12.5(0.2t − 1) + 21t

Step-by-step explanation:

Factor the denominator to simplify the expression: x^2 + x - 2 = (x - 1)(x + 2)

Use partial fraction decomposition to write the integrand as the sum of simpler terms: (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) = A/(x - 1) + B/(x + 2) + Cx + D, where A, B, C, and D are constants.

Multiply both sides of the equation by (x^2 + x - 2) to find the values of A, B, C, and D: x^3 + 2x^2 + 3x + 4 = A(x + 2) + B(x - 1) + (Cx + D)(x^2 + x - 2)

Substitute x = 1 and x = -2 into the equation to find two equations for A, B, C, and D:

x = 1: 4 + 2 + 3 + 4 = 9 = A(-2) + B + (C + D)(-1)

x = -2: -8 - 4 + 6 - 8 = -16 = A(1) + B(-2) + (C - 2D)(4)

Solve the system of equations to find the values of A, B, C, and D:

A = 9/3, B = 5, C = -11/3, D = 7/3

Use the partial fraction decomposition to integrate the integrand:

Integral of 1/(x - 1) = ln|x - 1|

Integral of 1/(x + 2) = ln|x + 2|

Integral of x = x^2/2

Integral of 1 = x

Evaluate the definite integral by subtracting the values of the antiderivatives at the limits of integration:

Integral from x=0 to x=1 of (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) = (ln|x + 2| - ln|x - 1| + x^2/2 - x)|x=1 - x=0 = (ln(3) - ln(-1) + 1/2 - 1) = (ln(3) + ln(1) + 1/2) = ln(3) + 1.

So the definite integral of (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) from x=0 to x=1 is equal to ln(3) + 1.

Answer:

Step-by-step explanation:

Answer: 5 (v+1)

Step-by-step explanation:

( i_i )

A salesman earns $490 as commission on his sales.

Percentage is defined as a given part or amount in every hundred. It is a fraction with 100 as the denominator and is represented by the symbol "%".

Given that, a salesman earns 7% commission on all the merchandise that he sells.

Last month he sold $7000 worth of merchandise.

Now, the commission is 7% of 7000

7/100 ×7000

= 0.07×7000

= $490

Therefore, a salesman earns $490.

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Answer:

18.2

Step-by-step explanation:

replace where x is with 3

[tex]3 + \frac{1}{2} + (5 \times 3) - \frac{3}{10} [/tex]

[tex]3 \frac{1}{2} + 15 - \frac{3}{10} [/tex]

[tex]15 - \frac{3}{10} = 14 \frac{7}{10} [/tex]

[tex]3 \frac{1}{2} + 14 \frac{7}{10} = 18 \frac{1}{5} [/tex]

[tex]3.5 + 15 - 0.3 = 18.2[/tex]

The standard deviation is 19.21 while the mean is 20.785.

a) Because E(Y) = 12, var(Y) = 9.

then, for 2Y+20

E(2Y+20)=2E(Y)+20=24+20=44

Var(2Y+20)=4(9)=36.

The average deviation is six.

b)E(X) = 80 var(X) = 144

so, for 3X

E(3X)=3(80)=240

Var(3X)=9Var(X)=9(144)=1296

The average deviation is 36.

c)0.25x+y

E(0.25x+y)=0.25E(x)+E(y)=0.25(80)+12=32

Var(0.25x+y)=0.0625 var(x)+var(y) +2(0.25)Cov(x,y)

Cov(x,y)=0 since x and y are independent.

Var(0.25x+y)=18

Standard deviation = 4.24 d) x – 5 y

E(x-5y)=E(x)-5E(y)=80-5(12)=20

Var(x-5y)=var(x)+25var(y)-2(5)Cov(x,y)

Because X and Y are unrelated, Cov(x,y) = 0

Var(x-5y)=369

The standard deviation is 19.21, while the mean is 20.785

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Answer:

The answer is 1720ㅤㅤㅤㅤㅤㅤㅤㅤ

The expression for the number of butterflies after 4 minutes 3([tex]2^4[/tex]) - 7.

A grouping of numbers in a specific order is known as a sequence. On the other hand, a series is described as the accumulation of a sequence's constituent parts.

An ordered list of numbers is a sequence. The three dots indicate that the established pattern should move forward.

