Title: Solving an equation involving tangent function

Synopsis: We will solve the equation 4tanθ+5=0 on the interval [0,360).

Abstract: To solve the equation 4tanθ+5=0, we need to isolate the variable (θ) on one side of the equation. We start by subtracting 5 from both sides, giving us 4tanθ=-5. Then we divide both sides by 4, obtaining tanθ=-5/4. Finally, we take the arctangent (or inverse tangent) of both sides to find θ, remembering to specify the interval [0,360).

Numbered list:

Start with the equation 4tanθ+5=0.

Subtract 5 from both sides to get 4tanθ=-5.

Divide both sides by 4 to obtain tanθ=-5/4.

Take the arctangent (or inverse tangent) of both sides to find θ.

Specify the interval [0,360) when giving the solution.

Analogy: Solving this equation is like solving a puzzle where you need to rearrange the pieces to get the right picture.

One-sentence summary: We solve the equation 4tanθ+5=0 by subtracting 5, dividing by 4, taking the arctangent, and specifying the interval [0,360).

To solve the equation 4tanθ+5=0, we need to isolate the variable (θ) on one side of the equation.

Subtracting 5 from both sides gives us: 4tanθ = -5

Dividing both sides by 4: tanθ = -5/4

Taking the arctangent (or inverse tangent) of both sides, we get: θ = arctan(-5/4)

Using a calculator, we find that θ is approximately -53.13 degrees or 306.87 degrees.

However, we need to specify the interval [0,360), which means we need to add 360 degrees to the negative solution, giving us 306.87+360 = 666.87 degrees (which is equivalent to 306.87-360 = -53.13 degrees in this interval).

Therefore, the solutions to the equation 4tanθ+5=0 on the interval [0,360) are approximately 306.87 degrees and 666.87 degrees.

Sure, here's just the math:

4tanθ + 5 = 0

Subtracting 5 from both sides:

4tanθ = -5

Dividing both sides by 4:

tanθ = -5/4

Taking the arctangent (or inverse tangent) of both sides:

θ = arctan(-5/4)

Using a calculator, we find that θ is approximately -53.13 degrees or 306.87 degrees.

Since we need to specify the interval [0,360), we add 360 degrees to the negative solution:

θ = -53.13 + 360 = 306.87 degrees

Therefore, the solutions to the equation 4tanθ+5=0 on the interval [0,360) are approximately 306.87 degrees and 666.87 degrees.

ChatGPT

Answer:

x = 26 degrees

Step-by-step explanation:

Supplementary angles mean that they sum up to be 180 degrees therefore

C + D = 180 degrees

128 degrees + D = 180 degrees

D = 180 degrees - 128 degrees

D = 52 degrees

If the measure of the angle D is two times the value of x then x is...

x = D/2

x = 52 degrees / 2

x = 26 degrees

Answer:

75.6%

Step-by-step explanation:

Reducing Debt: 2024's debt to equity ratio has increased from 67.6% to 75.6% over the past 5 years. Debt Coverage: 2024's operating cash flow is negative, therefore debt is not well covered.

The measure of line segment JK using the Secant Theorem is found as 7 units.

According to the intersecting secants theorem, the product of one whole secant segment but also its outer segment is equivalent to the product of the other whole secant segment as well as its external segment when two secants cross at an exterior point.

According to the Secant Theorem,

If two secants are drawn from an identical point through a circle, their products, A times their external segment and B times their external segment, are equal.

So, using Secant Theorem:

JK*JL = RM*RL

8* (2x + 7) = 24 * 7

Solving the equation:

x = 7

Thus, the  measure of line segment JK using the Secant Theorem is found as 7 units.

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Answer:

Step-by-step explanation:

3.1:

Using the compound angle expansion, we have:

sin 2x = sin(x + x) = sin x cos x + cos x sin x

sin 2x = 2 sin x cos x

Therefore, sin 2x = 2 sin x cos x.

