Use a = 7 and b = 28 to work out the value of these expressions.
The first one is done for you.
a+b
ab
09
(a + b)²
=
||
11
35

The third term of the geometric series is 7*(1/2)^2 = 3.5 and the second term is 7*(1/2)^1 = 3.5. When the third term is divided by the second term, the result is 1.

A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a constant. In this particular series, the first term is 7 and the sum is 4. This means that the constant is 1/2. To find the third term, we multiply the second term by the constant, which gives us 7*(1/2)^2 = 3.5. To find the result when the third term is divided by the second term, we divide 3.5 by 3.5, which gives us 1. This is true for any infinite geometric series with the same first term and sum, because the ratio between every two consecutive terms is always the same.

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The complete equation is  y =2x+7  after substituting the given values.

What is equation?

An equation is a condition on a variable such that two expressions in the variable should have equal value and  Substitution means replacing the variables (letters) in an algebraic expression with their numerical values.

According to the question.

We have a table which shows the relation between x and y.

Let the missing term be a and b.

The the given equation becomes

y = ax+b

For finding the value of a and b.

Substitute x = 0 and y = 7 in equation y = ax + b.

7 = a(0) + b

b= 7

Again, substitute  x = 1 and y = 9 in the equation y = ax+ b

9 = a(1)+b

9 = a+7

a=9-7

a=2

substitute the value of a and b in the equation y = ax + b. we get ,

y = 2x+7

Therefore, the complete equation is y = 2x+7

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The quotient when 21x^3-42x² + 3x is divided by 3x is 7x^2 - 14x + 3.

When we divide a number, the result we ultimately get is the quotient. Division is a mathematical operation that involves dividing items into equal groups. It is represented by the symbol (÷). For instance, three groups of 15 balls each need to be equally divided. Therefore, the division formula is 15 3 = 5 when we divide the balls into three equal groups. Here, the quotient is 5, so. This implies that there will be 5 balls in each group.

The given expression is:

21x^3 - 42x^2 + 3x

Divide the equation by 3x:

21x^3 - 42x^2 + 3x / 3x

Divide each term with 3x.

Using the rule of exponents we have a^n/ a^m = a^(n-m):

7x^2 - 14x + 3

Hence the quotient when 21x^3-42x² + 3x is divided by 3x is 7x^2 - 14x + 3.

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The degree of measure of the angle ACP is 31.5°

The term angle in math is defined as the figure that is formed when two rays are joined together at a common point.

Here we have given that the total area of the given diagram is the sum of two semicircles with arc AB and arc CB having radius R and r respectively

Then it can be written as

=> R = AC

And the value of r = DB

Then the value of R is written as

=> R = 2 × r

So here we know that the area of the semicircles are given as follows

Here we have to write the semicircle with arc AB is written as

=> A₁ = π × R²/2 =

=> A₁ = π × (2×r)²/2

Then the value of A₁ is 2πr²

Similarly for the semicircle with arc CB is calculated as,

=> A₂ = π × r²/2

=> A₂ = 1/2πr²

Therefore, the degree of angle is 31.5°

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Step-by-step explanation:

[tex]\tt 33-5(x-4)=3[/tex]

[tex]\tt 33-5x+20=3[/tex]

[tex]\tt 53-5x=3[/tex]

[tex]\tt -5x=3-53[/tex]

[tex]\tt 5x=50[/tex]

[tex]\tt x=10[/tex]

I hope it's helpful

Answer: d=162km

Step-by-step explanation:

the equation of this question is:

speed = distance / time

s=d/t

18=d/9

d=18*9

d=162km

Yes, for the given measurement of sides 8cm , 7cm, and 9cm the triangle formation is possible as sum of two sides is always greater than the third side.

As given in the question,

Given measurement of the side length of the triangle is equal to :

Measurement of Side 1 = 8cm

Measurement of Side 2= 7cm

Measurement of Side3 = 9cm

To form a triangle sum of the measure of two sides of a triangle should be greater than the measure of the third side of the triangle.

In the given triangle,

Side 1 + Side 2

= 8cm + 7cm

= 15cm > 9cm

Side 1 + Side 3

= 8cm + 9cm

= 17cm > 7cm

Side 3 + Side 2

= 9cm + 7cm

= 16cm > 8cm

Therefore, sum of measure of two sides is always greater than the third side it is possible to form a triangle with given measurement of sides.

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the product of x and a factor not depending on x

What is quadratic equation

it's a second-degree quadratic equation which is an algebraic equation in x. Ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term, is the quadratic equation in its standard form. A non-zero term (a 0) for the coefficient of x2 is a prerequisite for an equation to be a quadratic equation. The x2 term is written first, then the x term, and finally the constant term is written when constructing a quadratic equation in standard form. In most cases, the numerical values of letters a, b, and c are expressed as integral values rather than fractions or decimals.

The equation given 3x(m-6n)²

This expression contains two factors:

1 factor: 3x

2 factor: (m-6n)²

Hence the product of x and a factor not depending on x.

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Step-by-step explanation:

the slope-intercept form is

y = ax + b

"a" being the slope, "b" being the y-intercept (the y-value when x = 0).

we see that with every increase of x by 1 y increases by 3.

so, the slope "a" is 3.

to get b we use one of the given points, e.g. (1, 5) :

5 = 3×1 + b = 3 + b

b = 2

so, our function is

y = 3x + 2

Answer:

30 degrees.

Step-by-step explanation:

2 cos x = √3

cos x = √3/2

x = 30.

The quiz should have 5 3-point number of questions, 3 4-point questions, and 4 5-point questions, for a total of 35 points.

3-point questions: 5

4-point questions: 3

5-point questions: 4

1. The total number of questions is 9.

2. The total number of points for the quiz is 35.

3. There should be 1 more 3-point question than 5-point questions.

4. So, 5 of the questions should be worth 3 points each, for a total of 15 points.

5. That leaves 4 points remaining.

6. 4 points can be achieved by having 3 4-point questions (12 points) and 4 5-point questions (20 points).

7. Therefore, the number of 3-point, 4-point, and 5-point questions should be 5, 3, and 4 respectively.

The quiz should have 5 3-point questions, 3 4-point questions, and 4 5-point questions, for a total of 35 points.

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Answer:

0,277777777777777777

(ii) 2.5 cm, 3.5 cm, 6.0 cm and (iii) 2.5 cm, 4.2 cm, 8 cm cannot be the sides of a triangle.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

(i) 4.5 cm, 3.5 cm, 6.4 cm

4.5 + 3.5 = 8 > 6.4

4.5 + 6.4 =10.9 > 3.5

3.5 + 6.4 = 9.9 > 4.5

so these lengths can form a triangle.

(ii) 2.5 cm, 3.5 cm, 6.0 cm

2.5 + 3.5 = 6.0 = 6.0

2.5 + 6.0 = 8.5 > 3.5

3.5 + 6.0 = 9.5 > 2.5

so these lengths cannot form a triangle.

(iii) 2.5 cm, 4.2 cm, 8 cm:

2.5 + 4.2 = 6.7 < 8

2.5 + 8 = 10.5 > 4.2

4.2 + 8 = 12.2 > 2.5

so these lengths cannot form a triangle.

Therefore, (ii) and (iii) cannot be the sides of a triangle.

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By algebra properties, the rational equation 1 / [1 / (x + 2) + 1 / (x + 3)] is equivalent to the rational expression (x² + 5 · x + 6) / (2 · x + 5). (Correct choice: B)

In this problem we find a rational equation, whose simplified form has to be found by means of algebra properties. First, write the entire expression:

1 / [1 / (x + 2) + 1 / (x + 3)]

Second, expand the denominator by addition of fractions with distinct denominator:

1 / [[(x + 3) + (x + 2)] / [(x + 2) · (x + 3)]]

Third, use division of fractions:

[(x + 2) · (x + 3)] / [(x + 3) + (x + 2)]

Fourth, simplify both numerator and denominator:

(x² + 5 · x + 6) / (2 · x + 5)

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Answer: [tex]4x^4 -3x^3 y+8x^2 y^2 -7xy^3[/tex]

Step-by-step explanation:

In standard form, the terms are arranged in order from highest to lowest exponent of the variable that comes first alphabetically.

Since the remaining distance is negative, it implies that Sam has already exceeded his goal distance. Therefore, he doesn't need to walk any further to reach his goal.

To find the remaining distance that Sam needs to walk to reach his goal of (3)1/2 miles, we need to subtract the distance he has already walked from his goal distance.

Sam walks (1)1/8 miles before lunch and 3/4 miles after resting. To calculate the total distance he has walked, we add these two distances:

Total distance walked = (1)1/8 + 3/4

To add these fractions, we need to find a common denominator, which is 8 in this case

Total distance walked = (9/8) + (6/8)

= 15/8

= 1 (7/8)

Now, we subtract the total distance walked from Sam's goal distance:

Remaining distance = (3)1/2 - 1 (7/8)

To subtract fractions, we also need a common denominator. The common denominator is 2 in this case:

Remaining distance = (6/2 + 1/2) - (15/8)

= (12/8 + 1/8) - (15/8)

= 13/8 - 15/8

= -2/8

= -1/4

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The value of x (hypotenuse) by using trigonometry is 2√5.

What is trigonometry ?

The study of correlations between triangles' side lengths and angles is known as trigonometry. The field was created in the Hellenistic era in the third century BC as a result of the use of geometry in astronomical research.

The area of mathematics known as trigonometry examines the link between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are employed to analyze this connection.

The measurement of angles and issues relating to angles are covered in the fundamentals of trigonometry. Trigonometry has three fundamental operations: sine, cosine, and tangent. The cotangent, secant, and cosecant are three crucial trigonometric functions that may be derived from these three fundamental ratios or functions. These functions serve as the foundation for all the key ideas in trigonometry.

In the triangle height = √15  and hypotenuse as x

The value of the angle is  60°

by using trigonomtry we can write

sin60° = height/ hypotenuse

√3/2 = √15/ x

x = (√15 * 2)/√3

x = 2√5

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The exponential function that shows the relationship between y and t is y  = 220,000(1.062^t).

An exponential equation can be described as an equation with exponents. The exponent is usually a variable.

The general form of exponential equation is f(x) = e^x

Where:

The form of the exponential equation that can be used to determine the population after 1997 is:

FV = P (1 + r)^n

Where:

y=220,000 x ( 1+ 0.062)^t

y  = 220,000(1.062^t).

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Comparing the value of 1 lbs of the candy we observe that, store A sells the candy at a much cheaper price and hence it has the better buy.

A mixed fraction is one that is represented by both its quotient and remainder. A mixed fraction is, for instance, 2 1/3, where 2 is the quotient and 1 is the remainder. An amalgam of a whole number and a legal fraction is a mixed fraction.

Given the number of candies the store sells for a given amount:

Store A:

2 1/4 lbs candy = $13.50

1 lbs candy = x

[tex]x = \frac{13.50}{\frac{9}{4} } \\\\x = \frac{(13.50)(4)}{9}\\ \\x=6[/tex]

Hence, store A sells 1 lbs of candy for $6.

Store B:

1 1/2 lbs candy = $15.75

1 lbs candy = y

[tex]y = \frac{15.75}{\frac{3}{2} } \\\\y = \frac{(15.75)(2)}{3}\\ \\y = 10.5[/tex]

Store B sells 1 lbs of candy for $ 10,5.

Comparing the value of 1 lbs of the candy we observe that, store A sells the candy at a much cheaper price and hence it has the best buy.

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If the two circle centered at P and Q with radius 10 and 8 respectively , and if the common external tangent intersects line PQ at R , then the length of QR is 72  .

The radius of the circle that is centered at P is = 10 ;

the radius of circle that is centered at Q is = 8 ;

the common external tangent intersects the line PQ at R ,

Let XYR be external tangent with X on the Circle with center P , and at point Y on circle with center Q .

we extend , the line PQ to point R ,

we get , that triangle XPR is similar to triangle YQR ,

it means that , [tex]\frac{RQ}{RP} =\frac{QY}{PX}[/tex] ;

let the length of RQ be = x , then

length of RP will be = [tex]x+18[/tex]  ;

we get ; [tex]\frac{x}{x+18} =\frac{8}{10}[/tex] ;

⇒ [tex]10x=8x+144[/tex] ;

⇒ [tex]2x=144[/tex] ;

⇒ [tex]x=72[/tex] .

Therefore , the length of QR is 72 .

The given question is incomplete , the complete question is

A circle centered at P with radius 10 and a circle centered at Q with radius 8 are externally tangent. A common external tangent intersects line PQ at R. Find QR .

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According to the given statement If each can sold in accounted for as to its cost Last in first out.

The average is determined by adding together all of the numbers as well as dividing the total by the total number of figures provided. It represents the midpoint of the supplied data set. The numerical number that may display a lot of facts is the average value. We constantly encounter the average computation in our daily lives.

The four sorts of average that we recognize are mean, mode, median, and range. Although the others are our most popular "measures of central tendency," range is actually a measure of spread or dispersion.

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The units digit of an integer every square of an integer ends with (=has a units digit) 0, 1, 4, 5, 6, or 9.

This question can be answered in a variety of ways. Finding squares with the unit digits 0, 4, 5, and 6 is one way to demonstrate by means of elimination that the answer must be 2. The reason this method works is that you're presuming one of the answers to this specific multiple-choice question is true, but it's not actually that fulfilling.

A more effective method is to note that any number has the form (10m+i) such that 0i9. By squaring, you may determine exactly which numbers are feasible units digits in a square by getting 100m2+20mi+i2, etc. in turn.

Every square of an integer should have the units digit (0, 1, 4, 5, 6, or 9) as its final value.

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The random variable x is exponentially distributed, where x represents the waiting time to see a shooting star during a meteor shower. if x has an average value of 53 seconds, what are the parameters of the exponential distribution:  

                    X ~ Exp( μ = 53)

Random Variable:

A random variable is a variable that can take many values. This is because random events can have multiple outcomes. So don't confuse random variables with algebraic variables. Algebraic variables represent the values ​​of unknown quantities in computable algebraic equations. A random variable, on the other hand, can have a range of values ​​that could be the result of a random experiment.

Suppose two dice are rolled and a random variable X is used to represent the sum of the numbers. The minimum value of X is 2 (1 + 1) and the maximum value is 12 (6 + 6). Therefore, X can have any value between 2 and 12 (inclusive). If probabilities are assigned to each outcome, we can determine the probability distribution of X.

According to the Question:

We know that the random variable X who represents the waiting time to see a shooting star during a meteor shower follows an exponential distribution and for this case we can write this as:

            X ~ Exp( μ = 53)

But, also we can define the variable in terms of  like this:

            X ~ Exp (λ =1/λ =1/53)

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In grade 7, one common method to solve absolute value equations is by isolating the absolute value on one side of the equation and then solving for the two possible solutions.

Here is a step-by-step process to solve an absolute value equation:

Isolate the absolute value on one side of the equation by adding or subtracting the same value to both sides of the equation.

For example, |x| = 3 can be rewritten as x = 3 or x = -3

Split the equation into two separate cases, one for when the value inside the absolute value sign is positive, and one for when it is negative.

Solve each case separately.

Check your solutions by plugging them back into the original equation and making sure they are valid solutions.

Write your answer in interval notation if the solutions are continuous or list them as individual values if they are not.

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Answer to Problem 13 is  34.5  

Answer to Problem 15 is  22.9  

==================================================

Work Shown for Problem 13

sin(x)/17 = sin(91)/30

sin(x) = 17*sin(91)/30

sin(x) = 0.56658036

x = arcsin(0.56658036) or x = 180-arcsin(0.56658036)

x = 34.51210645 or x = 180-34.51210645

x = 34.51210645 or x = 145.48789355

x = 34.5 or x = 145.5

If x = 34.5, then the missing unmarked angle is 180-x-91 = 180-34.5-91 = 54.5 which is a valid angle (since it's between 0 and 180).

If x = 145.5, then the missing unmarked angle is 180-x-91 = 180-145.5-91 = -56.5; but this is NOT valid because the angle needs to be between 0 and 180 (i.e. negative angles aren't allowed)

In short, x = 34.5 is valid while x = 145.5 is not valid.

Therefore, the only possible answer is   34.5  

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

Work Shown for Problem 15

sin(x)/20 = sin(119)/45

sin(x) = 20*sin(119)/45

sin(x) = 0.38871987

x = arcsin(0.38871987) or x = 180-arcsin(0.38871987)

x = 22.87486940 or x = 180-22.87486940

x = 22.87486940 or x = 157.1251306

x = 22.9 or x = 157.1

If x = 22.9, then the missing unmarked angle is 180-x-119=180-22.9-119 = 38.1 which is valid since it's between 0 and 180.

If x = 157.1, then 180-x-119=180-157.1-119 = -96.1 which is NOT a valid angle since it's not between 0 and 180. This allows us to rule out the case that x = 157.1

The only possible answer is therefore   22.9  

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

Side notes:

By applying the Least Common Multiple theory, it can be concluded that Philip has to buy 2 packs of pencils and 3 packs of erasers.

Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that could be divided by those numbers.

The simplest way to find the LCM of two or more numbers is to list the multiples of each number.

Then, check the smallest multiple that appears in all of the multiple lists. This value is the LCM of those numbers.

From the question, we know that Philip wants to have the same number of pencils as erasers. Whilst they come in different packages. To do so, firstly we list the multiples of each package.

Pencil: 18 : 18, 36, 54, 72, 90, ...

Eraser: 12 : 12, 24, 36, 48, 60, ...

Now we check the smallest multiple that appears in all of the multiple lists, the LCM is 36.

36 pencils means 36 : 18 = 2 packs of pencils

36 erasers means 36 : 12 = 3 packs of erasers

Thus, Philip has to buy 2 packs of pencils and 3 packs of erasers

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All of the true statements that have to do with the expression 8y+3x+5 are:

An expression in mathematics is made up of a mixture of variables, integers, and functions (such as addition, subtraction, multiplication or division etc.) In some ways, phrases and expressions are comparable.

In the equation 5 is a constant, this is because the value would not have to change because it does not have a variable attached.

The terms of the expression are, 3x, 8 y and 5. This makes it a total of 3 terms in the expression.

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The constant of variation for this relation will be option (C.) 5

The ratio between two variables in a direct variation or the product of two variables in an inverse variation.

We solve the problem as follows.

Given that y is directly proportional x

y = kx, where k = proportionality constant

Constant of proportionality is the constant value of the ratio between two proportional quantities. Two varying quantities are said to be in a relation of proportionality when, either their ratio or their product yields a constant. The value of the constant of proportionality depends on the type of proportion between the two given quantities: Direct Variation and Inverse Variation.

According to the provided data,

y = 20

x = 4.

So,

20 = 4k

k = 20/4 = 5

Therefore constant of variation will be 5

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As per the given points, the distance is 0.08 meter.

The term distance in math is known as the length of the line joining the two points.

Here we have given that over a period of one year and the two points on opposite sides of a mid-ocean ridge moved a distance of 4 centimeters farther apart.

Here we have to calculate the distance between the two point in  meter.

Let us consider that x be the distance between the two points.

As we know that in each side there is 4 centimeter distance.

Then the total distance between the two points,

=> 4 x 2 = 8 centimeter.

Then the meter equivalent is calculated as,

=> 8/100 = 0.08

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