The first three terms of an arithmetic sequence are x, 2x + 4, 5x. Find the value of x.
P

Answer:

[tex]{ \sf{ \frac{dy}{dx} = y( {x}^{2} + 1)}} \\ \\ { \sf{ \frac{dy}{y} = ( {x}^{2} + 1)dx}} \\ \\ { \sf{ \int \frac{1}{y} dy = \int ( {x}^{2} + 1) \: dx}} \\ \\ { \sf{ ln(y) = \frac{ {x}^{3} }{3} + x + c }}[/tex]

Answer:

[tex]\large\text{$y& = e^{\frac{1}{3}x^3+x+\text{C}}$}[/tex]

Step-by-step explanation:

Given differential equation:

[tex]\dfrac{\text{d}x}{\text{d}y}=\dfrac{1}{y(x^2+1)}[/tex]

Rearrange the equation so that all the terms containing y are on the left side, and all the terms containing x are on the right side:

[tex]\begin{aligned}\implies \dfrac{\text{d}x}{\text{d}y}&=\dfrac{1}{y(x^2+1)}\\\\(x^2+1)\;\dfrac{\text{d}x}{\text{d}y}&=\dfrac{1}{y}\\\\(x^2+1)\;\text{d}x}&=\dfrac{1}{y}\;\text{d}y\\\\\dfrac{1}{y}\;\text{d}y&=(x^2+1)\;\text{d}x}\end{aligned}[/tex]

Integrate both sides:

[tex]\begin{aligned}\implies \displaystyle \int \dfrac{1}{y}\;\text{d}y &= \int (x^2+1)\;\text{d}x}\\\\\int \dfrac{1}{y}\;\text{d}y &= \int x^2\;\text{d}x}+\int 1\;\text{d}x}\\\\\ln y & = \dfrac{1}{3}x^3+x+\text{C}\\\\e^{\ln y} & = e^{\frac{1}{3}x^3+x+\text{C}}\\\\y& = e^{\frac{1}{3}x^3+x+\text{C}}\\\\\end{aligned}[/tex]

Therefore, the solution to the given differential equation is:

[tex]\large\text{$y& = e^{\frac{1}{3}x^3+x+\text{C}}$}[/tex]

Integration rules used:

[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Integrating $\frac{1}{x}$}\\\\$\displaystyle \int \dfrac{1}{x}\:\text{d}x=\ln |x|+\text{C}$\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating a constant}\\\\$\displaystyle \int n\:\text{d}x=nx+\text{C}$\\(where $n$ is any constant value)\end{minipage}}[/tex]

The relationship between the depth(d), the base area (b) and volume (V) is d = 2400/b

An equation is an expression that shows the relationship between two or more numbers and variables.

Equations are classified based on degree (value of highest exponents) as linear, quadratic, cubic and so on. Variables can be dependent or independent. Dependent variables depend on other variable while an independent variable do not depend.

Let d represent the depth of the tank, b represent the area of the base and V represent the volume.

The depth of water is inversely proportional to the area of its base at constant volume. Hence

d = V/b

When b = 200 ft² and d = 12:

12 = V/200

V = 2400

d = 2400/b

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Answer: 6931

Step-by-step explanation:

79% of all students are taking math this semesters.  So there is an additional 21% of students not taking math.

5476/x = 79/100

cross multiply and divide to get the total number of students.

5476 x 100 = 547600

547600 divided by 79 = 6931  

check 5476 divided by 6931 = 79%  

Further proof of correct answer:

21% of 6931 = 1455

1455 + 5476 equals 6931, so the answer calculated is correct.    

nswer:

cvents A and B are independent because P(A|B) = P(A)

pxplanation:

f aA and B are independent, then:

In this case,

P(A) = 0.4, P(B) = 0.2

If we multiply:

Which verify what the problem tell us.

Next, the definition of two independent events is P(A|B) = P(A)

An exponential equation that passes through the points (2,1.8) and (5,14.4) is y = (0.6708)2ˣ

Exponential equation:

If you have two points, (x1, y1) and (x2, y2), you can define the exponential function that passes through these points by substituting them in the equation

y = abˣ

and solving for a and b.

In general, you have to solve this pair of equations:

y1 = abˣ¹ and y2 = abˣ²

Given,

The points (2,1.8) and (5,14.4).

Here we need to find the exponential equation that passes through the points.

This yields the following pair of equations:

1.8 = ab²

14.4 = ab⁵

If you divide the second equation by the first, you get

b³ = 14.4/1.8

b³ = 8

b = ∛8

b = 2

so b = 2. It's possible for b to also be equal to -2, but in this case, assume it's positive.

Now, we can substitute this value for b in either equation to get a. It's easier to use the first equation, so:

1.8 = a² x (2)²

1.8 = a²  x 4

a²  = 1.8/4

a²  = 0.45

a = √0.45

a = 0.6708

Therefore, the exponential form of the equation is,

y = (0.6708)2ˣ

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Answer

Area of the figure = 105.12 ft²

Explanation

The figure shown is a combination of a rectangle with length 10 ft. and width 8 ft., with a semicircle that has diameter 8 ft., which translates to a radius of 4 ft.

The area of the figure = (Area of the rectangle) + (Area of the semicircle)

Area of rectangle = L × W

L = Length = 10 ft.

W = Width = 8 ft.

Area of the rectangle = 10 × 8 = 80 ft²

Area of semicircle = Half of the area of a circle = ½ × πR²

π = pi = 3.14 (According to the question)

R = Radius of the semicircle = 4 ft.

Area of semicircle = ½ × 3.14 × 4² = 25.12 ft²

Area of the figure = 80 + 25.12 = 105.12 ft²

Hope this Helps!!!

Answer:

[tex]g( - 16) = - 3[/tex]

Step-by-step explanation:

substitute

[tex]x = - 16 \: by \: g(x) = \frac{1}{4} x + 1 \\ g( - 16) = \frac{1}{4} x( - 16) + 1 \\ g( - 16) = - \frac{1}{4} x16 + 1 \\ g( - 16) = - 4 +1 \\ g( - 16) = - 3[/tex]

The amount paralegal earned from the job is 1860 dollars.

The amount allocated for savings is 1116 dollars.

Paralegal bills the law firm for 15.5 hours of work at a rate of $120 per hour.

Therefore, the law firm have to pay paralegal 120 dollars for every hour worked.

Therefore,

the amount earned by paralegal = 15.5 × 120

the amount earned by paralegal = 1860 dollars

Paralegal splits her earnings between utilities and savings in a ratio of 2 ∶ 3.

Therefore, let's find the amount of money allocated for her savings.

amount allocated for savings = 3 / 5 × 1860

amount allocated for savings = 5580 / 5

Therefore,

amount allocated for savings = 1116 dollars

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The measure of angle:-

angle 1 = 78 degree

angle 2 = 40 degree

angle 3 =  140 degree  

A triangle is a type of polygon which have only three vertices and no parallel sides.

 In this question we use property of triangle that :-

1:- Sum of interior angle is 180 degree

2:-Exterior angle of a triangle is equal to the sum of angle of opposite sides

A.T.Q:-

  lets vertices of triangle be A,B, and C    

  so,

       angle A + angle B +angle C = 180

       62 +78+ angle C = 180

      140 + angle C = 180

      angle C =40 degree

we know that , angle C and angle 3 makes 180 degree

                        angle C + angle3 =180     , 40+ angle 3= 180

angle 3 = 140 degree                                                                                     And ,  angle 2 = angle C

                :- angle 2=40                                                                               finally;

               angle 1 =  angle B

               angle 1 = 78 degree

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It is to be noted that the ONLY option that the customer can select from when purchasing the product is  "(M2, C2, A3)" Option C.

The above exercise is a test of logic, as it is with a simple calculation.

The study of formal logic within mathematics is known as mathematical logic. Mathematical logic research frequently focuses on the mathematical aspects of formal logic systems, such as their expressive or deductive capability.

Hence, from the above, we can see that:

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Full Question:

Apple has created a new i-Pad. There are 4 choices for memory size (M), 8 choices for color (C) and 3 apps (A) to select when you purchase the product. From the options below what is the ONLY option the customer can select?

(M1, C2, A4)

(M1, C9, A3)

(M2, C2, A3)

(M4, C2, A8)

The total amount to be paid on a 9 year, $60,500 loan at 7.3% p.a. is $100248.50.

It should be noted that simple interest is calculated as:

Interest = Principal × Rate × Time / 100

Interest = 60500 × 7.3 × 9 / 100

Interest = $39748.50

It should be noted that the amount will be the addition of the principal and the interest. This will. be:

= Principal + Interest

= $60500 + $39748.50

= $100248.50

The amount is $100248.50.

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We can tell the difference among linear, quadratic, and exponential functions just by looking at equation as Exponent functions are rapidly increasing or decreasing, and the highest power of variable for a linear function is 1. The highest power of variable for a quadratic function is 2.

Linear functions are functions that are used to model phenomena that increase or decrease at a constant rate. These types of functions are polynomial functions with a highest exponent of one on the variable. The graphs of these functions are in the shape of a straight line.

Exponential functions are functions that have the variable in the exponent. They increase or decrease slowly then quickly or quickly then slowly.

Quadratic functions are the functions that have the highest power of variables equal to 2.

For example,

Linear Equation : The population of a family of rabbits starts with 2 rabbits and doubles every month.

Expoential Equation : Compound interest is example of exponential function.

Quadratic Equation : When you throw a ball (or shoot an arrow, fire a missile or throw a stone) it goes up into the air, slowing as it travels, then comes down again faster and faster and a Quadratic Equation tells you its position at all times!

Therefore, we can conclude that linear function have highest power of variable as 1 , exponent functions are quickly increasing or decreasing, Quadratic functions have the power of variables equal to 2.

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Answer:

6(3+7)

Step-by-step explanation:

The factors of 18 are 1, 2, 3, 6, 9, 18.

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

The greatest common factor is six.

To put it in distributive form, divide both by 6:

6(3+7)

The labelled points on the intersected planes shown in the attachment are;

1) Lines AB and m are both coplanar on plane S and are perpendicular to each other intersecting at point B

2) Plane R and plane S intersect on line p

3) Line AB and line p are perpendicular to each other and both intersect with line n at point A

We want to label the points, lines, and planes showing the following;

Line AB and line m perpendicular to each other in plane S at point B.

Plane S intersecting plane R in line p.

Line AB both perpendicular to line p and intersecting line n at point A.

Therefore, the given parameters are;

1) Lines AB and m are both coplanar on plane S and are perpendicular to each other intersecting at point B

2) Plane R and plane S intersect on line p

3) Line AB and line p are perpendicular to each other and both intersect with line n at point A

The diagram showing the labelled points is as attached.

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The inverse of the conditional statement w > ~v is ~v < w

An inverse statement is seen as a type of conditional sentence having an immediate change that is made from original statement.

It is known to carry the negation of both the hypothesis and conclusion from the given original statement.

It is known to carry the opposite of the original statement that is given.

An inverse statement is formed by negating the hypothesis and conclusion of a statement.

It is expressed as;

Original statement:  If a, then b

Inverse statement: If not b, not a

Given the expression;

w > ~v

The inverse is represented as;

~v < w

Hence, the inverse is ~v < w

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Using transformations, the vector that describes the translation that J underwent is:

(x,y) -> (x - 1, y + 2).

The vertices of the transformed figure K are given as follows:

(4,0), (6,0), (6,-1), (5,-1), (5,-2), (4,-2).

The rule for a reflection about the line y = x is given by:

(x,y) -> (y,x).

Hence, taking the vertices of figure K, the vertices of the translated figure were given by:

(0,4), (0,6), (-1,6), (-1,5), (-2,5), (-2,4).

The vertices of the original figure J are given as follows:

(1,2), (1,4), (0,4), (0,3), (-1,3), (-1,3).

Hence the vector rule from J to the translated figure is given as follows:

(x,y) -> (x - 1, y + 2).

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The solution set for all possible side lengths for side L is;

⇒ L < 7

What is Triangle?

A triangle is a three sided polygon , which has three vertices and three angles.

Given that;

The longest side of a triangle must be less than the sum of the other two sides.

Suppose a triangle is made up of lengths 2 inches, 5 inches and L. inches, where, L is the longest side.

Now, By the given condition;

''The longest side of a triangle must be less than the sum of the other two sides.''

We can formulate;

L < 2 + 5

L < 7

So, The solution set for all possible side lengths for side L is;

⇒ L < 7

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The equation of the line is y = -1x + 3.

Parallel lines are any two or more lines that all lie in the same plane and never cross one another. They are equally spaced apart and have the same incline. In this essay, let's study more about parallel lines. No matter how far we extend a parallel line, it will never meet another parallel line.

The slope so rewrite the equation for the line as y = -x + 1.

The equation in “y = mx + b” form, the slope is always the number m in front of x.

The slope is -1.

First, by using “y = mx + b” form,

we know m is -1 so only need b.

To get b, plug in the (x, y) coordinates for the point (-1, 4)

(4) = -1×(-1) + b.

The y-coordinate 4 on the left side, and the x number -1 on the right.

b = 3.

Therefore, the equation of the line is y = -1x + 3.

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The inequality that represents having enough money to buy the gaming system is x ≥ 35.

The mathematical expression that represents the sentence " you charge $10 per dog" is x = 10 where x is the number of dogs.

x ≥ 35 means that the minimum number of dogs to wash is 35 to make $500.

It is a statement of an ordered relationship

- greater than,

- greater than or equal to,

- less than,

- less than or equal to between two numbers or algebraic expressions.

We have,

The cost of the gaming system = $500

Amount already saved = $150

Charges per dog = $10

The inequality that represents having enough money to buy the gaming system:

150 + 10x ≥ 500

10x ≥ 500 - 150

10x ≥ 350

x ≥ 35   where x is the number of dogs.

The mathematical expression that represents the sentence " you charge $10 per dog ":

x = 10 where x is the number of dogs.

x ≥ 35 means that the minimum number of dogs to wash is 35 to make $500.

Thus,

The inequality that represents having enough money to buy the gaming system is x ≥ 35.

The mathematical expression that represents the sentence " you charge $10 per dog" is x = 10 where x is the number of dogs.

x ≥ 35 means that the minimum number of dogs to wash is 35 to make $500.

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Answer:

252558855555335555422358236

Answer:

[tex]y=\frac{-1}{4}x+2[/tex]

Step-by-step explanation:

The function can be written in point-slope form following this equation:

[tex]y=m(x-x_{0} )+y_{0}[/tex]

where y and x are variables, and [tex]x_{0}[/tex] and [tex]y_{0}[/tex] are x and y values of any chosen point on the curve (line)

m (slope) can be calculated by the equation [tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1}}[/tex] where you need to find 2 chosen points on the line where [tex]x_{1}[/tex] and [tex]y_{1}[/tex] are the x and y values of the first set, and [tex]x_{2}[/tex] and [tex]y_{2}[/tex] are the x and y values of the second set.

solve for m and we get [tex]-\frac{1}{4}[/tex]

Plug it into the equation with any set of x and y values on the line into the point-slope form equation, then you get the answer

As a result of the sequences, the numbers for the first equation are 31,37,43, 64,128,256, and 37,42,47 for the second and third equations, respectively.

An ordered group of numbers with a shared difference between each succeeding word is known as an arithmetic sequence. For instance, the common difference in the arithmetic series 3, 9, 15, 21, and 27 is 6. An arithmetic progression is another name for an arithmetic sequence.

A geometric progression, sometimes referred to as a geometric sequence in mathematics, is a series of non-zero numbers where each term following the first is obtained by multiplying the preceding one by a constant, non-zero value known as the common ratio.

so for the first equation:

we see the common difference is constant :

second term/third term - first term/second term =7-1 or 13-7

                        =6

the next three terms in this series would be calculated by adding the common difference to consecutive last terms  =31,37,43

for the third equation

the common difference is a constant:

second term-first term=5

the next three terms in this series would be calculated by adding the common difference to consecutive last terms  =37,42,47

for the second equation we don't see a constant value as a common difference ,then we conclude that it is a Geometric progression

difference in series = second term/first term

                              =4/2

                              =2

so for the three consecutive terms we will multiply the last consecutive terms by 2=64,128,256

Therefore the numbers for First equation are 31,37,43 for the second equation are 64,128,256 and third equation are 37,42,47

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Answer:

true

Step-by-step explanation:

The probability that that my dog tears up the trash and it rains tomorrow is 0.03.

From the information, the dog has a 30% chance of tearing up the trash on a given day and there's a 10% chance of rain tomorrow.

Therefore, the that my dog tears up the trash and it rains will be:

= Probability of tearing up × Probability of rain

= 30% × 10%

= 0.3 × 0.1

= 0.03

The probability is 0.03.

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Answer:

p(2) = 2

f(3) = 7

Step-by-step explanation:

Given p(x) = -3(x - 4)² + 14 find p(2)

Simply substitute x with the value 4 in the above expression

p(2) = -3(2-4)² + 14

= -3(-2)² + 14

= -3(4) -+ 14

= -12 + 14

= 2

f(3) can directly read from the table

for x = 3, f(x) = 7

so f(3) = 7

15,0

Step-by-step explanation:

The equation of parallel line is y = 2x + 3

Given,

The line below is the function y=2x-1.

From the figure,

Identify the equation of a parallel line.

Now, According to the question:

We are given that line a represented by the equation

y = 2x -1

and, We know that

Slope- intercept form of line is given by

y = mx + c

Where m=Slope of line a

c = y- intercept

Comparing it  with the given equation then, we get

Slope of line a = 2

We know that

When two lines are parallel then their slopes are equal.

Hence, Option (b) slope is equal to the given function

It means, y = 2x - 1 and y = 2x +3

Value of slope m = 2 = 2

Hence, The equation of parallel line is y = 2x + 3

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If a car consumes 110 liters of fuel with in a distance of 250 km, then the rate of fuel consumption is 2.5 km/liter

The distance travelled by the car with 110 liters of fuel = 250km

To find the rate of fuel consumption we have to apply the unitary method

The distance travelled by the car with 1 liters of fuel = The distance travelled by the car with 110 liters of fuel /110

Substitute the values in the equation

The distance travelled by the car with 1 liters of fuel = 250/110

= 2.5 km

So a car consumes 1 liters of fuel to cover 2.5 km

The rate of fuel consumption = 2.5 km/liter

Hence, if a car consumes 110 liters of fuel with in a distance of 250 km, then the rate of fuel consumption is 2.5 km/liter

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Answer:

y=13

Step-by-step explanation: