How will the product change if one number is increased by a factor of 12 and the other is decreased by a factor of 4

The sum of the nth roots of unity is zero.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers are used in mathematics and science to represent quantities that have both a real and imaginary part.

The nth roots of unity are given by:

[tex]\omega ^0, \omega^1, \omega^2, ..., \omega^(n-1)[/tex]

where [tex]\omega = e^{(2\pi i/n)[/tex]is a complex number and i is the imaginary unit.

The sum of these roots is:

[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^{(n-1)[/tex]

To simplify this expression, we can use the formula for the sum of a geometric series:

[tex]a + ar + ar^2 + ... + ar^(n-1) = a(1 - r^n)/(1 - r)[/tex]

Let a = 1 and [tex]r = ω.[/tex] Then the sum of the nth roots of unity is:

[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - \omega^n)/(1 - \omega)[/tex]

Substituting [tex]ω = e^(2πi/n)[/tex], we get:

[tex]\omega^n = (e^(2\pi i/n))^n = e^2 \pi i = 1[/tex]

Therefore, the sum of the nth roots of unity is:[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - 1)/(1 - \omega) = 0/(1 - \omega) = 0[/tex]

Hence, the sum of the nth roots of unity is zero.

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The desalination plant can produce [tex]9.84*10^6[/tex] gallons of water in four days in scientific notation.

We must divide the daily production rate by the number of days in order to get the total volume of water that a desalination plant can generate in four days:

4. days at a rate of [tex]2.46*10^6[/tex] gallons equals [tex]9.84*10^6[/tex] gallons.

Hence, in four days, the desalination plant can produce [tex]9.84 * 10^6[/tex]gallons of water.

Large or small numbers can be conveniently represented using scientific notation, especially when working with measurements in science and engineering. A number is written in scientific notation as a coefficient multiplied by 10 and raised to a power of some exponent. For example, [tex]2.46*10^6[/tex] denotes 2,460,000, which is 2.46 multiplied by 10 to the power of 6.

The solution in this case is [tex]9.84*10^6[/tex], or 9,840,000, which is 9.84 multiplied by 10 to the power of 6. Large numbers can be written and compared more easily using this format, and scientific notation rules can be used to conduct computations with them.

In conclusion, the desalination plant has a four-day capacity of [tex]9.84 * 10^6[/tex] gallons of water production.

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Answer:

Step-by-step explanation:

Solution:

   [tex](x-2)^2+(y+3)^2=25[/tex]

Equation of general circle:

    [tex](x-a)^2+(y-b)^2=r^2[/tex]      where (a,b) is circle center and r is the radius.

Answer:

the missing values are 2, 8, and 3.

Answer:

Without knowing the exact length of Ted's pencil, we cannot give an exact decimal representation of its length. However, we do know that Ted's pencil is longer than Jan's pencil, which is 8.5 cm long.

If we assume that Ted's pencil is one centimeter longer than Jan's pencil, then its length would be 9.5 cm. In decimal form, this would be written as 9.5.

If we assume that Ted's pencil is two centimeters longer than Jan's pencil, then its length would be 10.5 cm. In decimal form, this would be written as 10.5.

So, the decimal that could represent the length of Ted's pencil depends on how much longer it is than Jan's pencil.

(A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

What is the rate of change?

The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.

A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.

Using the formula for an annual rate of change (r):

final value = initial value * [tex](1 - r)^t[/tex]

where t is the number of years and r is the annual rate of change expressed as a decimal.

Substituting the given values, we get:

$15,000 = $44,000 * (1 - r)¹⁴

Solving for r, we get:

r = 0.0804

So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.

B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:

r = 0.0804 * 100% = 8.04%

C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.

