Write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression.
csc ß- sin ß
Choose the correct answer below.
OA. 1
B. cos ßcot B
OC. sin²B
OD. 1-sin² B
OE. sin² ßtan² ß
OF. cos B
...

The derivate of function f(x)=(5x⁶+4x³)⁴is F'(x) = 4(30x³+12x²)(5x⁶+4x³)³

The derivative of a function f(x) represents its rate of change and is denoted by f'(x) or df/dx.

The derivative Formula is given as:

f1 (x)=lim△x→0 f(x+△x) − f(x)△x.

Take the derivative: F'(x) = [tex]\frac{d}{dx} (5x^6+4x^3)^5[/tex]

Substitute the inner function by (5x⁶+4x³)

Apply the chain rule:   [tex]\frac{d}{dx} (g)^4*\frac{d}{dx} (5x^6+4x^3)^4[/tex]

Calculate the derivative: [tex]4g^3*\frac{d}{dx} (5x^6+4x^3)^4[/tex]

Substitute back: [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]

Result after applying the chain rule: F'(x) = [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]

Calculate the derivative of the sum or difference:

F'(x) =[tex]4(5x^6+4x^3)^3*(\frac{d}{dx} (5x^6))+\frac{d}{dx} (4x^3))[/tex]

Take out the coefficients F'(x) = [tex]4(5x^6+4x^3)^3*(5*\frac{d}{dx} (x^6)+4\frac{d}{dx}(x^3))[/tex]

Apply the power rule: F'(x) = 4(5x⁶+4x³)³*(5*6x³+4*3x²)

Multiply the monomials: F'(x) =  4(5x⁶+4x³)³*(30x³+12x²)

Rewrite the expression: F'(x) = 4(30x³+12x²)*(5x⁶+4x³)³

So, the answer is F'(x) =4(30x³+12x²)*(5x⁶+4x³)³

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The solution to the equation x^2-5x+7=0 are x=5/2+3/2i and x=5/2-3/2i

A mathematical statement that has an "equal to" symbol between two expressions with equal values is called an equation. as in 3x + 5 Equals 15, for instance. Equations come in a variety of forms, including linear, quadratic, cubic, and others. In this tutorial, let's study more about math equations.

There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form.

According to the given question, we see

by solving the inequality we get

x=5/2+3/2i and x=5/2-3/2i

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Answer:

K

Step-by-step explanation:

K represents negative 1 ⅓.

to find the opposite of a number you must look to see if its negative. if its negative then its opposite would be a positive. If its a positive then its opposite would be a negative. Opposites are always the number in opposite signs like 7 and -7.

I found K to be 1⅓ because there are 2 spaces in between numbers so it would be ¹ ⅓ ⅔ 1. so I went to the number 1 and went one space back/where K because I know it would be 1 ⅓

The number of seconds that the ball attains its height round to the nearest hundredth will be 4.06 seconds.

For the maximum value of the function if f''(x) < 0 and for the minimum value of the function if f''(x) > 0.

The maximum and minimum value of the function is at f'(x) = 0.

The function is given below.

h(t) = -16t² + 130t + 2

Differentiate the function with respect to t. Then we have

h'(t) = -32t + 130

h'(t) = 0

- 32t + 130 = 0

t = 130 / 32

t = 4.0625 seconds

The number of seconds that the ball attains its height round to the nearest hundredth will be 4.06 seconds.

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The coordinates of the linear equation for (m, s) are; (0, 4), (2, 3), (-2, 5), (8, 0)

This is a one to one mapping and example of a function

We are given the equation as;

2m + 4s = 16

Thus;

When m = 0, we have;

2(0) + 4s = 16

4s = 16

divide both sides by 4 to get;

s = 4

When s = 3;

2m + 4(3) = 16

2m + 12 = 16

2m = 16 - 12

Divide both sides by 2 to get;

m = 4/2

m = 2

When m = -2

we have;

2(-2) + 4s = 16

-4 + 4s = 16

4s = 20

divide both sides by 4 to get;

s = 5

When s = 0, we have;

2m + 4(0) = 16

2m + 0 = 16

2m = 16

Divide both sides by 2 to get;

m = 16/2

m = 8

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Answer:

4,4

Step-by-step explanation:

44x1/10

Answer:

5 & -4

4 & -3

3 & -2

2 & -1

Step-by-step explanation:

5 + (-4) = 1

4 + (-3) = 1

3 + (-2) = 1

2 + (-1) = 1

Answer:

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

Given sequence from zeros to 6th terms:

It is a GP with the common ratio of 1/2 as each term is half the previous one.

Use the nth term equation, considering the first term is t(1) = 8, and common ratio r = 1/2:

Answer:

a)  Geometric sequence

[tex]\textsf{b)} \quad t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

Step-by-step explanation:

An Arithmetic Sequence has a constant difference between each consecutive term.

A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.

From inspection of the given table, t(n) halves each time n increases by 1.

Therefore, the sequence is geometric with a common ratio of 1/2.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the initial value (when n = 1) and the common ratio into the formula to create an equation for the nth term:

[tex]\implies t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

The correct option is E. the roots of the auxiliary equation are √6 and -√6.

Differential equations are defined as the equations in which consist of one or more terms which consist of the a dependent variable and it's derivatives with respect to the independent variable.

Given differential equation,

[tex]\frac{d^{2} y}{dx^{2} }[/tex] - 6y = 0

This can be also written as,

y'' - 6y = 0, where y'' indicates the second derivative of y.

Auxiliary equation is the polynomial equation which is found by replacing the right member of a standard differential equation by zero.

To find the roots of auxiliary equation can be found by,

Let m = y', then m² = y''

m² - 6 = 0

m² = 6

m = ± √6

So the roots are √6 and -√6.

Hence the roots of the auxiliary equation of the given differential equation are √6 and -√6.

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The domain and range of the piecewise function is;

B) Domain: x < 2 or x ≥ 3.; range: y > -3

The piecewise function is given as;

f(x) = {-¹/₂x - 2, if x < 2

        {2x - 4, if x ≥ 3

The domain of a function is the set of all possible input values of the function and as such in this case the domain is x < 2 or x ≥ 3.

The range is defined as the set of all possible output values of the function and in this case, we have;

For -¹/₂x - 2; If x < 2

At x = 2; -¹/₂(2) - 2 = -3

Range is; y > -3

Similarly, for 2x - 4, if x ≥ 3

At x = 3; 2(3) - 4 = 2

Range is; y ≥ 2

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The triangle has length of the side h is equal to 97.1 inches approximately to the nearest tenth, using the sine rule.

The sine rule is a relationship between the size of an angle in a triangle and the opposing side.

For the triangle ΔGHI, by sine rule;

g/sinG = h/sinH = i/sinI

We first derive the angle G as follows:

151° + 7° + G = 180° {sum of interior angles of a triangle}

G = 180° - 158

G = 22°

Hence by sine rule;

75/sin22° = h/sin151°

h = (75 × sin151°)/sin22° {cross multiplication}

h = 97.1

Therefore, by application of the sine rule we have the length of the side h = 97.1 inches approximately to the nearest tenth

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Answer:

  2, 4, 6

Step-by-step explanation:

You want three consecutive even integers such that the sum of twice the first and three times the second is 4 more than twice the third.

Let x represent the middle integer. Then the first is (x-2) and the third is (x+2). The given relation is ...

  2(x -2) + 3x = 2(x +2) +4

Simplifying the equation gives ...

  2x -4 +3x = 2x +4 +4

  5x -4 = 2x +8 . . . . . . . collect terms

  3x = 12 . . . . . . . . . . . add 4-2x

  x = 4 . . . . . . the middle integer

The integers are 2, 4, 6.

__

Additional comment

It often simplifies the equation if the variable is used to represent the middle of the consecutive integers in problems like this. In this particular problem, there is not a great advantage.

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The set of 4 integers are is given by the equation A = { 1 , 2 , 3 , 4 }

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the set of 4 integers be represented as set A

Now , the equation will be

Let the first number be 1

The absolute value of 1 is | 1 | = 1

Let the second number be 2

The absolute value of 2 is | 2 | = 2

Let the third number be 3

The absolute value of 3 is | 3 | = 3

Let the fourth number be 4

The absolute value of 4 is | 4 | = 4

And , | 1 | < | 2 | < | 3 | < | 4 |

Hence , the set of integers are { 1 , 2 , 3 , 4 }

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Answer:

-3

Step-by-step explanation:

rise over run method

3 up and 1 to the left which makes the slope negative

(A) f(-1) = 1

(B) The range of the function f(x) is [-4, ∞).

What is the range of the function?

In mathematics, a function's range can refer to one of two closely related ideas: Its codomain is a function. The picture of the action Given two sets X and Y, a binary relation f between them is a function if and only if there is precisely one y in Y such that f connects x to y for every x in X.

For the given function we have to find the value of f(-1) and to find the range of the function.

