In a circle, the length of an an arc intercepted by a central angle is 12mm, and the radius of the circle is a 8 mm. What is the measure, in radians, of the angle?​

For the given mathematical statements:

For L:  n represents jerry's run time.

For M: n represents  the age of his father.

For N: n represents amount her sister has.

For O: n represents age of his younger brother.

To write the variables n represents each expression,

Take the following steps:

→ Identify the variable in each expression.

→ Determine what the variable represents by looking at the context of the problem and the information provided in the expression.

→ Write down the meaning of the variable for each expression.

Therefore,

L. Lisa can run the 100-yard dash 5 seconds faster than Jerry. Lisa's time is n-5 seconds.

Here,

n represents jerry's run time.

M:  Pedro is 3 times younger than his father. Pedro is n + 3 years old.

Here,

n represents  the age of his father,

N: Lucia has $10.00 less than her sister. Lucia has n-10 dollars.

Here,

n represents amount her sister has.

O: Jack is 2 times taller than his younger brother. Jack is 2 x n inches tall.

Here,

n represents age of his younger brother.

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From the given data mean is 11.86, median is 10.5 and inter quartile range is 5.

The given data is 6, 7, 7, 8, 8, 10, 10, 11, 11, 12, 12, 12, 20, 20.

2) Histogram (Plotted below)

3) Mean = (6+7+7+8+8+10+10+11+11+12+12+12+12+20+20)/14

= 166/14

=  11.86

4) Box plot (Plotted below)

5) Median = (10+11)/2

= 21/2

= 10.5

6) Q1 = 7

7) Q3 = 12

8) IQR=Q3-Q1

= 12-7

= 5

Therefore, from the given data mean is 11.86, median is 10.5 and inter quartile range is 5.

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The length of the missing hypotenuse is approximately 8.06 feet

Let's create a real-world scenario involving a right triangle. Imagine you are a construction worker tasked with building a roof for a house. The roof needs to have a triangular shape and one of the corners needs to form a right angle.

The two sides that meet at the right angle measure 4 feet and 7 feet respectively. Now, you need to figure out the length of the third side (the hypotenuse) so that you can properly cut the roofing material. To solve for the missing hypotenuse, we can use the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the lengths of the legs (in this case, 4 feet and 7 feet) and c is the length of the hypotenuse (which we are solving for).

Plugging in the values:

4^2 + 7^2 = c^2

16 + 49 = c^2

65 = c^2

Taking the square root of both sides:

c = √65 feet (approx. 8.06 feet)

. This will allow you to properly cut the roofing material to fit the triangular shape of the roof.

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Answer:

A, B, and E. Hope this helps!

Answer:

35/36

Step-by-step explanation:

5/9 + 5/12 =

We need to get a common denominator of 9 and 12, which is 36.

5/9 * 4/4 = 20/36

5/12 * 3/3 = 15/36

20/36 + 15/36

35/36

The lengths of the sides of the triangle are:

AC = 6.2 m

AB = 11.88 m

BC = 3.2 m

Let x be the length of each of the legs of the isosceles triangle.

And let y be the length of the base of the triangle.

From the given statement, we have,

The perimeter of the triangle is 15.6 m,

therefore,

⇒ x + x + y = 15.6

Simplifying:

⇒ 2x + y = 15.6

The base of the triangle is 3 m shorter than a leg,

so,

⇒ y = x - 3

Now we can substitute this expression for y into the previous equation:

⇒ 2x + (x - 3) = 15.6

Simplifying:

⇒ 3x - 3 = 15.6

Adding 3 to both sides:

⇒ 3x = 18.6

Dividing by 3:

⇒ x = 6.2

So each leg of the triangle has length 6.2 m.

Now use the expression for y that we found earlier:

⇒ y = x - 3

      = 6.2 - 3

      = 3.2

So the base of the triangle has length 3.2 m.

Use the Pythagorean theorem to find the lengths of AC and AB:

AC is one of the legs of the triangle, so its length is 6.2 m.