Substituting the known values, we get:

value in 2009 = $15,000 * (1 - 0.0804)³

value in 2009 = $11,628.40

Rounding to the nearest $50, we get:

value in 2009 = $11,650

Hence, (A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

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Answer:

[tex] \frac{13y ^{2} - 59y - 90 }{(y + 10)(y - 10)} [/tex]

Step-by-step explanation:

6.

the figure can be considered as a combination of a 16×8 rectangle and a 10×8 right-angled triangle on the left side.

for both, perimeter and area, we need to find the length of the missing third side of the right-angled triangle, as this is a part of the long baseline of the overall figure.

as it is a right-angled triangle, we can use Pythagoras :

c² = a² + b²

"c" being the Hypotenuse (the side opposite of the 90° angle, in our case 10 ft). "a" and "b" being the legs (in our case 8 ft and unknown).

10² = 8² + (leg2)²

100 = 64 + (leg2)²

36 = (leg2)²

leg2 = 6 ft

that means the bottom baseline is 16 + 6 = 22 ft long.

a.

Perimeter = 10 + 16 + 8 + 22 = 56 ft

b.

Area is the sum of the area of the rectangle and the area of the triangle.

area rectangle = 16×8 = 128 ft²

area triangle (in a right-angled triangle the legs can be considered baseline and height) = 8×6/2 = 24 ft²

total Area = 128 + 24 = 152 ft²

7.

it was important that the bottom left angle of the quadrilateral was indicated as right angle (90°). otherwise this would not be solvable.

but so we know, it is actually a rectangle.

that means all corner angles are 90°.

therefore, the angle AMT = 90 - 20 = 70°.

the diagonals split these 90° angles into 2 parts that are equal in both corners of the diagonal, they are just up-down mirrored.

a.

so, the angle HTM = AMT = 70°.

b.

MEA is an isoceles triangle.

so, the angles AME and EAM are equal.

the angle AME = AMT = EAM = 70°.

the sum of all angles in a triangle is always 180°.

therefore,

the angle MEA = 180 - 70 - 70 = 40°

c.

both diagonals are equally long, and they intersect each other at their corresponding midpoints.

so, when AE = 15 cm, then AH = 2×15 = 30 cm.

TM = AH = 30 cm.

so we're assuming there are 360 days in a year, so 83 days is really just 83/360 of a year, so

[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$586.21\\ r=rate\to 6.3\%\to \frac{6.3}{100}\dotfill &0.063\\ t=years\dotfill &\frac{83}{360} \end{cases} \\\\\\ I = (586.21)(0.063)(\frac{83}{360}) \implies I \approx 8.51[/tex]

Let L be the original length of the rectangle and W be the original width of the rectangle. We know that:

(L - 6) = (W + 3) (1) (since the length is decreased by 6cm and the width is increased by 3cm, the result is a square)

The area of the rectangle is LW, and the area of the square is (L - 6)(W + 3). We also know that the area of the square is 27cm^2 smaller than the area of the rectangle. So we can write:

(L - 6)(W + 3) = LW - 27 (2)

Expanding the left side of equation (2), we get:

LW - 6W + 3L - 18 = LW - 27

Simplifying and rearranging, we get:

3L - 6W = 9

Dividing both sides by 3, we get:

L - 2W = 3 (3)

Now we have two equations with two unknowns, equations (1) and (3). We can solve this system of equations by substitution. Rearranging equation (1), we get:

L = W + 9

Substituting this into equation (3), we get:

(W + 9) - 2W = 3

Simplifying, we get:

W = 6

Substituting this value of W into equation (1), we get:

L - 6 = 9

So:

L = 15

Therefore, the area of the rectangle is:

A = LW = 15 x 6 = 90 cm^2.

Answer:

252

Step-by-step explanation:

lol I take rsm too and I just guessed and checked

It can be expressed in interval notation as:[0, 12]

The practical domain for the function g(x) is [0,12], as this is the time during which the football is in the air, from the time it is thrown until it hits the ground.

The practical domain for the function g(x) would be the time interval during which the football is in the air, since it only makes sense to talk about the height of the football while it is airborne.

From the problem statement, we know that the football is thrown in the air at time x = 0, reaches its maximum height of 28 feet at time x = 6, and hits the ground at time x = 12. Therefore, the practical domain for the function g(x) is:
0 <= x <= 12
This means that the function g(x) is defined and meaningful for any value of x between 0 and 12, inclusive. Beyond this domain, the function does not have a practical interpretation because the football is either not yet thrown or has already hit the ground.

The function g(x) is the height of a football x seconds after it is thrown in the air. The football reaches its maximum height of 30 feet in 4 seconds, and hits the ground at 10 seconds.
What is the practical domain for the function f(x)
Write your answer in interval notation.

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The solution to the problem is that the number "x" must be greater than or equal to -12.

Let's define the variable as "x," representing the unknown number. According to the problem statement, one fourth of the number is no less than -3.

Mathematically, we can express this statement as:

x/4 ≥ -3

To solve for "x," we can start by multiplying both sides of the inequality by 4 to eliminate the fraction:

x ≥ -3*4

x ≥ -12

Therefore, the solution to the problem is that the number "x" must be greater than or equal to -12.

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Answer:

30

Step-by-step explanation:

its right

order of operations

Answer:

The answer is 30...

Step-by-step explanation:

Apply the rule BODMAS...

3(40)/(4)

120/4

30

Answer:

Step-by-step explanation:

500/0.4 = 1250

The rate that yields maximum revenue is $9 per month, and the maximum revenue is $1,296,000.

To find the rate that yields maximum revenue, we need to find the price that maximizes the revenue. Let x be the number of $0.50 decreases in the monthly fee, and let y be the number of subscribers who sign up at the new rate. Then we have the following equations:

y = 4800 + 120x (number of subscribers)

p = 24 - 0.5x (price per month)

r = xy(p) = (4800 + 120x)(24 - 0.5x) (revenue)

To find the rate that yields maximum revenue, we need to take the derivative of the revenue function concerning x, set it equal to zero, and solve for x:

r' = 120(24 - x) - (4800 + 120x)(0.5) = 0

x = 60

Therefore, the optimal number of $0.50 decreases is 60, and the corresponding price per month is $24 - 0.5(60) = $9. The number of subscribers at this rate is 4800 + 120(60) = 12000.

Finally, the maximum revenue is given by r = xy(p) = (4800 + 120(60))(24 - 0.5(60)) = $1,296,000.

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The quadratic function that fits the given points is: f(x) = (4/3)x² - (10/3)x + 1

By using the simplification formula

f(x) = ax² + bx + c

where a, b, and c are constants

a(-1)² + b(-1) + c = 1/3

a(0)² + b(0) + c = 1

a(1)² + b(1) + c = 3

a(2)² + b(2) + c = 9

Simplifying each

a - b + c = 1/3

c = 1

a + b + c = 3

4a + 2b + c = 9

We can solve this using any method like substitution, elimination

a - b + c = 1/3

a + b + c = 3

2a + 2c = 9/3

Adding the first two equations 2a + 2c = 10/3

Subtracting the third equation

b = 5a/3 - 2

a - (5a/3 - 2) + 1 = 1/3

a = 4/3

Finally, we can substitute a = 4/3 and b = 5a/3 - 2, and c = 1 into the standard form of the quadratic equation to get: f(x) = (4/3)x² - (10/3)x + 1.

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Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).

A polyhedron with two triangular sides and three rectangles sides is referred to as a triangular prism. It is a three-dimensional shape with two base faces, three side faces, and connections between them at the edges.

Given :

We must calculate the area of each face of the triangular prism and put them together to determine its surface area.

The areas of the triangular faces are equal, so we may calculate one of their areas and multiply it by two:

One triangular face's area is equal to (1/2) the sum of its base and height, or (1/2) 6 x 8 x 6, or 24 square units.

Both triangular faces' surface area is 2 x 24 or 48 square units.

Finding the area of the rectangular faces is now necessary:

One rectangular face's area is given by length x breadth (10 x 6) = 60 square units.