From graph we can see that,

f(-1) = 1

The range of the function is the interval in which function cover the x - axis.

The range of the function is [-4, ∞).

Hence,

(A) f(-1) = 1

(B) The range of the function f(x) is [-4, ∞).

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Answer:

Step-by-step explanation:

The length of the two diagonals of a rectangle are said to be equivalent.

From the Pythagorean theorem, the diagonal (a) will be the hypotenuse in the equation and the length (b) and width (c) will be its opposite and adjacent.

From the figure given, the length of half of two diagonal of rectangles are given.

We can simply solve for the value of x

TO = RO

3x + 1 = 2x + 4 (combine like terms by first transposing it to the other side of the equation and changing its sign from (+) to (-) or vice versa).

3x - 2x = 4 - 1

x = 3

Answer:

Copy the table into your notebook and include columns to find the endpoint (upper value) for each interval. Figure out the cumulative frequency and

Step-by-step explanation:

The standard form is used to show horizontal lines and vertical lines, while the slope intercept form shows the slope and y intercept of the linear equation.

An equation is an expression relating two or more numbers and variables.

The standard form of a linear equation is:

Ax + By = C

where A, B and C are constants

The slope - intercept form of a linear equation is:

y = mx + b

Where m is the rate of change(slope) and b is the y intercept.

The standard form is used to show horizontal lines (A=0) and vertical lines (B=0), while the slope intercept form shows the slope and y intercept.

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1. A city wants to reduce its carbon footprint by switching to renewable energy sources. The city's current energy consumption is measured in gigawatt hours (GWh) and they want to reduce it by 20% in the next 5 years. The city's current energy consumption is 2,000 GWh per year. What is the target energy consumption in GWh per year that the city should aim for to achieve their goal?

2. A company wants to reduce its water consumption by 50% in the next 10 years. The company currently consumes 500,000 gallons of water per day. How many gallons of water per day should the company aim to consume in 10 years to achieve their goal?

The exponential function will be y=  -2(3)^x.

What are exponential functions?

An exponential function is a Mathematical function in the form f (x) = a^x, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828.

Now,

Given points for exponential function are (2, –50), (4, –1250).

and exponential function is y=ab^x

To find a and b put points in the function

-50=ab^2 -->1

-1250=ab^4 --->2

From equation 1

a=-50/b^2

Put value of a into 2

-1250=-50/b^2*b^4

b^2=25

b=5

For b=5, a=-50/5^2=-2

Hence, a=-2

Therefore,

               The exponential function will be y= -2*5^x.

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Answer: If the model kit for the Dornier Do17 airplane has a wingspan of 10 inches and the scale of the kit is 1 inch to 72 inches. This means that for every inch on the model, the equivalent measurement on the actual Dornier Do17 airplane is 72 inches.

We know that the wingspan of the model is 10 inches and we want to find the actual wingspan of the airplane so we'll use the scale factor of 1:72

To find the actual wingspan, we'll multiply the wingspan of the model by the scale factor.

10 inches * 72 inches/1 inch = 720 inches

Therefore, the actual wingspan of the Dornier Do17 airplane is 720 inches or 60 feet.

Step-by-step explanation:

Answer: The answer provided is based on the following calculation:

To make a 75% syrup, Ella should add enough water to the sugar to create a solution that is 25% sugar and 75% water.

To calculate how much water to add, you can use the formula: (desired % of sugar) / (100 - desired % of sugar) * (amount of sugar).

In this case, it would be: (0.75) / (100 - 0.75) * (0.5 lb) = 0.67 lb of water.

To calculate the total amount of syrup, you add the amount of sugar and the amount of water: 0.5 lb + 0.67 lb = 1.17 lb.

So Ella should add 0.67 lb of water to make a 75% syrup and will have a total of 1.17 lb syrup.

Step-by-step explanation:

Answer:

for the 75% syrup: 0.167lbs

For the 1.5% syrup: 32.83 lbs

Therefore , the solution of the given problem of triangle comes out to be ∠P = 30° , PQ = 4√3  and  PN = 2.

An triangle is a polygon since it has four or more parts. It is a simple geometric shape. A square having angles A, B, and C is referred to as a triangle ABC. When the sides are not collinear, Euclidean geometry generates a single plane and square. If a triangle has three sides and three corners, it is a polygon. The corners of a triangle are where its three sides meet. The sum of the angles in a triangle is 180 degrees.

