Solve the initial value problem 2yy' + 3 = y^2 + 3x with y(0) = 9. 1. To solve this, we should use the substitution u = _____ With this substitution, y = _____
y' = _____
Enter derivatives using prime notation (e.g., you would enter y' for dy/dx). 2. After the substitution from the previous part, we obtain the following linear differential equation in x, u, u'. 3. The solution to the original initial value problem is described by the following equation in x, y.

Using the arithmetic operation of multiplication, two cities having a distance of 2.5 inches on map have an actual distance of 50 miles.

What is arithmetic operation?

A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.

The value of 1 inch on scale = 20 miles.

The distance between two cities on map = 2.5 inches

To find the distance between two cities in actual use the arithmetic operation of multiplication.

Let the original distance value in miles be x.

The equation for distance will be -

x = 2.5 × 20

Solve the equation -

x = 2.5 × 20

x = 50

Therefore, the actual distance in miles is 50 miles.

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The length of NQ for this problem is given as follows:

NQ = 5.

In this problem, we have two similar triangles, meaning that their side lengths are proportional.

The lengths for this problem are given as follows:

The similar triangles are given as follows:

PQR and PNO.

The equivalent side lengths are given as follows:

Hence the proportional relationship is given as follows:

5/(5 + x) = (x - 1)/8

Applying cross multiplication, we have that:

(x + 5)(x - 1) = 40

x² + 4x - 5 = 40

x² + 4x - 45 = 0.

Hence:

(x + 9)(x - 5) = 0.

The positive solution is:

x - 5 = 0

x = 5.

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1. Yes. A dilation is performed on the figure. The ratio of the two figures is same. 2. No. The ratios of the sides of the figures are not equal and no transformation is applied.

The transformation, or f: X X, is the name given to the function, f, that maps to itself. After the transformation, the pre-image X becomes the picture X. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation. A function can be moved in one way or another using translation, rotation, reflection, and dilation. A function can also be scaled using rotation around a point. Two-dimensional mathematical figures move about a coordinate plane according to transformations.

1. Yes. A dilation is performed on the figure. The ratio of the two figures is same.

2. No. The ratios of the sides of the figures are not equal and no transformation is applied.

3. Yes. A translation is performed on the figure to shift the transformation with 2 units.

4. No. The ratios of the sides of the figure are not same.

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The type of slope for each function is given as follows:

Zero, Positive, Positive, Undefined.

There are four types of lines, and their slopes are classified as follows:

There is a mistake in the options or in the graph, as both graphs b and c are increasing lines.

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B.The first class’s histogram has greater spread (variability) since it is derived from a smaller number of tosses per student.

The question is asking which statement is true regarding the two histograms of -values. The first class has 250 students and they are each instructed to toss a coin 20 times. The second class has 200 students and they are told to toss a coin 40 times. The first class is tossing the coin less times than the second class, so the spread (variability) is greater for the first histogram since it is derived from a smaller number of tosses per student.

The first class has 250 students and they are each instructed to toss a coin 20 times. The second class has 200 students and they are told to toss a coin 40 times.

The first class is tossing the coin less times than the second class, so the spread (variability) is greater for the first histogram since it is derived from a smaller number of tosses per student.

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The area of the major sector in the given diagram is 115.5 sq.unit.

The region having a greater area is known as Major Sector while the one which has a smaller area is the minor sector.

A sector with a central angle greater than 180° is called a major sector and the formula is given below:

/360° ₓ r²

The radius is 7m and is 270°

Therefore 270/360 × 3.142 × 7² =  115.5sq.unit.

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The correct answer is OB. The statement does not make sense as a and b are real numbers in the imaginary number of the form a + bi. Hence, the imaginary numbers are defined and interpreted in the set of real numbers also.

Imaginary numbers are defined as the set of numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit.

It is a mathematical concept that extends the real numbers system, by introducing a new number i which is defined as the square root of -1. It allows us to solve equations which are not solvable in real numbers.

The word "imaginary" is used to denote the presence of the imaginary unit i, which is not a real number but is used to extend the real numbers. It does not mean that these numbers are undefined.

Hence, the statement "The word imaginary in imaginary numbers tells that these numbers are undefined" is not true, as the imaginary numbers are defined and interpreted in the set of real numbers also.