Given:

Initially we have 3 butterflies

After one minute =  3(2)

                             = 6 butterflies

After two minutes = 3(2)(2)

= 3(2²) butterflies

After three minutes = 3(2³) butterflies

After four minutes = 3([tex]2^4[/tex]) butterflies

as, she captured 7 butterflies

Then, after 4 min the number of butterflies on screen

= 3([tex]2^4[/tex]) - 7

= 3(16) - 7

= 48-7

= 41

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The sales representative has 24 possible sequences of cities to visit.

The multiplication rule of counting states that if there are n choices for the first event, m choices for the second event, p choices for the third event, and q choices for the fourth event, then there are a total of n x m x p x q possible sequences of choices. Applying this to the situation of the sales representative visiting four cities, we can say that there are 4 x 3 x 2 x 1 = 24 possible sequences of the cities.

To break this down further, the sales representative has 4 choices for the first city, Omaha, Dallas, Wichita, and Oklahoma City. Once the first city has been chosen, there are then 3 choices for the second city, Omaha, Dallas, and Wichita. From these 3 choices, there are then 2 choices for the third city, Omaha and Dallas. And then finally, there is 1 choice for the fourth city, Omaha. Multiplying these numbers together gives us 4 x 3 x 2 x 1 = 24 possible sequences of cities.

Therefore, the sales representative has 24 possible sequences of cities to visit.

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In order to determine cash flows from operating activities, firms may use a method in which every item on the income statement is adjusted from accrual accounting to cash accounting which is called Modified cash basis.

This is referred to as an accounting method that combines elements of the two primary bookkeeping practices known as  cash and accrual accounting.

In this scenario, different adjustments are made so as to get the accurate value of the income generated by the company or organization.

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The number 9,839,038 after indicated rounding off gives the number 9,839,000.

Rounding off of a number is defined as the simplification of the number by keeping the value of the number same but is made closer to the next number.

Rounding can be done for whole numbers as well as decimals.

Given is a whole number 9,839,038.

We have to round it to the place shown where the digit is 9.

Look at the next digit right to 9.

It is 0, which is less than 5.

So replace all the digits next to 9 by 0.

We get the number 9,839,000.

Hence the rounded number is 9,839,000.

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The numerical form of the statement is (5/2)n - 17 = 48.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

Let, 'n' be the unknown number,

So, the statement The difference between 5/2 of a number and 17 is 48

can be written as,

(5/2)n - 17 = 48.

5n - 34 = 96.

5n = 96 + 34.

5n = 130.

n = 130/5.

n = 26.

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The two equations which can be used to represent the situation are;

r + s = 1787

4r + 3s = 5,792

The correct answer choice is option B and E

Reserved seat for basketball game = r

Students seat for basketball game = s

Cost of reserved seat tickets = $4

Cost of students tickets = $3

Total number of people who attended the basketball game= 1,787 people

Total amount earned for tickets sales= $5,792

r + s = 1787

4r + 3s = 5,792

Therefore, the basketball game situation can be represented by the equation r + s = 1787; 4r + 3s = 5,792

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Answer:

26

Step-by-step explanation:

2 squared equals 4. So 30 - 4 = 26

hope it helps!

Answer:

Step-by-step explanation:

To find the x-coordinates of the points where the curve with the equation y = x³ - 4x² + 5x + 4 has a gradient of 1, we need to find the points where the derivative of the equation y = x³ - 4x² + 5x + 4 is equal to 1. The derivative of the equation y = x³ - 4x² + 5x + 4 is given by:

dy/dx = 3x² - 8x + 5

So, we need to find the solutions of the equation:

3x² - 8x + 5 = 1

This is a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula. To use the quadratic formula, we can write:

x = (-b ± √(b² - 4ac)) / (2a)

Where a = 3, b = -8, and c = 4. Plugging in these values, we get:

x = (-(-8) ± √((-8)² - 4 * 3 * 4)) / (2 * 3)

x = (8 ± √(64 - 36)) / 6

x = (8 ± √(28)) / 6

x = (8 ± 2√7) / 6

So, the two solutions are:

x = (8 + 2√7) / 6 and x = (8 - 2√7) / 6

These are the x-coordinates of the points where the curve with the equation y = x³ - 4x² + 5x + 4 has a gradient of 1.