Here,

Given :  A right triangle,

=> ∠P + 60° + 90° = 180°

=>  ∠P = 180° - 150°

=> ∠P = 30°

For PQ

=> Tan 60°  =  √3

=> PQ /4 = √3

=> PQ = 4√3

For PN ,

=> Sin 60°  = √3/2

=>  4√3  / PN =  √3/2

=> PN = 2

Therefore , the solution of the given problem of triangle comes out to be

∠P = 30° , PQ = 4√3  and  PN = 2.

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P(A|B) read as the probability of the event A given the event B is equal to 0.5000 rounded up to four decimal place.

Conditional probability is known as the possibility of an event or outcome happening, based on the existence of a previous event or outcome. The conditional probability P(A|B) occur only in the case of dependent events. It gives the conditional probability of A given that B has occurred.

P(A|B) = P(A∩B) / P(B)

P(A∩B) = Probability of happening of both A and B

When a die is rolled, the sample space = {1, 2, 3, 4, 5, 6}.

A is the event the blue die shows a number greater than 2, So;

A = {3, 4, 5, 6}

B is the event the red die shows a number less than 5. So;

B = {1, 2, 3, 4}.

Then A∩B = {3, 4}

P(A∩B) = 2/6 = 1/3

P(B) = 4/6 = 2/3

P(A|B) = 1/3 ÷ 2/3

P(A|B) = 1/3 × 3/2 {changing division to multiplication}

P(A|B) = 1/2

Therefore, the conditional probability of the event A given the event B denoted by P(A|B) is equal to 1/2 or 0.5000 to four decimal place.

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End Balance: $3,839.25 + $2,579.20 - $2,190.05 = $5,735.15, so the reconciled balance of $5,735.15.

This theory is used to compare the bank statements to the check register in order to properly reconcile the two balances.

The theory used in this question is the bank reconciliation theory.

Step 1: Begin by looking at the two balances - the check register balance and the bank statement balance.

In this case, the check register balance is $3,839.25 and the bank statement balance is $3,450.10.

Step 2: Add up all of the deposits that are not reported on the bank statement. In this case, the deposits are $2,000, $135.67, $254.77, and $188.76.

Step 3: Subtract all of the checks that are not reported on the bank statement. In this case, the checks are $567.89, $23.83, $598.33, and $1,000.

Step 4: Add the deposits and subtract the checks from the check register balance.

In this case, the check register balance ($3,839.25) plus the deposits ($2,579.20) minus the checks ($2,190.05) gives a reconciled balance of $5,735.15.

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Here, on solving the provide question to us, we got to know that - equation  210 feet represents  = 825/4

Since equations are essentially questions, efforts to systematically find answers to those questions have been the driving force behind the development of mathematics. From straightforward algebraic equations that just require addition or multiplication to differential equations, exponential equations that use exponential expressions, and integral equations, there are many different types of equations that range in complexity.

Here,

15 feet represents = 1/8  of an inch

1 feet represents = 1/8 X 15

The growth of mathematics has been fueled by efforts to methodically find answers to equations, which are basically questions. There are many distinct forms of equations that range in complexity, from simple algebraic equations that just need addition or multiplication to differential equations, exponential equations that employ exponential expressions, and integral equations 210 feet represents = 1/8 X 15 X 210 = 825/4

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Therefore , the solution to the given problem of derivative comes out to be 2cos(mx+ny) for both the cases of partial derivative.

The definition of a derivative is the simultaneous change that occurs at a certain place. Implicit & explicit functions are typically distinguished from one another. Explicit functions are those where the value of the known individual entity "x" directly determines the value of the known dependent variable "y."

Here,

Given : f(x,y)= sin^(2) (mx+ny)"

Partial derivative with respect to x

=> dx/dy = 2sin (mx+ ny) *m

=> d²x/dy = 2cos(mx+ny)

Partial derivative with respect to y

=>  dx/dy = 2sin (mx+ ny) *n

=> d²x/dy = 2cos(mx+ny)

Therefore , the solution to the given problem of derivative comes out to be 2cos(mx+ny) for both the cases of partial derivative.

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The total length of the race track that Antonio remote controlled car runs on is 70 m.

An equation is an expression that shows how numbers and variables relate with each other.

Equations can either be linear, quadratic and so on depending on the degree of the polynomial

Let x represent the length of the racetrack.

Antonio car stops 10m past the starting line. After he fixes a wheel, the car goes 2/5 the racetrack’s total distance. It can be represented by the equation:

(2/5)x + 10 = 38

(2/5)x = 28

x = (28 * 5)/2

x = 70

The length of the race track is 70 m.

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