In summary, the statement is not making sense as the imaginary numbers are defined in the set of complex numbers.

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the relationship between the monthly rent (p) and the demand (x) is p(x) = -15x + 1560.

A real estate office handles an apartment complex with 80 units. When the rent per unit is $360 per month, all 80 units are occupied. However, when the rent is $405per month, the average number of occupied units drops to 77. Assume that the relationship between the monthly rent (p) and the demand (x) is linear.

Since the equation will be linear, we only need two coordinates to write the equation.

They are given, we can assign the values x = no.of units and y = rent:

x1 = 80, y1 = 360

and

x2 = 77, y2 = 405

Find the slope: m = (y2-y1)/(x2-x1)

m = (405-360)÷(77-80) = 45÷(-3)= -15

The slope m= -15

A) Write the equation of the line giving the demand (x) in terms of the rent (p)

Use the point/slope equation.

Using the point/slope formula: y - y1 = m(x - x1)

y - 360= -15(x - 80)

y - 360 = -15x + 1200

y = -15x + 1200 + 360

y = -15x + 1560

or

p(x) = -15x + 1560

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Answer:

Step-by-step explanation:

The solution to the equation is x = 7.

What is the linear equation in one variable?

A linear equation in one variable is an equation of the form ax + b = 0, where a and b are constants and x is the variable. The general form of a linear equation in one variable is ax + b = 0, where a and b are constants and x is the variable. The graph of a linear equation in one variable is a straight line.

The first step in solving the equation 2x - 4 = 10 using algebra tiles would be to add 4 to both sides of the equation to get 2x = 14. The next step would be to divide both sides of the equation by 2 to get x = 7.

Hence, the solution to the equation is x = 7.

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Answer: To find the distance Alexandria is from her starting point, you can use the Pythagorean theorem.

The theorem states that in a right triangle, the square of the hypotenuse (the distance between the starting point and the final point) is equal to the sum of the squares of the other two sides (the distance traveled north or south and the distance traveled east or west).

In this case, the distance traveled north or south is 480 meters (the distance traveled due south) and the distance traveled east or west is 427 meters (the distance traveled due east) - 171 meters (the distance traveled due west) = 256 meters.

So, using the Pythagorean theorem:

Distance = √(480^2 + 256^2)

Distance = √(230400 + 65536)

Distance = √(295936)

Distance = 544.98 meters

So Alexandria is 544.98 meters away from her starting point.

Step-by-step explanation:

The word problem "8 times the sum of 2 and a number" is equal to 8(2 + n).

In Mathematics, an algebraic expression simply refers to a mathematical equation that can be used for showing the relationship which exist between two (2) or more variables, numerical quantities, as well as in conjunction with different mathematical operations.

Assuming the variable n represents the unknown numerical value (number), we would translate the word problem into an algebraic expression as follows;

Algebraic expression = 8 × (2 + n)

Algebraic expression = 8(2 + n).

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+4.533 is the percent error for the company's estimate

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.

Given, Bay City Balloons studied the demand in Bay City before opening a hot air balloon company. The owners expected to book 375 hot air balloon rides in their first ninety days of business. Instead, they booked 392 rides in that period.

Since percentage error:

Percent error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage. In other words, the percent error is the relative error multiplied by 100.

Thus in our case

Percentage error = ((392 - 375)/ 375) * 100

Percentage error = 4.533

therefore, the percent error for the company's estimate is +4.533.

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We need 32 rows in the truth table that would use to evaluate a logical statement in five variables.

To find out the number of rows in a truth table with 5 variables, we should know what is a truth table.

A truth table is a study of a logic function by listing all possible values the function can attain. The top row illustrates logical variables and combinations in increasing complexity leading up to the final function.

The number of rows that a truth table needs is decided by the number of basic statement letters involved in the set formulas that will be involved in the set of formulas.

The formula to calculate the number of rows is equal to 2ⁿ, where n is the number of basic statement letters involved.

Number of rows = 2ⁿ

If the number of variables in a truth table is 5, then the number of rows = 2⁵

2⁵ = 2×2× 2× 2× 2

= 32

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Answer: Because instead it equals 0.005

Step-by-step explanation:

The reason 5/100 is not equal to 0.5 is that they are written in different forms.

5/100 is written in the form of a fraction, where the numerator (5) represents the number of parts out of a whole, and the denominator (100) represents the total number of parts in the whole. So, 5/100 represents 5 parts out of 100 parts, which is equal to 0.05.

On the other hand, 0.5 is written in decimal form. It represents the number of parts out of 1, where the digits to the right of the decimal point represent the number of tenths, hundredths, thousandths and so on. So, 0.5 represents 5 tenths, or 5 parts out of 10 parts.

In conclusion, while they represent the same value, they are written in different forms and it's important to know how to read and convert them correctly.

Assume f(x) has a continuous second derivative. The expression "f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx" is TRUE.

To show that f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx, we will use Taylor's theorem with a remainder in the integral form.

Taylor's theorem states that for any function f(x) that has a continuous kth derivative, we can express the function at any point x = b as:

f(b) = f(a) + f'(a)(b-a) + (1/2!)f''(a)(b-a)^2 + ... + (1/k!)f^(k)(c)(b-a)^k

where c is some value between a and b.

If we take the limit as k approaches infinity, we get:

f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx

This is known as the Taylor's theorem with a remainder in the integral form.

This equation shows that f(b) is equal to f(a) plus the linear approximation of f'(a)(b-a) plus the remaining error term ∫ₐᵇf"(x)(b-x)dx which is the integral of the second derivative of f(x) times (b-x) from a to b.

This is true because f"(x) is continuous on the interval [a, b], so the integral of f"(x) over that interval exists. And because the second derivative of f(x) is continuous, the integral can be evaluated.

Therefore, assuming that f(x) has a continuous second derivative, we can conclude that f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx

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Answer:

1/3

Step-by-step explanation:

Using the fractions given, make common denominators.

After you do the first hint, look for a pattern in the top separate from a pattern in the bottom.

A statement, rule, or legislation that establishes the link between the independent variable and the dependent variable (the dependent variable). f ( 6 ) = 82 .

Domain of [tex]$2 x^2+5 \sqrt{x-2}:\left[\begin{array}{cc}\text { Solution: } & x \geq 2 \\ \text { Interval Notation: } & {[2, \infty)}\end{array}\right]$[/tex]

Range of [tex]$2 x^2+5 \sqrt{x-2}:\left[\begin{array}{cc}\text { Solution: } & f(x) \geq 8 \\ \text { Interval Notation: } & {[8, \infty)}\end{array}\right]$[/tex]

Interception zones on the axis [tex]$2 x^2+5 \sqrt{x-2}$[/tex] : None .

Asymptotes of [tex]$2 x^2+5 \sqrt{x-2}$[/tex] : None .

Points of Extremity [tex]$2 x^2+5 \sqrt{x-2}$[/tex] : Minimum (2,8) .

[tex]f(x) = 2x^{2} +5\sqrt{(x-2} \\f (6) = 2 *(6^{2} ) + \sqrt{6 - 2} \\ = 82[/tex]

the type of conduct or activity appropriate to a person, thing, or institution; the reason why something is made or existing; role. any formal event or occasion that is held in public or among friends. a component that affects or is influenced by other components: Availability and demand determine price.

As a set of inputs with one output for each, a function is defined as a relationship between them. A function, expressed simply, is an association between inputs where each input is connected to one and only one output. Each function has a range or codomain. f(x), where x is the input, is a common way to refer to a function.

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Answer:

0 > x

Step-by-step explanation:

The open circle indicates that x cannot be equal to zero, which rules out the last two options. Since the line is going to the right (towards numbers greater than 0), x must be greater than zero, which means that the first option is correct.

Answer:

x > 0

Step-by-step explanation:

we see the blue line. it covers everything that is larger than 0.

the dot on 0 is empty, which indicates that the point (here 0) itself is not included.

therefore, the second answer option (x > 0) is correct.

The class width is 10.

A histogram is a representation of statistical data that makes use of rectangles to illustrate the frequency of data items in a series of equal-sized numerical intervals. The independent variable is represented along the horizontal axis and the dependent variable is plotted along the vertical axis in the most popular type of histogram.

Given:

Order the data increasingly

45,45,46,59,59,63,64,65,65,66,67,69,69,70,70,70,71,71,71,72,72, 73,73,73,74,74,75,76,76,76,77,77,78,79,80,80,80,80,80,80,81,81

81,81,82,82,83,83,83,83,83,84,84,85,85,85,87,87,89,89,90,93,94

94,94,96,99,101,101,109

Now, the range: maximum-minimum

= 109-45 = 64

Now, Divide the range by the number of classes you wish to distribute the data,

= 64/6

= 10.66 (round to 11)

Create intervals of type [a,b] starting with the first datum, and then add the class width to create the interval classes.

[45,56), [56,67), [67,78), [78,89), [89,100), [100,111)

Then, Create a frequency table that includes class boundaries (the smallest and largest data in the class), midpoints (adding the endpoints and dividing by 2), frequencies (the amount of data that fall into the class), relative frequencies (the frequency of the class divided by the total number of data, in this case 70), and cumulative frequencies (the sum of the frequencies of the class plus the previous ones)

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d = 4

b = -3. c = 1

a = 3 and d = 1

What are three polynomials factorized below?

To get the missing values in the table, we will factorize the given expression and compare the factored expression with the expression containing the missing constants.

1) For the expression x²+2x-8, on factorizing we have;

x²+2x-8

= (x²+4x)-(2x-8)

Factoring out the common terms from both parenthesis;

= x(x+4)-2(x+4)

= (1x+4)(1x-2)

=  (1x-2)(1x+4)

Comparing the resulting expression with (ax+b)(cx+d)

a = 1, b = -2, c = 1 and d = 4

2) For the expression 2x³+2x²-24x

Factoring out the common term we will have;

= 2x(x²+x-12)

= 2x(x²-3x+4x-12)

= 2x{x(x-3)+4(x-3)}

= 2x{(x+4)(x-3)}

=  2x(1x-3)(1x+4)

Comparing the resulting expression with 2x(ax+b)(cx+d)

a = 1, b = -3. c = 1 and d = 4

3) For the expression 6x²-15x-9 we will have;

On simplifying,

= 6x²+3x-18x-9

= 3x(2x+1)-9(2x+1)

= (3x-9)(2x+1)

Comparing the resulting expression with (ax+b)(cx+d)

a = 3, b = -9, c = 2 and d = 1

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The distance of the line segment joining the two points is [tex]2\sqrt{2}[/tex].

The shortest distance (length of the straight line segment's length connecting both given points) between points ( p,q) and (x,y) is:

[tex]D = √[(x-p)² + (y-q)²] D = \sqrt{(x-p)^2 + (y-q)^2} \: \rm units.[/tex]

Given;

The two points;

(2,0) (0,-2)

Now,

D=[tex]\sqrt{(0-2)^2+(-2-0)^2}[/tex]

=2[tex]\sqrt{2}[/tex]

Therefore, the distance between two points will be 2[tex]\sqrt{2}[/tex]

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The Expressions with solution are shown below.

A mathematical operation such as subtraction, addition, multiplication, or division is used to combine terms into an expression. In a mathematical expression, the following terms are used:

Given:

As, the expression is contains the variables, numbers and operation.

So, the expression can be

1. 3x + 4 + 6x

= 9x + 4 (adding the like terms)

2.  x+5y  

3. 4(2y+3)

= 8y + 12

4. 3y×2

= 6y

5. 20x / 5x

= 4

6. 42 / 12x

= 7/2x

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A quadratic function in vertex form with a fraction can be graphed by first converting the fraction to decimal form.

A quadratic function is a function of the form f(x) = ax² + bx + c, where a, b, and c are numbers with a not equal to zero. It should be noted that the graph of a quadratic function is a curve called a parabola.

Then, using the vertex and the slope of the parabola, plot the vertex point and at least two more points on either side of the vertex. Finally, use these points to graph the parabola. The equation of a quadratic in vertex form is y = a(x-h)^2 + k where (h,k) is the vertex.

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If f(-8) = 1, then the point (-8, 1) is on the graph of f.

The numeric value of the function at x = -8 is given as follows:

f(-8) = 1.

The format of each point on the coordinate plane is given as follows:

(x,y).

In which y is the numeric value of the function at x, meaning that the point can also be written as follows:

(x, f(x)).

Considering that when x = -8, f(x) = 1, the point is given as follows:

(-8,1).

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The exact value of cot A is .20/21

The denominator means moving the radical term square root or cube root to the numerator such that a denominator is a whole number.

We have been given that sin( A ) = 21/29 and angle A is in quadrant 1. We are asked to find the exact value of  cot (A) in simplest radical form.

We know that sine relates opposite side of right triangle with hypotenuse.

Since = opposite

            hypotenuse

This means that opposite side is 21 units and hypotenuse is 29 units.

We know that cotangent relates adjacent side of right triangle with adjacent side.

Cot = Adjacent

         opposite

Now we will find adjacent side using Pythagoras theorem as:

Adjacet² = hypotenuse² - opposite²

Adjacent² = 29² - 21²

Adjacent² = 841 - 441

Adjacent² =400

Let us take positive square root on both sides:

√Adjacent² = √400

 Adjacent = 20

Therefore, adjacent side of angle A is 20 units.

Cot (A) = 20/21

The exact value of cot A is .20/21

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a. The frequency which she hears is 440Hz.

b. The person in the car would hear a frequency of 391.98 Hz when the ferry leaves the dock and heads directly away from the driver.

(a) The frequency that the person in the car hears is affected by the motion of the car and the sound source (the ferry). The effect is called the Doppler effect, which causes the perceived frequency of sound to change depending on the observer and the relative motion of the source.

If the car is moving toward the ferry, the frequency will be higher than the actual frequency of the sound. The perceived frequency is given by the formula:

f' = f (v + v_s) / (v + v_s)

where

f = the actual frequency of the sound (400 Hz)

v = the velocity of sound (approximately 340 m/s)

[tex]v_s[/tex] = the velocity of the car (60 km/h = 16.67 m/s)

So,

f' = 400(340 + 16.67) / (340 - 16.67) = 400(356.67) / 323.33 = 440 Hz

Hence, the frequency which she hears is 440Hz.

(b) The perceived frequency when the ferry leaves the dock and heads directly away from the driver is also affected by the Doppler effect.

f' = f (v - v_s) / (v - v_s)

where

f = the actual frequency of the sound (400 Hz)

v = the velocity of sound (approximately 340 m/s)

v_s = the velocity of the ferry (15 km/h = 4.17 m/s)

So,

f' = 400(340 - 4.17) / (340 + 4.17) = 400(335.83) / 344.17 = 391.98 Hz

The person in the car would hear a frequency of 391.98 Hz when the ferry leaves the dock and heads directly away from the driver.

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There are 16128 ways to form the committee when 2 of the men refuse to serve together, 23040 ways to form the committee when 2 of the women refuse to serve together, and 5040 ways to form the committee when 1 man and 1 woman refuse to serve together.

(a) When two of the men refuse to serve together, there are 6 men to choose from for the first 3 positions, 4 for the next, and 2 for the last. So there are 6 × 4 × 2 = 48 ways to choose the 3 men.

There are 8 women to choose from for the first 3 positions, so there are 8 × 7 × 6 = 336 ways to choose the 3 women. Thus there are 48 × 336 = 16128 ways to form the committee when 2 of the men refuse to serve together.

(b) When two of the women refuse to serve together, there are 8 women to choose from for the first position, 6 for the next, and 4 for the last. So there are 8 × 6 × 4 = 192 ways to choose the 3 women.

There are 6 men to choose from for the first 3 positions, so there are 6 × 5 × 4 = 120 ways to choose the 3 men. Thus there are 192 × 120 = 23040 ways to form the committee when 2 of the women refuse to serve together.

(c) When 1 man and 1 woman refuse to serve together, there are 7 women to choose from for the first 3 positions and 6 men to choose from for the last 3 positions.

So there are 7 × 6 × 5 × 4 × 3 = 5040 ways to form the committee when 1 man and 1 woman refuse to serve together.

The answer to each of the questions is the number of different committees that can be formed under the given conditions.

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--The given question is incomplete; the complete question is

"From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How many different committees are possible if (a) 2 of the men refuse to serve together? (b) 2 of the women refuse to serve together? (c) 1 man and 1 woman refuse to serve together?"--