To find AB, we can draw the altitude from A to BC, splitting the triangle into two right triangles. Let h be the length of this altitude.

Then we have:

⇒ h + (y/2) = x

Substituting the values we found earlier:

⇒ h + 1.6 = 6.2

Simplifying:

⇒ h = 6.2 - 1.6

    h = 5.94

So the length of AB is 2h, or:

AB = 2h

     = 2(5.94)

     = 11.88

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The angle of intersecting arc theorem indicates that the angle marked ? in the drawing is 190°

The angle of intersecting chords theorem states that the angle formed by two chords when they intersect in a circle is equivalent to half the sum of the arcs they intercept on the circumference of the circle.

The angle of intersecting chords theorem states that we get;

The angle QPR formed = (1/2) × ((Arc QR) + (Arc NS))

Therefore, we get;

m∠QPR = 125° = (1/2) × ((Arc QR) + 60°)

2 × 125° = ((Arc QR) + 60°)

250° = (Arc QR) + 60°

(Arc QR) = 250° - 60° = 190°

The measure of the arc QR = 190°

m[tex]\widehat{QR}[/tex] = 190°

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k, a, f, and c are the exterior angles of the given figure.

In a figure, an exterior angle refers to an angle formed by one side of the trapezium and the extension of an adjacent side.

In other words, an exterior angle is formed when you extend one of the sides of the trapezium and look at the angle it creates with the adjacent side.

The angle b degrees, g , d and 70 degrees are interior angles.

k, a, f, and c are the exterior angles of the given figure.

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Unemployment can be defined as members of the labour force that do not have a job but are actively seeking employment.

Structural unemployment is an unemployment that occurs as a result of changes in the economy. These changes can be as a result of changes in technology, polices or competition. Structural unemployment tends to be permanent.  Ned is structurally unemployed due to changes in his industry

The period of time a person is unemployed from the period he leaves his current job and the time he gets another job. Eg. The period from when graduates till when he gets a job, he is frictionally unemployed

Cyclical unemployment: it occurs as a result of fluctuations in the economy. Unemployment would be high in a downturn and low in a boom .Jane is not unemployed because she is not seeking employment.

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There are countless possibilities for shapes that are similar but not congruent to shape 1, as long as they maintain the same proportions and shape.

There are many shapes that are similar but not congruent to shape 1. Similar shapes have the same shape, but their sizes may be different. In contrast, congruent shapes have the same shape and size.

One example of a similar shape to shape 1 is a rectangle that is twice as long and half as wide. Another example could be a square that is three times larger in area than shape 1.

A parallelogram that has the same base as shape 1 but is three times as tall is also similar but not congruent.

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Answer:

6

Step-by-step explanation:

Put the numbers from least to greatest. The cross out the numbers starting from the left to the right.

Answer:

7

Step-by-step explanation:

domain-only real numbers

range-only one number 7

When there is not enough evidence to conclude a significant linear relationship between variables, it is not appropriate to use a regression equation to make predictions. In such cases, it is more reasonable to rely on the mean of the y values to make predictions.

The mean of the y values provides a simple and neutral estimate of the average response variable value, irrespective of any relationship with the predictor variable. This approach assumes that, on average, future observations will be similar to the past observations in terms of the response variable. While this may not account for potential variations or patterns, it represents a conservative estimate that is not influenced by an unsupported assumption of a linear relationship.

It is important to note that using the mean as a prediction in this context assumes that there are no other relevant predictors or factors influencing the response variable. If there are additional variables that might affect the relationship, you may need to consider alternative approaches, such as exploring other models or using additional information to make predictions.

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Answer:

9 × √3 cm or approximately 15.59 cm

Step-by-step explanation:

(look at the picture)

Answer:

Step-by-step explanation:

a>b

1≤b,a≤25

smallest value of b=1

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

That's 6th grade math but the answer is -5 3/5, this is because subtracting a positive number from a negative adding them together and adding a negative sign.

The value of y in this linear system as per the given data, the value of y in the linear system is 18.

To find the value of y inside the linear device, we are able to use the given equations:

Equation 1: x + y + z = 62

Equation 2: x - y = 12

Equation three: 2x + y + z = ninety two

To remedy this gadget, we can dispose of the variable x via subtracting Equation 2 from Equation 1:

(x + y + z) - (x - y) = 62 - 12

Simplifying, we get:

2y + z = 50

Next, we can subtract Equation 2 from Equation 3 to eliminate the variable x once more:

(2x + y + z) - (x - y) = 92 - 12

Simplifying, we get:

x + 2y + z = 80

(x + 2y + z) - (2y + z) = 80 - 50

Simplifying, we get:

x = 30

30 - y = 12

Solving for y, we get:

y = 18

Thus, the correct option is B.

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The volume of the cuboctahedron is approximately [tex]32768 cm^3 - 512 * sqrt(3) cm^3.[/tex]

By using the clue provided, we may determine the volume of the original cube and then deduct the volume of the cut corners.

All of the edges of the original cube measure 32 cm. As a result, its volume can be determined as follows:

[tex]Volume of cube = (Edge length)^3 = (32 cm)^3 = 32^3 cm^3 = 32768 cm^3[/tex]

Let's now determine the volume of the corners that were eliminated. Each corner has a similar equilateral triangle with sides that are 16 cm long.

The volume of one equilateral triangle can be calculated using the formula:

Volume of equilateral triangle =[tex](sqrt(3) / 4) * (Side length)^2[/tex]

Plugging in the values, we have:

Volume of one corner [tex]= (sqrt(3) / 4) * (16 cm)^2[/tex]

Since there are 8 corners that are removed, we multiply the volume of one corner by 8 to get the total volume of the removed corners:

Total volume of corners = [tex]8 * [(sqrt(3) / 4) * (16 cm)^2][/tex]

Simplifying the expression:

Total volume of corners = [tex]8 * (sqrt(3) / 4) * (16 cm)^2[/tex]

=[tex]2 * sqrt(3) * (16 cm)^2[/tex]

Now, we can subtract the volume of the corners from the volume of the original cube to find the volume of the cuboctahedron:

Volume of cuboctahedron = Volume of cube - Total volume of corners

[tex]= 32768 cm^3 - 2 * sqrt(3) * (16 cm)^2[/tex]

Calculating the expression:

Volume of cuboctahedron ≈ [tex]32768 cm^3 - 2 * sqrt(3) * (16 cm)^2[/tex]

≈ [tex]32768 cm^3 - 2 * sqrt(3) * 256 cm^2[/tex]

≈ [tex]32768 cm^3 - 512 * sqrt(3) cm^3[/tex]

Therefore, the volume of the cuboctahedron is approximately[tex]32768 cm^3 - 512 * sqrt(3) cm^3.[/tex]

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Answer:

Therefore, the 4-point DFT of the signal [3, 2, -5, 4] is [4, 5 - 9i, 0, -1 + i].

Step-by-step explanation:

To find the 4-point Discrete Fourier Transform (DFT) of the given signal [3, 2, -5, 4], we can use the formula for DFT:

X(k) = Σ[n=0 to N-1] x(n) * exp(-j * 2π * k * n / N)

where X(k) is the DFT coefficient at frequency index k, x(n) is the input signal, N is the length of the signal, and j is the imaginary unit.

For the given signal [3, 2, -5, 4], with N = 4, the 4-point DFT coefficients can be calculated as follows:

For k = 0:

X(0) = (3 * exp(-j * 2π * 0 * 0 / 4)) + (2 * exp(-j * 2π * 0 * 1 / 4)) + (-5 * exp(-j * 2π * 0 * 2 / 4)) + (4 * exp(-j * 2π * 0 * 3 / 4))

= 3 + 2 - 5 + 4

= 4

For k = 1:

X(1) = (3 * exp(-j * 2π * 1 * 0 / 4)) + (2 * exp(-j * 2π * 1 * 1 / 4)) + (-5 * exp(-j * 2π * 1 * 2 / 4)) + (4 * exp(-j * 2π * 1 * 3 / 4))

= 3 + 2 - 5i - 4i

= 5 - 9i

For k = 2:

X(2) = (3 * exp(-j * 2π * 2 * 0 / 4)) + (2 * exp(-j * 2π * 2 * 1 / 4)) + (-5 * exp(-j * 2π * 2 * 2 / 4)) + (4 * exp(-j * 2π * 2 * 3 / 4))

= 3 - 2 - 5 + 4

= 0

For k = 3:

X(3) = (3 * exp(-j * 2π * 3 * 0 / 4)) + (2 * exp(-j * 2π * 3 * 1 / 4)) + (-5 * exp(-j * 2π * 3 * 2 / 4)) + (4 * exp(-j * 2π * 3 * 3 / 4))

= 3 - 2 + 5i - 4i

= -1 + i

Therefore, the 4-point DFT of the signal [3, 2, -5, 4] is [4, 5 - 9i, 0, -1 + i].

Answer:

[tex]\textsf{11.} \quad x^2 + y^2 = 2304[/tex]

[tex]\textsf{12.} \quad \textsf{Yes, the house is within the broadcast range of the radio station.}[/tex]

[tex]\textsf{13.} \quad \textsf{Yes, the plant is within the range of the sprinkler.}[/tex]

Step-by-step explanation:

Since the base of the Capitol Dome is circular, we can use the equation of a circle to write the equation in standard form for the outside boundary of the dome.

[tex]\boxed{\begin{minipage}{4.5 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h,k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

In this case, since the origin is at the center of the dome, the center coordinates are (0, 0):

The diameter of the dome is given as 96 feet. Since the diameter of a circle is twice its radius, the radius of the circular base of the dome is:

Substitute the values of h, k and r into the circle equation formula:

[tex](x - 0)^2 + (y - 0)^2 = 48^2[/tex]

Simplify:

[tex]x^2 + y^2 = 2304[/tex]

Therefore, the equation in standard form for the outside boundary of the dome is:

[tex]\boxed{x^2 + y^2 = 2304}[/tex]

[tex]\hrulefill[/tex]

To determine if your house is within the broadcast range of the radio station, we can calculate the distance between your house and the radio station using the distance formula. If the distance is less than or equal to the broadcast radius of 40 miles, then your house is within the broadcast range.

Given that your house is located 32 miles west and 21 miles north of the radio station, we can use these coordinates to calculate the distance.

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance Formula}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\\\where:\\ \phantom{ww}$\bullet$ $d$ is the distance between two points. \\\phantom{ww}$\bullet$ $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]

Define the two points:

Substitute these values into the distance formula and solve for d:

[tex]\begin{aligned}d&=\sqrt{(-32-0)^2+(21-0)^2}\\&=\sqrt{(-32)^2+(21)^2}\\&=\sqrt{1024+441}\\&=\sqrt{1465}\\&=38.27\; \sf miles\;(2\;d.p.)\end{aligned}[/tex]

Therefore, the calculated distance between your house and the radio station is approximately 38.27 miles.

Since the distance is less than the broadcast radius of 40 miles, we can conclude that your house is within the broadcast range of the radio station.

[tex]\hrulefill[/tex]

To determine if the plant is within the range of the sprinkler, we can calculate the distance between the plant and the sprinkler using the distance formula. If the distance is less than or equal to the sprinkler's radius, then the plant is within the sprinkler's range.

Given that the sprinkler has a radius of 8 feet, we can use these coordinates to calculate the distance.

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance Formula}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\\\where:\\ \phantom{ww}$\bullet$ $d$ is the distance between two points. \\\phantom{ww}$\bullet$ $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]

Define the two points:

Substitute these values into the distance formula and solve for d:

[tex]\begin{aligned}d&=\sqrt{(4-0)^2+(6-0)^2}\\&=\sqrt{(4)^2+(6)^2}\\&=\sqrt{16+36}\\&=\sqrt{52}\\&=7.21\; \sf feet\;(2\;d.p.)\end{aligned}[/tex]

The calculated distance between the plant and the sprinkler is approximately 7.21 feet.

Since the distance is less than the sprinkler's radius of 8 feet, we can conclude that the plant is within the range of the sprinkler.

The equation for the outside of the dome in standard form is x² + y² = 2304 and your house is within the broadcast range.

The center of the dome is at the origin of the coordinate plane, and the diameter of the dome is 96 feet, which means the radius is half of the diameter, so it is 48 feet.

The center of the dome is at the origin (0, 0), and the radius is 48 feet.

Therefore, the equation for the outside of the dome in standard form is:

x² + y² = 48²

x² + y² = 2304

Now let us determine if your house is within the broadcast range of the radio station, we need to check if it is within the radius of the broadcast.

The radio station has a broadcast radius of 40 miles. Your house is located 32 miles west and 21 miles north of the radio station.

Using the Pythagorean theorem, we can calculate the distance between your house and the radio station:

Distance = √((32)² + (21)²)

Distance = √(1024 + 441)

Distance = √1465

Distance = 38.27 miles

Since the distance between your house and the radio station is less than the broadcast radius of 40 miles, your house is within the broadcast range.

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The price of a ticket should be $1.20 to make the lottery a fair game.

To find the expected gain from one ticket, we first need to calculate the total amount of money won by all the tickets. The winning ticket receives $5000, while the five tickets that each win $1000 collectively receive $5000.

Similarly, the ten tickets that each win $200 collectively receive $2000. Therefore, the total amount of money won by all the tickets is $12,000.

To find the expected gain from one ticket, we divide the total amount of money won by the number of tickets sold. So, the expected gain from one ticket is $1.20.

To make the lottery a fair game, the price of a ticket should be equal to the expected gain from one ticket. However, in reality, the price of a ticket is usually higher to cover the cost of organizing and running the lottery.

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The area of the circle created by the paint is given by the expression [tex]4πt^2[/tex] square units.

To find the area of the canvas, we need to multiply the lengths of its sides.

Given that the sides of the canvas measure 3x + 5 and 2x + 1 inches, the area can be calculated as follows:

Area = (3x + 5) * (2x + 1)

Using the distributive property, we can expand the expression:

Area = 6[tex]x^2[/tex]+ 3x + 10x + 5

Simplifying further, we have:

Area = 6[tex]x^2[/tex] + 13x + 5

Therefore, the area of the canvas is given by the expression 6x^2 + 13x + 5 square inches.

Now let's calculate the area of the circle created by the paint. The area of a circle can be calculated using the formula A = πr^2, where r represents the radius of the circle.

In this case, the paint is spreading at a rate of r(t) = 2t, where t represents time in minutes. So, after time t minutes, the radius of the circle will be 2t.

Substituting the radius into the area formula, we get:

A[r(t)] = π(2t)^2

       = 4[tex]πt^2[/tex]

Therefore, the area of the circle created by the paint is given by the expression 4πt^2 square units.

To determine if the area of the circle will be at least 300 square inches in 5 minutes, we need to substitute t = 5 into the area formula and compare it with 300.

A[r(5)] = 4π[tex](5)^2[/tex]

       = 4π(25)

       = 100π

Since π is a constant value approximately equal to 3.14159, we can calculate:

A[r(5)] ≈ 100(3.14159)

       ≈ 314.159

The area of the circle after 5 minutes is approximately 314.159 square inches, which is larger than 300 square inches. Therefore, the circle will be larger than 300 square inches in 5 minutes.

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Answer: 4/5

Step-by-step explanation:

Area of the rectangle is 15*8 or 120

There is a certain triangle, where 3^2+4^2=5^2

This is that exact triangle. How? Well, 5*2=10 and 3*2=6

In other words, the base is 4*2 or 8

8*6/2=24

24/120=1/5

So the chance of it NOT being inside the triangle is 4/5

The solution of the system of linear equations is

(3, -1)

The graphical solution of linear equations involves representing the equations as lines on a coordinate plane and finding the point of intersection, if any, which corresponds to the solution of the system of equations.

To graphically solve a system of two linear equations, you would typically plot the lines represented by each equation on the same coordinate plane.

The point(s) where the lines intersect would then represent the solution(s) to the system.

Examining the graph, the point of intersection is at

(3, -1)

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Answer:

B.) 3/4

Step-by-step explanation:

There are 13 diamond cards in a deck of 52. Divide 52 by 13 to get 4. 4 helps you get a simplified version of the equation.  4/4 subtracted by 1/4 is 3/4 that's what you have left.

Answer:

2,-3

Step-by-step explanation:

The correct answer is 2 and -3 .

Given ,

Equation : x²+x-6=0

Now to find the value of x we need to find the roots/ solutions of the equation by either factor method or by quadratic formula.

So,

x² + x - 6 = 0

By factor method,

x² + 3x - 2x - 6 = 0

Take the common terms ,

x(x + 3) -2(x + 3) = 0

(x-2)(x + 3) = 0

Thus the roots of x are 2 and -3 .

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Answer:

Step-by-step explanation:

The derivative is f'(x) = (4x - 1) / (2x² - x).

To find the derivative of the function f(x) = ln(2x² - x), we can use the chain rule.

Begin by identifying the function to be differentiated, which is f(x) = ln(2x² - x).

Apply the chain rule, which states that if we have a composition of functions, f(g(x)), the derivative is given by (f'(g(x))) g'(x).

Let's consider the inner function g(x) = 2x² - x.

To differentiate g(x), we need to apply the power rule and the constant rule. The derivative of 2x² is 4x, and the derivative of -x is -1.

Now, we differentiate the outer function f'(g(x)). The derivative of ln(x) is 1/x.

Finally, we multiply the derivatives obtained in steps 3 and 4. Thus, the derivative of f(x) = ln(2x² - x) is:

f'(x) = (1 / (2x² - x)) * (4x - 1)

Simplifying further, we have:

f'(x) = (4x - 1) / (2x² - x)

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The modulus is √7 / 5 and the argument is π/6 for the given complex expression.

To find the modulus and argument of the complex expression ((1+i)^2) / ((1-√3i)^4), we can simplify the expression and then calculate the modulus and argument separately.

Let's simplify the expression first:

((1+i)²) / ((1-√3i)[tex]4[/tex])

Expanding the numerator:

(1+i)^2 = (1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i

Expanding the denominator:

(1-√3i)^4 = (1-√3i)(1-√3i)(1-√3i)(1-√3i)

= (1 - 2√3i + 3i^2)(1 - 2√3i + 3i^2)

= (1 - 2√3i - 3)(1 - 2√3i - 3)

= (-2 - 2√3i)(-2 - 2√3i)

= 4 + 8√3i - 12i - 12√3

= -8 - 12√3 - 4i(3 + 2√3)

= -8 - 12√3 - 12i - 8√3i

= -20 - 20√3i

Now, let's calculate the modulus:

Modulus = |(2i) / (-20 - 20√3i)|

= |2i| / |-20 - 20√3i|

= 2 / |20 + 20√3i|

= 2 / √[tex](20^2[/tex] + (20√3)^2)

= 2 / √(400 + 2400)

= 2 / √2800

= 2 / (10√28)

= √28 / 10

= √7 / 5

Next, let's calculate the argument:

Argument = arg((2i) / (-20 - 20√3i))

= arg(2i) - arg(-20 - 20√3i)

= π/2 - arg(-20 - 20√3i)

To find the argument of -20 - 20√3i, we can use the arctan function:

arg(-20 - 20√3i) = arctan((-20√3) / -20)

= arctan(√3)

= π/3

Therefore, the argument of[tex]((1+i)^2) / ((1-\sqrt{3} i)^4)[/tex]is:

Argument = π/2 - π/3

= π/6

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