Answer:

x = -1/18

Step-by-step explanation:

[tex]\frac{5}{6} +6x=\frac{1}{2}[/tex]

[tex]6x=\frac{1}{2} -\frac{5}{6}[/tex]

[tex]6x=\frac{6-10}{12}[/tex]

[tex]6x=\frac{-4}{12}[/tex]

[tex]6x=-\frac{1}{3}[/tex]

[tex]x=\frac{-\frac{1}{3} }{6} =\frac{-1}{(3)(6)}[/tex]

[tex]x=-\frac{1}{18}[/tex]

Hope this helps

Survey was wrong: holiday, systematic sample, 6 similar flights. To improve: questionnaire, various times/locations, entrance, different days.

The problems with the survey conducted by the airline are:

1) Conducting the survey on a holiday weekend is not representative of the entire population of air travelers;

2) The survey uses systematic sampling which may not accurately reflect the entire population;

3) The survey was only conducted on 6 similar flights and does not represent the entire population of air travelers.

To improve the survey, the following steps can be taken:

1) Ask travelers to fill out a questionnaire;

2) Conduct the survey at the entrance to the airport;

3) Conduct the survey during different times of the year;

4) Use various days of the week to conduct the survey;

5) Conduct the survey using flights to and from various locations.

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Solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain.

What is meant by domain and range?

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Solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain.

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.

All of the values that can go into a relation or function (input) are called the domain. All of the values that come out of a relation or function (output) are called the range.

To find the domain and range, we simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain. To calculate the range of the function, we simply express x as x=g(y) and then find the domain of g(y).

Given equation: y=f(x)

We can solve for the value of x by rearranging the equation to the form x = [tex]f^{-1y}[/tex]

Domain:

x = [tex]f^{-1y}[/tex]

The domain of f(x) is all the possible values of x for which y is defined.

Range:

y = g(x)

The range of g(x) is all the possible values of y for which x is defined.

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The equation below that does not show equivalent expressions when y = 2, is  5y + 25  = 5 ( y + 4 ).

option C.

The equation below that does not show equivalent expressions when y = 2, is determined by solving for y in all the given equation.

y + 4y  = y ( 1 + 4 )

y + 4y = y + 4y

5y = 5y

y = y

y + y + y = 3y

3y = 3y

y = y

5y + 25  = 5 ( y + 4 )

5y + 25 = 5y + 20

5y + 5 = 5y

5 ( y + 1 ) = 5y

y + 1 = y

2 (y + 2 ) = 2y + 4

2y + 4 = 2y + 4

2y = 2y

y = y

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The differential equation (y^2 cos(x−y) −x^2 sin(x−y)) dx+(2xycos(x−y) +x^2 sin(x−y)) dy =0 is not exact, because it cannot be written in the form F(x,y)dx + G(x,y)dy = 0 for some functions F and G.

b. The differential equation (2x+12y^2)dx+(12x+2y)dy=0 is exact, because it can be written in the form F(x,y)dx + G(x,y)dy = 0 for F(x,y) = 2x + 12y^2 and G(x,y) = 12x + 2y.

To find a general solution in an implicit form, we can integrate both sides:

∫ (2x + 12y^2) dx + ∫ (12x + 2y)dy = C

x^2 + 6y^2 + 12xy = C

c. The differential equation (12x^2 + 2y) dx+(2x + 12y^2)dy=0 is not exact, because it cannot be written in the form F(x,y)dx + G(x,y)dy = 0 for some functions F and G.

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The difference between density-dependent and independent is given below.

By controlling population density-related factors including disease, competition, and prediction, density dependence regulates the population.

A huge population is often where density-dependent operates.

The rate of increase and decrease determines this.

Food, housing, prediction, competition, and disease are factors that depend on population density.

Density Independence: Natural disasters and the weather are examples of things that govern the population without taking density into account.

It is possible to use density independence in both small and big populations.

Both small and large populations can be served by density-independent.

Fluctuations, fires, droughts, extremely high temperatures, and tornadoes are examples of density-independent variables.

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The total amount spent by Ivan is given by the equation A = $ 40.20

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 total amount spent by Ivan be represented as A

Now , the equation will be

The cost of one large necklace = $ 5.90

The cost of one small necklace = $ 4.90

The number of large necklaces = 1

The number of small necklaces = 7

So , the cost of 7 small necklaces = 7 x cost of one small necklace

Substituting the values in the equation , we get

The cost of 7 small necklaces = 7 x $ 4.90

The cost of 7 small necklaces = $ 34.30

And , the total amount spent by Ivan = cost of one large necklace + cost of 7 small necklaces

On simplifying the equation , we get

The total amount spent by Ivan A = 5.90 + 34.30

The total amount spent by Ivan A = $ 40.20

Hence , the equation is A = $ 40.20

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Answer:

-0.4444444444 or just -0.4

Answer:

Given Figure has set of Stacked cubes.

To Draw the Base Plan, lets see its right side, Left Side and Front side view.

Right side:

It has 3 cubes in base.

2 cubes in 2nd layer clearly seen in figure and only 1 cube in 3rd layer.

Left Side:

It has 2 Cubes in base.

Both Cubes attached to 1st and 2nd cube of right side cubes.

Front Side:

It can be seen that, here base only has 2 rows of cubes.

Therefore,  1st Figure is correct Base Plan (or enclosed figure)

Note: Numbers in enclosed figure just shows the total no. of cubes in that stack.

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Answer:

1.a)  x = g(2)

1.b)  See below

2.a)  See below

2.b) α = 60°

3.a)  2 + h

3.b)  2

4.a)  Interval [3, 4] = 3 m/s

       Interval [3, 3.1] = 2.1 m/s

4.b)  2 m/s

Step-by-step explanation:

Part (a)

Given function f(x):

[tex]f(x)=\dfrac{x}{x-2}[/tex]

If the range of f is (-∞, 1) U (1, ∞), then a value of y in the range of f is y=2.

Set the function equal to the value of y and solve for x:

[tex]\begin{aligned}y=2 \implies \dfrac{x}{x-2}&=2\\x&=2(x-2)\\x&=2x-4\\-x&=-4\\x&=4\end{aligned}[/tex]

The solution written as a function x = g(y) is:

Part (b)

Given function g(y):

[tex]g(y)=\dfrac{2y}{y-1}[/tex]

Verify that g is the inverse function of f by calculating g(f(x)):

[tex]\begin{aligned}\implies g(f(x)) &=\dfrac{2f(x)}{f(x)-1}\\\\&=\dfrac{2\left(\frac{x}{x-2}\right)}{\left(\frac{x}{x-2}\right)-1}\\\\&=\dfrac{2\left(\frac{x}{x-2}\right)}{\left(\frac{x-(x-2)}{x-2}\right)}\\\\&=\dfrac{\left(\dfrac{2x}{x-2}\right)}{\left(\dfrac{2}{x-2}\right)}\\\\&=\dfrac{2x}{2}\\\\&=x \quad (\text{for all $x\neq 2$})\end{aligned}[/tex]

Similarly, calculate f(g(y)):

[tex]\begin{aligned}\implies f(g(y))&=\dfrac{g(y)}{g(y)-2}\\\\&=\dfrac{\left(\frac{2y}{y-1}\right)}{\left(\frac{2y}{y-1}\right)-2}\\\\&=\dfrac{\left(\frac{2y}{y-1}\right)}{\left(\frac{2y-2(y-1)}{y-1}\right)}\\\\&=\dfrac{\left(\dfrac{2y}{y-1}\right)}{\left(\dfrac{2}{y-1}\right)}\\\\&=\dfrac{2y}{2}\\\\&=y \quad \text{(for all $y \neq 1$)}\end{aligned}[/tex]

Hence verifying that g is the inverse function of f.

[tex]\boxed{\begin{minipage}{5 cm}\underline{Trigonometric Identities}\\\\$\tan x=\dfrac{\sin x}{\cos x}$\\\\\\$\cos \left(\arctan \left(x\right)\right)=\dfrac{1}{\sqrt{1+x^2}}$\\\\\\$\sin\left(\arctan \left(x\right)\right)=\dfrac{x}{\sqrt{1+x^2}}$\\\end{minipage}}[/tex]

Part (a)

If  y = arctan(x)  then  tan(y) = x.

Use the trigonometric identities to express sin(y) and cos(y) in terms of x.

[tex]\boxed{\begin{aligned}\tan(y) &= x\\\implies \dfrac{\sin y}{\cos y}&=x\\\sin y&=x \cos y\\ \sin y&=x \cos (\arctan x)\\ \sin y&=\dfrac{x}{\sqrt{1+x^2}}\end{aligned}}[/tex]

[tex]\boxed{\begin{aligned}\tan(y) &= x\\\\\implies \dfrac{\sin y}{\cos y}&=x\\\\\dfrac{\sin y}{x}&= \cos y\\\\ \cos y&=\dfrac{\sin(\arctan x)}{x}\\\\ \cos y&=\dfrac{\dfrac{x}{\sqrt{1+x^2}}}{x}\\\\\cos y&=\dfrac{x}{\sqrt{x^2+1}x}\\\\\cos y&=\dfrac{1}{\sqrt{x^2+1}}\end{aligned}}[/tex]

Part (b)

Given linear equation:

[tex]y=\sqrt{3}x+1[/tex]

As the slope of a linear equation y = mx + b is m, then the slope (m) of the given line is √3.

If m = tan(α) then:

[tex]\begin{aligned}\implies m&=\sqrt{3}\\\tan (\alpha)&=\sqrt{3}\\\alpha&=\arctan \sqrt{3}\\\alpha&=60^{\circ}+180^{\circ}n\\\end{aligned}[/tex]

If α is the angle the line forms with the x-axis, then α = 60°.

Given function:

[tex]f(x)=x^2-1[/tex]

Part (a)

[tex]\begin{aligned}\dfrac{f(1+h)-f(1)}{h} &=\dfrac{((1+h)^2-1)-(1^2-1)}{h}\\\\&=\dfrac{(1+2h+h^2-1)-0}{h}\\\\&=\dfrac{2h+h^2}{h}\\\\&=2+h\end{aligned}[/tex]

Part (b)

Part a used differentiating from first principles to find the gradient of f(x) at x = 1.  As h gets smaller, the gradient of the straight line gets closer and closer to the gradient of the curve.  Therefore, as h gets close to zero, (2 + h) gets close to 2.  Therefore, the slope of the tangent line to the graph of f(x) at the point where x = 1 is 2.

A particle is moving on the x-axis and its position at time is given by

[tex]x(t)=t^2-4t+3[/tex]

Evaluate the position of the particle at t = 3, t = 4 and t = 3.1:

[tex]x(3)=3^2-4(3)+3=0[/tex]

[tex]x(4)=4^2-4(4)+3=3[/tex]

[tex]x(3.1)=(3.1)^2-4(3.1)+3=0.21[/tex]

Average velocity formula

[tex]\overline{v}=\dfrac{s(t_2)-s(t_1)}{t_2-t_1}[/tex]

Therefore, the average velocity of this particle over the interval [3, 4]:

[tex]\overline{v}=\dfrac{x(4)-x(3)}{4-3}=\dfrac{3-0}{1}=3\;\; \rm m/s[/tex]

The average velocity over the interval [3, 3.1] is:

[tex]\overline{v}=\dfrac{x(3.1)-x(3)}{3.1-3}=\dfrac{0.21-0}{0.1}=2.1\;\; \rm m/s[/tex]

Part (b)

To find an equation for velocity, differentiate the given equation for displacement:

[tex]v(t)=\dfrac{\text{d}x}{\text{d}t}=2t-4[/tex]

To calculate the instantaneous velocity of the particle at time t = 3, substitute t = 3 into the equation for velocity:

[tex]v(3)=2(3)-4=2\;\; \rm m/s[/tex]

Answer:

Step-by-step explanation:

ito answer: Combine 130 and x2.C=x230−2x+1530

Answer: Find the length of one side and square it

Example: Lets say one side of the square has a length of 7. The area would be 49 because 7x7=49

The height of the right cone will be 57 units.

The area a cone takes up in a three-dimensional plane is known as its volume. A cone's base is circular, so it is made of a radius and a diameter.

Given that the volume of a right cone is 4275π cubic units. The circumference of the cone is 30π.

The height of the cone will be calculated as below:-

First, calculate the radius of the base of the cone,

Circumference = 2πr

30π = 2πr

r = 15 units

The height of the cone:-

Volume = 4275π

( 1 / 3 ) πr²h =  4275π

h = ( 4275 x 3 ) / r²

h = ( 4275 x 3 ) / ( 15)²

h = 57 units

Hence, the height will be 57 units.

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