3.2:

Using the result from 3.1:

3.2.1: sin 100° = 2 sin 50° cos 50° = sin 100° = 0.766

3.2.2: sin 46° = 2 sin 23° cos 23° = sin 46° = 0.719

3.2.3: sin 40° = 2 sin 20° cos 20° = sin 40° = 0.642

3.3:

Using the identity sin(a ± b) = sin a cos b ± cos a sin b:

3.3.1: 2 sin 19° cos 19° = sin 38° = sin (45° - 7°) = sin 45° cos 7° - cos 45° sin 7° = (1/√2)cos 7° - (1/√2)sin 7° = -sin (7° - 45°) = -sin 38°

Therefore, 2 sin 19° cos 19° = -sin 38°.

3.3.2: 2 cos 40° sin 40° = sin 80° = sin (45° + 35°) = sin 45° cos 35° + cos 45° sin 35° = (1/√2)cos 35° + (1/√2)sin 35° = sin (35° + 45°) = sin 80°

Therefore, 2 cos 40° sin 40° = sin 80°.

3.3.3: sin 25° cos 155° = (1/2)(sin 180°) = 0

Therefore, sin 25° cos 155° = 0.

Answer:

Step-by-step explanation:

The area of the figure is 1120

A: (4,-2)

B: (1,-1)

C: (1,-4)

Hope this helps! Just graph it.

Answer:

Step-by-step explanation:

For a right angle triangle we use Pythagoras theorem

hypotenuse²=x²+(x+18)²

                    =x²+x²+36x+324

                    =2x²+36x+324

Answer: The top 1% of the world's population holds more than 45% of global wealth.

Step-by-step explanation:

To find the percentage of global wealth concentrated in the hands of the top 1%, we need to divide the wealth of the top 1% by the total amount of global wealth, and then multiply by 100 to get the percentage.

Percentage of global wealth held by top 1% = (Wealth of top 1% / Total global wealth) x 100

We are given that the total amount of global wealth is $418.3 trillion, and the wealth of the top 1% is more than $190 trillion.

So, the percentage of global wealth held by the top 1% is:

(190 trillion / 418.3 trillion) x 100 = 45.38%

Therefore, the top 1% of the world's population holds more than 45% of global wealth.

Answer:

45.42%

Step-by-step explanation:

To find:-

Answer:-

We are here given that, the total global wealth is 418.3T dollars and more than 190T dollars is in hands of top 1% of the people. We are interested in finding out the percentage of global wealth in the hands of top 1% . We can find the percentage as ,

[tex]:\sf\implies \pink{ \% =\dfrac{Wealth\ owned\ by \ 1\% }{Total\ wealth}\times 100}\\[/tex]

On substituting the respective values, we have;

[tex]:\sf\implies \% =\dfrac{\$ 190T}{\$ 418.3T}\times 100\\[/tex]

[tex]:\sf\implies \pink{\% = 45.42\% } \\[/tex]

Hence the required percentage is 45.42% .

the answer is (a) [tex]\frac{1}{6}b-\frac{5}{6}[/tex]

In order to combine the two terms that have a 3b common denominator, we can first do so to simplify the statement 1/3b + 2/3b - 5/6(b+1):

1/3b + 2/3b = 3/3b = 1/b

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added before a variable and then multiplied by it.

from the question:

In order to combine the two terms that have a 3b common denominator, we can first do so to simplify the statement 1/3b + 2/3b - 5/6(b+1):

1/3b + 2/3b = 3/3b = 1/b

We can now reintroduce this value into the first expression:

1/b - 5/6(b+1)

We can distribute the 5/6 term in order to further simplify this:

1/b - 5/6b - 5/6

Now that the first two terms share a common denominator of 6b, we may combine them:

6/6b - 5/6b - 5/6

Simplifying this gives us:

1/6b - 5/6

Therefore, the answer is (a) 1/6b - 5/6.

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The probability of the sum of 4 is 3/36

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

rolling a 6-sided die twice

So, the sample space are

S1 = {1, 2, 3, 4, 5, 6}

S2 = {1, 2, 3, 4, 5, 6}

For the sum of 4, we have]

(1, 3), (2, 2) and (3, 1): Count of 3

The sample size of the event is

Size = 6 * 6

Size = 4

Using the above as a guide, we have the following:

P = Count/Size

So, we have

P = 3/36

Evaluate

P = 1/12

Hence, the probability is 1/9

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Answer:

Step-by-step explanation:

domain is negative infinity, positive infinity

range is -3, positive infinity

Answer:

F(x) is all values that x can be. so the answer is the domain is (-∞,4)

Step-by-step explanation:

The solution of each equation are;

1) x = 6, x = 0

2) x = 3, x = 2

3) x= 4 ±√6

4) x = 6 ± √24/2

An extraneous solution is a solution obtained in a mathematical equation that does not satisfy the original equation, even though it appears to be a valid solution when plugged into the equation.

1) (√6x)^2 = x^2

6x = x^2

x^2 - 6x = 0

x = 6, x = 0

2) √5x - 6 = x

5x - 6 = x^2

x^2 - 5x + 6 = 0

x = 3, x = 2

3) √2x - 1 = x - 3

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

2x - 1 = x^2 - 6x + 9

x^2 - 6x - 2x + 9 + 1 = 0

x^2 -8x + 10 = 0

x= 4 ±√6

4) x = 2 + √2x - 11

x -2 = √2x - 11

x^2 - 4x + 4 = 2x - 11

x^2 -4x -2x + 4 + 11 = 0

x^2 - 6x + 15 = 0

x = 6 ± √24/2

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The following table lists Telkom's rates for international calls. Number minutes needed for [tex]x[/tex] to be [tex]17[/tex]and [tex]y[/tex] to be [tex]14.85[/tex].

Many numbers have deeper symbolic connotations in addition to their spiritual significance, and it is crucial to grasp these meanings in order to comprehend how they influence other cultures' perceptions of the physical world.

According to Wilder, any sequence of consecutive threes is probably a sign of good fortune or a shift in your job, achievement, aspirations, or emphasis. According to Wilder, the number 333 is frequently a signal that you should have faith in your ability to make the next move.

Let[tex]y=k*b[/tex]

When [tex]x=3, y=4.95, x=5, y=8.25[/tex]

[tex]4.95=3k+b[/tex]

[tex]8.25=5k+b[/tex]

[tex]k=1.65[/tex]

[tex]b=0[/tex]

[tex]y=1.65x[/tex]

when [tex]x=9[/tex]

[tex]y=1.65*9 = 14.85[/tex]

when [tex]y=28.05[/tex]

[tex]x=y/1.65=28.05/1.65=17[/tex]

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The complete and the correct form of question is:

The International calls tariffs of Telkom are given in the following table:

Number of minutes         3                  5                   9                     y

Cost                               R4.95          R8.25              X                R28.05

Determine the cost of 9 minute call

The statements that are true about the diagram given above = Line JL is an angle bisector and L is the midpoint of a segment in the diagram. That is option B and F.

An angle bisector is defined as the lone the

at is drawn through an angle such that it divides the angle into two equal parts.

From the diagram above, line JL divided the angle < J into two equal angles while diving the line segment KI into two equal parts such as KL and LI.

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Answer: 1

Step-by-step explanation: From the above data, we observe that 1 is the greatest factor of itself and the smallest factor of any other number. Therefore, 1 is the factor of every number.

Considering the complex-conjugate root theorem, the statement is given as follows:

If a polynomial has one root in the form a - √5, it has a second root in the

form of a + √5.

The complex conjugate theorem for rational roots states that if a polynomial with real coefficients has a complex root of the form a + bi (where a and b are real numbers and i is the imaginary unit), then its complex conjugate a - bi is also a root of the polynomial.

In the case of a rational root, we have that:

If [tex]a + \sqrt{b}[/tex] is a root, then the number [tex]a - \sqrt{b}[/tex] is also a root, which justifies the answer for this problem.

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Answer:

Rodolfo has two checks, one for $108.93 and the other for $372.47. Let's add the amounts of the checks to get the total amount of the deposit:

$108.93 + $372.47 = $481.40

Next, Rodolfo wants to receive $75 cash back from his deposit. We need to subtract $75 from the total amount of the deposit to get the net deposit:

$481.40 - $75 = $406.40

Therefore, Rodolfo's net deposit is $406.40.

Step-by-step explanation:

Answer:

Step-by-step explanation:

To find Rodolfo's net deposit, we need to add the values of his two checks and subtract the amount of cash he wants to receive back:

Net deposit = (Value of Check 1) + (Value of Check 2) - (Amount of Cash Back)

Net deposit = $108.93 + $372.47 - $75

Net deposit = $406.40 - $75

Net deposit = $331.40

Therefore, Rodolfo's net deposit is $331.40.

The value of f(9), the y-intercept, and the equation will be -3, 15, and f(x) = -2x + 15, respectively.

Given that:

Points, (-1, 17) and (4, 7)

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

The equation of the line passing through (-1, 17) and (4, 7) is given as,

(y - 17) = [(7 - 17)/(4 + 1)](x + 1)

y - 17 = -2x - 2

f(x) = -2x + 15

The value of f(9) is calculated as,

f(9) = -2(9) + 15

f(9) = - 18 + 15

f(9) = - 3

The graph is given below.

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The effective interest rate on the loan is also 32%.

What is effective interest rate ?

Effective interest rate is the actual interest rate that is paid or earned on an investment, loan, or other financial product. It takes into account not only the nominal interest rate, which is the stated rate of interest, but also the frequency of compounding or other factors that affect the overall return.

According to the question:
To find the effective interest rate, we first need to calculate the amount of interest Kremena paid on the loan. We can do this by subtracting the amount she received from the amount of the loan:

Interest = Loan amount - Amount received

Interest = $500 - $460

Interest = $40

Next, we can use the formula for simple interest to calculate the interest rate:

I = P * r * t

where I is the interest, P is the principal (loan amount), r is the interest rate, and t is the time in years.

Since Kremena took out the loan for 3 months, which is a quarter of a year, we have:

I = P * r * t

$40 = $500 * r * (3/12)

$40 = $500 * r * 0.25

r = $40 / ($500 * 0.25)

r = 0.32 or 32%

So the interest rate on the loan is 32%. However, this is the nominal interest rate, which does not take into account the effects of compounding. To find the effective interest rate, we need to take into account the fact that Kremena only received $460, which means that she effectively paid more interest than she would have if she had received the full $500.

To calculate the effective interest rate, we can use the formula:

[tex]re = (1 + r/n)^n - 1[/tex]

where re is the effective interest rate, r is the nominal interest rate, and n is the number of compounding periods per year. Since the problem specifies simple interest, there is no compounding, so we can take n = 1. Plugging in the values, we get:

[tex]re = (1 + 0.32/1)^1 - 1[/tex]

re = 0.32 or 32%

So the effective interest rate on the loan is also 32%.

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The estimated salary fοr an emplοyee with 18 years οf experience is $77,201.24.

BODMAS stands fοr Brackets, Order, Divisiοn/Multiplicatiοn, Additiοn/Subtractiοn, and it represents the οrder οf mathematical οperatiοns tο be perfοrmed when evaluating an expressiοn. It ensures that expressiοns are evaluated in a cοnsistent and unambiguοus manner.

Tο calculate the estimated salary fοr an emplοyee with 18 years οf experience using the prοvided fοrmula, we can substitute 18 fοr "Years οf Experience" and sοlve fοr the salary:

Estimated Salary = 29,608.40 + 2632.98(18)

Estimated Salary = 29,608.40 + 47,592.84

Estimated Salary = 77,201.24

Therefοre, the estimated salary fοr an emplοyee with 18 years οf experience is $77,201.24.

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Answer:

Assuming the ball is a perfect sphere, we can use the formula for the volume of a sphere to calculate the amount of air it can hold.

The formula for the volume of a sphere is:

V = (4/3)πr^3

where V is the volume, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

To calculate the radius of the ball from its diameter of 15 inches, we divide by 2:

r = d/2 = 15/2 = 7.5 inches

Substituting this value into the formula, we get:

V = (4/3)π(7.5)^3 ≈ 1767.15 cubic inches

Therefore, the ball can hold approximately 1767.15 cubic inches of air to the nearest tenth.

Step-by-step explanation: