Investigation • Getting to the root

ASAP PLEASE!!!

The proof of "if a | b and b | a, where a and b are integers, then either a = b or a = -b" is shown below .

The representation "a|b and b|a ", means that "a" is a divisor of "b" and "b" is a divisor of "a".

which means , there exists an integer "c" ; such that b=ac    ...equation(1) and an integer "d" such that a = bd    .....equation(2)

Substituting the first equation into the equation(2) , we get:

⇒ a = bd = (ac)d = adc ;

Since a|a, it must be that either a = adc or adc = -a.

But  "c" and "d" are integers, So , "adc" must also be an integer.

Hence, it must be that adc = a.

Thus , a = adc = a. This implies that d = c = 1, and

So ⇒ a = b .....Equation(3) .

Similarly , if adc = -a, then we have: a = -a

Which implies that a = 0, but since "a" and "b" are integers, they cannot both be 0. So , it must be that a = b   ...equation(4) .

Therefore , On combining equation(3) and equation(4)  , we proved that "if a|b and b|a, where a and b are integers, then either a = b or a = -b" .

The given question is incomplete , the complete question is

Prove/show that if a|b and b|a, where a and b are integers, then either a = b or a = -b.

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The function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+3 inches is f(r) = 12π(r² + 3r + 3).

In this question, we are asked to determine a function for a spherical balloon that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+3 inches.

Volume of spherical balloon = V(r) = 4/3 π r³

If the radius of balloon is r+3 inches = V(r+3) = 4/3 π (r+3)³

The amount of air required is the difference between V(r) and V(r+3):

So, f(r) = V(r+3) - V(r) = 4/3 π (r+3)³ - 4/3 π (r)³

                                  = 4/3 π ((r+3)³ -r³)

                                  = 4/3 π (r+3 -r)((r+3)² + r(r+3) + r²)

                                  = 4π(r² + 9 + 6r + r²+3r + r²)

                                  = 4π(3r² + 9r + 9)

                             f(r) = 12π(r² + 3r + 3)

Hence, the function representing is f(r) = 12π(r² + 3r + 3).

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The equation for the line that goes through (-10, 6) is given as y = -4x - 34

A parallel line has the same slope as the original line, but a different y-intercept. So, we want to find an equation of the form y = mx + b where m = -4 (the slope of the original line) and b is the y-intercept that gives the desired point (-10, 6).

We can find b by plugging in the coordinates of the point (-10, 6) into the equation y = mx + b:

6 = -4 * (-10) + b

6 = 40 + b

Subtracting 40 from both sides, we get:

-34 = b

So, the equation for the parallel line that goes through (-10, 6) is:

y = -4x - 34

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The difference quotient of f; that is, [tex]\frac{f(x+h)-f(x)}{h}[/tex], h ≠ 0 for the function f(x) = x^2 - 9x + 6 is 2x - 9.

The function is f(x) = x^2 - 9x + 6.

We have to determine the differential equation; that is [tex]\frac{f(x+h)-f(x)}{h}[/tex].

To find the value of f(x+h) substitute x+h in place of x the the function.

f(x+h) = (x+h)^2 - 9(x+h) + 6

Solving

F(x+h) = x^2 + 2xh + h^2 - 9x - 9h + 6

Now putting the value

[tex]\frac{f(x+h)-f(x)}{h} = \frac{(x^2 + 2xh + h^2 - 9x - 9h + 6)-(x^2 - 9x + 6)}{h}[/tex]

Simplify

[tex]\frac{f(x+h)-f(x)}{h} = \frac{x^2 + 2xh + h^2 - 9x - 9h + 6-x^2 + 9x - 6}{h}[/tex]

[tex]\frac{f(x+h)-f(x)}{h} = \frac{2xh + h^2 - 9h}{h}[/tex]

Taking common h on both side, we get

[tex]\frac{f(x+h)-f(x)}{h} = \frac{h(2x + h - 9)}{h}[/tex]

[tex]\frac{f(x+h)-f(x)}{h}[/tex] = (2x + h - 9)

Now setting the limit h tends to 0

[tex]\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/tex] = [tex]\lim_{h\rightarrow 0}[/tex](2x + h - 9)

Now putting h = 0, we get

[tex]\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/tex] = 2x - 9

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The complete question is:

Find the difference quotient of f; that is, find [tex]\frac{f(x+h)-f(x)}{h}[/tex], h ≠ 0 for the following function. Be sure to simplify.

f(x) = x^2 - 9x + 6

The value of k is "i".

What is cube and square?

Any number multiplied by itself is a square number. 0 through 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are the square numbers. A number multiplied by itself three times is a cube number. Up to 100, the cube numbers are 1, 8, 27, and 64. Cubes in three dimensions can be used to represent cube numbers.

Given that the value of a cube and a square, A, is zero when A = -1, we can write the equation:

a^3 + A^2 = 0

Substituting A = -1 into this equation gives us:

a^3 + (-1)^2 = 0

Simplifying the right-hand side of this equation:

a^3 + 1 = 0

So,

a^3 = -1

Taking the cube root of both sides:

a = -1^(1/3)

The value of k is the cube root of -1, which is a complex number and is represented as an imaginary number, i.e., "i".

Therefore, the value of k is "i".

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The complex number -1, which is represented as an imaginary number, or I has the value of k, which is the cube root of that number.

Any number multiplied by itself is a square number. 0 through 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are the square numbers. A number multiplied by itself three times is a cube number. Up to 100, the cube numbers are 1, 8, 27, and 64. Cubes in three dimensions can be used to represent cube numbers.

Given that the value of a cube and a square, A, is zero when A = -1, we can write the equation:

[tex]a^3+A^2=0[/tex]

Substituting A = -1 into this equation gives us:

[tex]a^3+(-1)^2=0[/tex]

Simplifying the right-hand side of this equation:

[tex]a^3+1=0[/tex]

So,

[tex]a^=-1[/tex]

Taking the cube root of both sides:

[tex]a=(-1)^{\frac{1}{3} }[/tex]

The value of k is the cube root of -1, which is a complex number and is represented as an imaginary number, i.e., "i".

Therefore, the value of k is "i".

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Answer:

Option 1 is correct

Step-by-step explanation:

The scale factor is equal to 4 which is calculated by determining the coordinates.

Let's begin by determining the coordinates.

These are the ones we begin with!

D (1,1)

E (2,2)

F (4,2)

G (3,1)

Now we must locate the best image cords.

D' (4,4)

E' (8,8)

F' (16,8)

G' (12,4)

We're looking for the scale factor, so we'll see what number is multiplied by those coordinates to get the prime coordinates.

D is (1,1) and D' is (4,4), but we know that 1x4 equals 4, so 4 is the scale factor by which we multiply all of the numbers.

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8/4=12/x is the proportions could be used to find the width, w, of his model.

Completed question :

grace is making a scale model of her favorite car. the actual car is 8 feet long and 4 feet wide. grace wants her model to be 12 inches in length. which proportion could be used to find thethe width, w, of his model?

A:8/4=12/w B:8/4=w/12 C: 8/12=w/4 D:12/8=4/w

Given data :

length = 8 feet

width = 4 feet

Length that grace wants = 12 feet

8/4=12/x

8feet long           12 inches

4feet wide           x = width which is not given

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The value of the bulldozer after 12 years is 28,950.

Since it is given that the company depreciates the bulldozer linearly over its useful life,

The value of the bulldozer, as a function of time, should be in the form; y=m(t)+c.

where,

y= value of bulldozer

m and c = are some constants

And the boundary conditions are given as,

when t=0, y=78,750

= m(0)+c=78,750

= c = 78,750

and also given,

when t=16, y=12,350

=m(16 )+78,750=12,350

=m=12,350-78,750/16

=m=-4,150

so, the equation comes as,

y=78,750 - 4,150(t)

Value of bulldozer after 12 years would be,

y=78,750 - 4,150(12)

y=28,950

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The required percent error of the measurement is 0.8%.

Error bar, are the line through a point on a graph, axes, which emphasizes the uncertainty or variation of the corresponding coordinate of the point.

The percent error of Adrian's measurement can be calculated as follows:

(|Measured value - Actual value| / Actual value) * 100

= (|12.8 - 12.7| / 12.7) * 100

= (0.1 / 12.7) * 100

= 0.786%

Rounded to the nearest tenth of a percent, the error is 0.8%.

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Answer:6.6%

Step-by-step explanation:

Part a) A. The population must be normally distributed

Part b) P(overbar x < 68.2​) = 0.7967

Part c)P(overbar x ≥ 65.6​)=  0.3745

a) Population is normally distributed -

b) P(overbar x < 68.2​)

Estimate the z score (z).

z = (x -  μ) / s

z = (68.2 - 64) / 19/√14

z = 0.83

Use z table,

P(overbar x < 68.2​) = P(z < 0.83)

P(overbar x < 68.2​) = 0.7967

c)  P(overbar X  ≥  65.6)

Estimate the z score (z).

z = (x -  μ) / s

z = (65.6 - 64) / 19/√14

z = 0.32

Use z table,

P(overbar x ≥  65.6)  =  P(z  ≥  0.32)

                                  = 1 -  P(z  <  0.32)

                                 = 1 - 0.6255

                                 = 0.3745

P(overbar x ≥  65.6)  = 0.3745

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The K-Means algorithm terminates when. a user-defined minimum value for the summation of squared error differences between instances and their corresponding cluster center is seen.

The K-Means algorithm terminates when

A) a user-defined minimum value for the summation of squared error differences between instances and their corresponding cluster center is seen.

B) the cluster centers for the current iteration are identical to the cluster centers for the previous iteration.

C) the number of instances in each cluster for the current iteration is identical to the number of instances in each cluster of the previous iteration.

D) the number of clusters formed for the current iteration is identical to the number of clusters formed in the previous iteration.

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The initial amount in  Lea's account is 1212.12. Option A is the correct option.

What is the principal amount?

The amount you owe at any given time is the principal amount. When you first took out the loan, it is exactly your loan amount. The principal amount drops as you make EMI payments, and when it reaches zero, your loan is closed.

Given that  Lea earned $800 in simple interest on her investment after 11 years.

The formula of simple interest is

I = Prt.

P = Principal, r = rate of interest, t = time.

The rate of interest is r = 6% = 0.06, interest I = 800, time = 1 years.

Now putting the value of I, r,t in I = Prt:

800 = P×0.06×11

800= 0.66 P

Divide both sides by 0.66:

P = 800/0.66

P = 1212.1212

P = 1212.12(approx)

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We get to the conclusion that the response is 10x after examining the two sides.

The above equation's roots are 5 + 3i and 5 - 3i.

This indicates that x - (5+3i) and x - are the two factors of the provided polynomial (5 - 3i). The original statement must be the sum of the elements. We can thus write:

x² - (answer) + 34 = (x - 5 -3i)(x - 5 + 3i).........(1)

Simplifying the right hand side:

[tex]& (x-5-3 i)(x-5+3 i) \\[/tex]

[tex]& =x^2-5 x+3 x i-5 x+25-15 i-3 x i+15 i-9 i^2 \\[/tex]

[tex]& =x^2-10 x+25-9 i^2 \\[/tex]

[tex]& =x^2-10 x+25-9(-1) \\[/tex]

[tex]& =x^2-10 x+25+9 \\[/tex]

[tex]& =x^2-10 x+34[/tex]

Thus, we can write the Equation 1 as:

[tex]x^2-\left(\text { answer) }+34=x^2-10 x+34\right.[/tex]

Our analysis of the two sides leads us to the conclusion that the response is 10x.

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Answer: 1 7/10

Step-by-step explanation:

Answer: 4.5 pounds of flour equals to 72 ounces of flour. There are 14 days in 2 weeks, so 14 times 72 equals 1008 ounces of flour. If the baker uses all their 72 ounces of flour in one day to make loaves of bread, they will make 6 loaves of bread. That is also 84 loaves of bread if they make loaves of bread for 2 weeks straight.

Step-by-step explanation:

Answer:

Step-by-step explanation: Well, one lbs (pound) is equal to 16 oz (ounces)

So take the 4.5 lbs and multiply it by the 16 oz. That'll give you 224 oz

Then take the 224 oz and multiply that by 14, which is the number of days from two weeks. 224 multiplied by 14 is 3,136

To find the loaves of bread, take the 3,136 and divide it by 12, which gives you 261.3, or 261 1/3

Protons are a type of subatomic particle with a positive charge. With more electrons than protons, the particle is negatively charged.

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The solution of dy/dt = (t*y^-.5)/3 with y(0)=1 is y(1) = 1.

The solution to the given differential equation is given by the following formula:

y(t) = [tex](3/t^2)^2/3 + C[/tex]

where C is the constant of integration.

To find the value of y(1), we need to substitute t = 1 in the above formula.

y(1) =[tex](3/1^2)^2/3 + C[/tex]

y(1) = 9 + C

Since y(0) = 1, we can find the value of C by substituting t = 0 in the above formula.

y(0) = ([tex]3/0^2)^2/3 + C[/tex]

y(0) = 9 + C

1 = 9 + C

C = -8

Substituting C = -8 in the formula for y(1),

y(1) = 9 + (-8) = 1

Therefore, y(1) = 1.

Thus, the solution of [tex]dy/dt = (t*y^-.5)/3 , y(0)=1[/tex] is y(1) = 1.

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If the expression is the square of a binomial, then it must have the form [tex]$(ax + b)^2 = a^2x^2 + 2abx + b^2$[/tex].

Comparing the coefficients of like terms, we have:

[tex]a^2 = x^2$, so $a = x[/tex]

[tex]2ab = 18x$, so $b = 9[/tex]

Therefore, the missing number is [tex]$\boxed{9}$[/tex].

The missing number was found by using the fact that the expression must have the form [tex]$(ax + b)^2 = a^2x^2 + 2abx + b^2$[/tex] if it is the square of a binomial.

By comparing the coefficients of like terms in this equation and in the expression given, [tex]$x^2 18x \boxed{\phantom{00}}$[/tex], we were able to solve for the values of [tex]$a$[/tex] and [tex]$b$[/tex] as follows:

[tex]$a^2 = x^2 \Rightarrow a = x$[/tex]

[tex]$2ab = 18x \Rightarrow b = 9$[/tex]

Finally, we see that the coefficient of the [tex]$x$[/tex] term in the expression is [tex]$2ab = 2 \cdot x \cdot 9 = 18x$[/tex], which matches the given expression. Hence, the missing number is [tex]$\boxed{9}$[/tex].

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The height of the cone is - 3.4 centimeters.

Given data :

The total surface area of the cone is calculated by the formula -

Total surface area = πrl² + πr²

Keep the values in the formula to find the value of length or hypotenuse.

90π = πr(l² + r)

90 = 5(l² + 5)

18 = l² + 5

l² = 13

Now, find the height of the cone.

length² = perpendicular² + base²

13 = perpendicular² + 5²

Perpendicular² = 25 - 13

Perpendicular² = 12

Perpendicular = 3.4 centimeters.

Thus, the height of the cone is 3.4 centimeters

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The probability that the system works under these conditions is 0.81 (0.90 x 0.90 = 0.81).

This question uses the concept of independent probabilities, Independent probabilities refer to the likelihood of two or more events occurring independently of each other.

In other words, the probability of one event occurring does not influence the probability of another event occurring.

This means that if the probability of one event occurring is known, the probability of the other event occurring can be calculated independently.

given below

Step 1: Determine the probability that both components A and B will work.

Step 2: Multiply the probability that component A will work by the probability that component B will work.

Step 3: The result of Step 2 is the probability that the system will work under these conditions.

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Using Priya's method to calculate (0.0015) x (0.024) gives 0.0000360

To multiply decimals using Priya's method, first, multiply as if there is no decimal. Next, count the number of digits after the decimal in each factor. Finally, put the same number of digits behind the decimal in the product.

Now, 15 × 24 = 360

0.0015 has 4 numbers after the decimal and  0.024 has 3 numbers after the decimal. The total is 7.

Moving the decimal point 7 spaces to the left, we get 0.0000360.

Thus,  (0.0015) x (0.024) = 0.0000360

Note: 0.0000360 = 0.000036

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"Owning a home provides a partial tax shelter for housing-related expenses" the following is an advantage of purchasing and owning a home.

Hence, option (B) is correct choice.

The Benefits of Owning a Home:

Interest rates are low.

Creating equity: The difference between what you can sell the house for and what you owe is your equity. As you pay down your mortgage, your equity rises. Over time, more of your monthly payments will be applied to the loan balance rather than the interest, resulting in more equity.

Federal tax breaks: Mortgage interest is deductible on the first $750,000 of the house's purchase price, as are home equity loan interest, property taxes up to $10,000 if married ($5,000 if married filing separately), and various closing fees at the time of purchase.

Greater privacy: Because you own the property, you may renovate it to your desire, something tenants do not have.

Home office: The work-at-home craze may persist after the epidemic has passed, implying that more of us will require a home office. The appropriate configuration affects both comfort and productivity.

Monthly payments that are predictable: A fixed-rate mortgage requires you to pay the same amount each month for principle and interest until the loan is paid off. Rents are subject to rise with each annual lease renewal. Changes in property taxes or homeowner's insurance can affect monthly payments, although not as often as rent hikes.

Stability: People tend to stay in a house they own for a longer period of time, if only because purchasing, selling, and relocating is difficult.

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Answer:

B. Owning a home provides a partial tax shelter for housing related expenses

Hope this helps! Have an amazing day/night/afternoon!

KL and MN are chords of circle Q. They intersect at point J. the value of x is 5.9.

A circle's chord is the line segment that connects any two locations on the circle's circumference. It should be noted that a circle's diameter is defined as the lengthiest chord that runs through the centre of the circle.

KL and MN are chords of circle Q. They intersect at point J .MKN = (10x + 2 )° , mLM = (3x–4)°, and m∠KJM=103°.

(10x + 2 )° +  (3x–4)° +103° = 180 degree

13x + 2 + 103 = 180 degree

13x = 180 - 103 = 77

x = 77 / 13 = 5.92 .

KL and MN are chords of circle Q. They intersect at point J. the value of x is 5.9.

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Under the unbiased expectations theory of the term structure of interest rates, the expected one-year interest rate one year from now, E(2r1), is 10%.

The unbiased expectations theory states that the expected future rate (E(2r1)) is equal to the current rate (1R2). Therefore, the expected one-year interest rate one year from now, E(2r1), is 10%.

The unbiased expectations theory is a principle in finance that states that the expected future rate of return (E(2r1)) is equal to the current rate of return (r2). This theory suggests that market participants have rational expectations and their expectations are reflected in the current market prices. In other words, the theory posits that the current market prices incorporate all available information and reflect the best prediction of future prices.

In the example given, if the expected one-year interest rate one year from now is 10%, it means that market participants believe that the one-year interest rate in the future will be 10%. This expectation is reflected in the current market prices and is incorporated into the current rate of return.

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The solution is Option B.

The system of inequalities is x² - 2y ≥ 7000 and 2x + 5y < 1000

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

Given data ,

Let the first inequality equation be represented as A

Let the second inequality equation be represented as B

Now , value of a company’s stock is represented by the expression x² - 2y

And , company’s goal is to maintain a stock value of at least $ 7,000

So , the inequality is x² - 2y ≥ 7000

And , the company’s purchases are modeled by 2x + 5y

Also , the company's purchases should be below $ 1,000

So , the inequality is 2x + 5y < 1000

Hence , the inequalities are solved

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The expression that is equivalent to x^(2/5) is

Information given in the problem

the expression x^(2/5)

This equation represents exponents in fraction form and the basics for the calculation is from

a^(b/c) =

[tex]\sqrt[c]{a^{b} }[/tex]

Rewriting the expression is done as below

x^(2/5)

= [tex]\sqrt[5]{x^{2} }[/tex]

comparing with [tex]\sqrt[c]{a^{b} }[/tex]

gives the following values

a will be equal to x

b will be equal to 2

c will be equal to 5

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The perimeter of the entire shape that is made up of a semi circle and an equilateral triangle would be = 116.66in

The perimeter of a shape is defined as the total area of all the sides surrounding that shape.

The perimeter of the object can be calculated by the addition of the perimeter of the triangle and the circle.

The perimeter of the triangle = a+b+c = 19+19+19 = 57in

The perimeter of the circle = 2πr

where π = 3.14

r = 19/2 = 9.5 in

perimeter = 2×3.14×9.5 = 59.66in

The perimeter of the whole object = 57+59.66 = 116.66in.

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Answer: 1/6 of a pound

Step-by-step explanation: Hello!

You know that Miss Collier bought a block of cheddar cheese that weighs 2/3 of a pound and she split it into four equal pieces.  This can be represented by the equation below:

2/3 divided by 4 = weight of a single slice of cheese

2/3 divided by 4/1 = weight of a single slice of cheese

Take the reciprocal of 4/1.  4/1 turns into 1/4.  Then multiply the two fractions.

2/3 x 1/4 = weight of a single slice of cheese.  

Multiply the numerators and denominators.  2 x 1 = 2, 3 x 4 = 12.

2/12 = weight of a single slice of cheese.

You can simplify this by dividing the numerator and denominator by 2.  2/2 = 1, 12/2 = 6.

The final answer is: 1/6 of a pound.