120 square units are the area of both rectangular faces or 2 by 60.

Hence, the triangular prism's total surface area is:

Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).

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a) The total volume equals the sum of the volumes.

[tex]500 = x + y[/tex]

The total octane amount equals the sum of the octane amounts.

[tex]89(500) = 87x + 92y[/tex]

[tex]44500 = 87x + 92y[/tex]

b)

As x increases, y decreases.

c) Use substitution or elimination to solve the system of equations.

[tex]44500 = 87x + 92(500-x)[/tex]

[tex]44500 = 87x + 46000 - 92x[/tex]

[tex]5x = 1500[/tex]

[tex]x = 300[/tex]

[tex]y = 200[/tex]

The required volumes are 300 gallons of 87 gasoline and 200 gallons of 92 gasoline.

The formula used to calculate the surface area of a triangular prism is:  

S = bh + 2ah  

Where S = surface area, b = length or side of the triangle, and h = height of the triangle.  

For right triangles, the height can be calculated as:  

c = a^(2) \+ b^(2)  

c = the hypotenuse  

a and b = the two sides of the triangle

Answer:

the answer is A) -2x² + 4x + 30

Step-by-step explanation:

To find the equation of the quadratic function f(x), we can use the standard form of a quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants.

We can plug in the values of x and f(x) from the table to get three equations:

a(-1)^2 + b(-1) + c = 24

a(0)^2 + b(0) + c = 30

a(1)^2 + b(1) + c = 32

Simplifying each equation, we get:

a - b + c = 24

c = 30

a + b + c = 32

We can substitute c = 30 into the first and third equations to get:

a - b + 30 = 24

a + b + 30 = 32

Simplifying these equations, we get:

a - b = -6

a + b = 2

Adding these two equations, we get:

2a = -4

Dividing by 2, we get:

a = -2

Substituting a = -2 into one of the equations above, we get:

-2 - b = -6

Solving for b, we get:

b = 4

Therefore, the equation that represents f(x) is:

f(x) = -2x^2 + 4x + 30

So the answer is A) -2x² + 4x + 30

If the roots of a quadratic equation are given, we can write the equation in factored form as [tex](x - r1)(x - r2) = 0[/tex] , where r1 and r2 are the roots.the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

A quadratic equation can be written in the form:

[tex]ax^2 + bx + c = 0[/tex]

where a, b, and c are constants. Since the roots of the equation are 5 and 2, we can write:

[tex](x - 5)(x - 2) = 0[/tex]

Expanding this equation gives:

[tex]x^2 - 7x + 10 = 0[/tex]

To make the leading coefficient of this equation 4, we can multiply both sides by 4/1, which gives:

[tex]4x^2 - 28x + 40 = 0[/tex]

Therefore, the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

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Answer:

area=28cm^2

Step-by-step explanation:

4×5=20

20÷2=10 area of the bigger triangle

4×2=8

8÷2=4 area of the smaller triangle

10+10=20 for area of the the 2 bigger triangles

4+4=8 for the area of the 2 smaller triangles

area of kite= 20+8= 28

Answer:

A: [tex]2x^3+5x^2-8x[/tex]

Step-by-step explanation:

In order to get the answer to this, you have to combine like terms to simplify the answer.

Go through and organize the x's by the size of the exponents. (this step isn't necessary but it can help you visualize it if you are having trouble with that)

[tex]x^3+x^3+2x^2+3x^2-7x-x[/tex]

When the variables are raised to the same degree, the coefficients can be added together.

[tex]2x^3+5x^2-8x[/tex]

Answer:

581,58 in^3

Step-by-step explanation:

Given:

Cylinder shaped cups

d (diameter) = 4,2 in

r (radius) = 0,5 × 4,2 = 2,1 in

h (height) = 7 in

π = 3.14

.

First, let's find how much juice can he pour into 1 cup:

.

We need to find the base of the cylinder:

.

[tex]a(base) = \pi {r}^{2} = 3.14 \times( {2.1})^{2} = 13.8474[/tex]

.

Now, we can find the volume of one cup:

V = a (base) × h

[tex]v = 13.8474 \times 7 ≈96.93[/tex]

Multiply this number by 6 and we'll get the answer (since there's 6 cups):

96,93 × 6 = 581,58

The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

Probability refers to the likelihood that an occurrence will occur.

In actual life, we frequently have to make predictions about the future. We might or might not be conscious of how an event will turn out.

When this occurs, we proclaim that there is a possibility that the event will occur. In conclusion, probability has a broad range of amazing uses in both business and this rapidly developing area of artificial intelligence.

Simply dividing the favorable number of possibilities by the total number of possible outcomes using the probability formula will yield the chance of an event.

According to our question-

Total number of books that arrived that day = 23 + 41 = 64.

Let E be the event of arriving at a non-fictional book.

The event of arriving at a non-fictional book is n(E) = 23.

Total sample space, n(S) = 64.

The probability of a non-fictional book to arrive is = n(E) / n(S)

= 23 / 64 = 0.359375

= 0.359375 × 100

= 35.9375 ≈ 36.

Hence, The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

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The side of the given square is 36W^2 + 12W +1 of is 6W +1 and 81W^2 -72W + 16 is (9W-4).

The area is calculated by multiplying the length of a shape by its width.

and the unit of the square is a square unit.

Given Area of the square :

1) [tex]36W^{2}+12W+1[/tex]

Area of square = [tex]36W^{2}+12W+1[/tex]

[tex]side^{2}[/tex] = [tex]36W^{2} + 6W +6W +1\\6W (6W +1) +1 (6W +1)\\(6W +1)(6W+1)\\(6W +1 )^{2} \\side^{2} = (6W + 1)^{2} \\side = 6W + 1[/tex]

[tex]Area of the square = 81W^{2} - 72W + 16\\(side)^{2} = (9W-4)(9W-4)\\(side)^{2} = (9W-4)^{2} \\side = (9W-4)[/tex]

Therefore the side of the square is 6W+1 and 9x-4.

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The total surface area of the given hemispherical scoop is: 30.41 cm²

The formula for the total surface area of a hemisphere is:

TSA = 3πr² square units

Where:

π is a constant whose value is equal to 3.14 approximately.

r is the radius of the hemisphere.

Since the steel is 0.2cm thick and the outside of the scoop has a radius of 2cm, then we can say that:

TSA = 3π(2 + 0.2)²

= 3π(2.2)²

= 30.41 cm²

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Answer: Let's start by using the identity:

tan(arctan(x)) = x

to simplify the expression inside the sine function. So, we have:

arctan(u) / sqrt(3) = tan(arctan(u) / sqrt(3))

Now, using the trigonometric identity:

tan(x/y) = sin(x) / (cos(y) + sin(y))

with x = arctan(u) and y = sqrt(3), we get:

tan(arctan(u) / sqrt(3)) = sin(arctan(u)) / (cos(sqrt(3)) + sin(sqrt(3)))

Simplifying further, we know that:

sin(arctan(u)) = u / sqrt(1 + u^2)

and

cos(sqrt(3)) + sin(sqrt(3)) = 2cos(sqrt(3) - pi/4)

So, the expression becomes:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2cos(sqrt(3) - pi/4)]

Simplifying the denominator, we have:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / (2(cos(sqrt(3))cos(pi/4) + sin(sqrt(3))sin(pi/4)))

Using the values for cosine and sine of pi/4, we get:

cos(pi/4) = sin(pi/4) = 1/sqrt(2)

So, we have:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2(sqrt(3)/2 + 1/2)]

Simplifying further:

sin(arctan(u) / sqrt(3)) = u / (sqrt(1 + u^2) * (sqrt(3) + 1))

Therefore, the algebraic expression for sin(arctan(u) / sqrt(3)) is:

u / (sqrt(1 + u^2) * (sqrt(3) + 1))

Step-by-step explanation: