Could someone kindly give me a detailed explanation with step-by-step instructions for the solution? I would greatly appreciate it.
Let f: N → Q be defined by f(x) = x / 1 + x for any x ∈ N. Prove or disprove each of the following:
(a) The function f is injective.
(b) The function f is surjective.
Answers
A. The function f is injective.
B. The function f is not surjective, as there are values in Q (e.g., y = 1) for which there is no corresponding x ∈ N.
How to prove it is injectiveTo prove that the function f is injective, we need to show that if f(x) = f(y), then x = y.
Let f(x) = f(y). Then we have:
x / (1 + x) = y / (1 + y)
Cross-multiplying, we get:
x(1 + y) = y(1 + x)
x + xy = y + xy
Now, subtract xy from both sides:
x = y
Since x = y when f(x) = f(y), we can conclude that the function f is injective.
(b) To prove or disprove that the function f is surjective, we need to determine whether for every y ∈ Q, there exists an x ∈ N such that f(x) = y.
Consider the case where y = 1/2. We need to find x ∈ N such that:
f(x) = x / (1 + x) = 1/2
Cross-multiplying, we get:
2x = 1 + x
Subtract x from both sides:
x = 1
In this case, there exists an x ∈ N (x = 1) such that f(x) = 1/2. However, we need to consider all y ∈ Q. Let's try another case.
Consider the case where y = 1. There is no x ∈ N that satisfies the following equation:
f(x) = x / (1 + x) = 1
Cross-multiplying, we get:
x = 1 + x
Subtract x from both sides:
0 = 1
This is a contradiction, so there is no x ∈ N such that f(x) = 1. Therefore, the function f is not surjective, as there are values in Q (e.g., y = 1) for which there is no corresponding x ∈ N.
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Related Questions
show that a) cos3∅=cos²∅-3cos∅sin²∅
Answers
The correct proof of the equation is cos3θ =cos(2θ + θ) = cos²θ - 3cosθsin²θ (Proved)
Solving trigonometry identity
Given the trigonometry identity below:
cos3∅=cos²∅-3cos∅sin²∅
We are to prove that both sides of the equation are equal.
cos 3θ = cos(2θ + θ)
Using the double angle formula for cosine, we can expand the first term:
cos(2θ + θ) = cos2θ cos θ - sin2θ sinθ
Since cos2θ = cos²θ - sin²θ
cos(2θ + θ) = cos²θ - sin²θ - 2cosθsinθsinθ
cos(2θ + θ) = cos²θ - sin²θ - 2cosθsin²θ
On simplifying, we can see that;
cos3θ =cos(2θ + θ) = cos²θ - 3cosθsin²θ (Proved)
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This diagram shows the blue prints a contractor drew for a ramp. In order for the ramp to meet safety regulations, the angle created by the bottom of the ramp and the ground should be no more than 18°.
How many degrees should the contractor lower the ramp by in order to meet safety regulations?
Enter your answer, rounded to the nearest tenth, in the box.
Answers
The contractor should lower the ramp by approximately 4.62° to meet safety regulations.
We have,
To determine the angle created by the bottom of the ramp and the ground, we can use the trigonometric function of sine.
In the given triangle, the height is 6 ft and the base is 16 ft.
The angle we want to find is opposite the height (let's call it θ).
The sine of an angle is defined as the ratio of the opposite side to the hypotenuse.
sin(θ) = opposite/hypotenuse
sin(θ) = 6/16
sin(θ) = 0.375
To find the angle θ, we can take the inverse sine (also known as arcsine) of 0.375:
θ = arcsin(0.375)
θ ≈ 22.62°
To meet safety regulations, the angle should be no more than 18°. Therefore, the contractor needs to lower the ramp by:
22.62° - 18° ≈ 4.62°
Thus,
The contractor should lower the ramp by approximately 4.62° to meet safety regulations.
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100 Points! State the amplitude, period, and phase shift for each function. Then graph the function. Photo attached. Thank you!
Answers
The amplitude and the period of the function f(x) = cos(5θ) are 1 and 2π/5
Here, we have,
to determine the amplitude and period of the function
From the question, we have the following parameters that can be used in our computation:
f(x) = cos(5θ)
A sinusoidal function is represented as
f(x) = Acos(B(x + C)) + D
Where
Amplitude = A
Period = 2π/B
So, we have
A = 1
Period = 2π/5
Evaluate
A = 1
Period = 2π/5
Hence, the amplitude is 1 and the period is 2π/5
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complete question:
100 Points! Algebra question. Photo attached. Find the amplitude, if it exists, and period of the function. Then graph the function. Thank you!
Identify as a monomial, binomial, trinomial, or polynomial.
x^3 + y^2 - 3
7y^2
x^3 + 2x^2 - x + 10
x + 7
18 points!
Answers
The given expressions can be identified as:
1. Trinomial
2. Monomial
3. Polynomial
4. Binomial
Answer to the questionThe given expressions can be identified as:
1. x^3 + y^2 - 3: This is a trinomial since it consists of three terms.
2. 7y^2: This is a monomial since it consists of only one term.
3. x^3 + 2x^2 - x + 10: This is a polynomial since it consists of four terms.
4. x + 7: This is a binomial since it consists of two terms.
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write an equation for the line shown
Answers
Answer:
y = 1x + -3
Step-by-step explanation:
y=mx + b where m is the slope and b is the y intercept.
The y intercept is -3.
Let's calculate the slope. Pick 2 points. I'll choose (0,-3) and (3,0).
The slope should be positive; our line is pointing UP and as x increases, y increases.
Slope = rise/run = (y2-y1)/(x2-x1)
= (-3-0)/(0-3)
= -3/-3 = 1
So y = 1x + -3
We can try another point from the graph to check that we're right. Let's use (1,-2)
y = 1(1) +-3
y = 1-3
y = -2, which is exactly our coordinate from the graph.
Rewrite yxyxy in exponent form
Answers
Step-by-step explanation: there are 3 y's so the answer is 3y
and there are 2 x's so therefore the answer is 2x
and then you just add them together to get 3y + 2x
100 Points! Algebra question. Photo attached. Find the missing variable. Round your answers to the nearest hundredth. Please show as much work as possible. Thank you!
Answers
Using the z-score, mean and sample mean, the standard deviation of the sample is 7.57
What is z-score?A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. The z-score is positive if the value lies above the mean and negative if it lies below the mean.
The formula is given as;
z = x - μ / σ
z = z -scoreμ = mean x = sample meanσ = standard deviationSubstituting the values into the formula;
-2.13 = 19.3 - 29 / σ
Cross multiply both sides;
-2.13σ = 19.3 - 29
-2.13σ = -9.7
Divide both sides by the coefficient;
-2.13σ / -2.13 = -9.7 / -2.13
σ = 7.57
The standard deviation is 7.57
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-5=[tex]\frac{y-7}{9}[/tex]
Answers
The solution of expression is,
⇒ y = - 39
We have to given that,
An expression is,
⇒ - 5 = (y - 7) / 9
Now, We can simplify as,
An expression is,
⇒ - 5 = (y - 7) / 9
Multiply by 9,
⇒ - 5 × 9 = (y - 7) × 9 / 9
⇒ - 45 =y - 7
Add 7 both side,
⇒ - 45 + 7 = y
⇒ y = - 39
Therefore, The solution of expression is,
⇒ y = - 39
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how many batches of cupcakes does lihle need to make to produce 120 cupcakes
Answers
To produce 120 cupcakes, Lihle needs to make 5 batches if one batch produces 24 cupcakes. However, it is important to plan for a little more than needed in case of any failures or discrepancies, so making an extra batch might be helpful. It is also important to consider the amount of frosting needed for 120 cupcakes, which can vary depending on the recipe. As a basic estimate, 1 cup of frosting is suggested for 12 cupcakes.
Some answers use the Pythagorean theorem and some use 1/2bh I'm confused at this point
Answers
Answer:
Step-by-step explanation:
The areas of the triangles are listed below:
Triangle A: A = 30 cm²
Triangle B: A = 14 cm²
Triangle C: A = 15 cm²
Triangle D: A = 40.997 cm²
How to find the areas of the triangles
In this problem we have a composite figure in the form of a square and formed by three right triangles (A, B, C) and a non-right triangle (D). The area of each triangle must be found, the areas of the each right triangle can be determined by this formula:
A = 0.5 · b · h
Where:
A - Areab - Baseh - HeightAnd the area of the non-right triangle can be found by Heron's formula:
A = √[s · (s - a) · (s - b) · (s - c)]
s = 0.5 · (a + b + c)
Where:
s - Semiperimeter.a, b, c - Side lengthsFirst, determine the areas of the right triangles:
Triangle A
A = 0.5 · (10 cm) · (6 cm)
A = 30 cm²
Triangle B
A = 0.5 · (7 cm) · (4 cm)
A = 14 cm²
Triangle C
A = 0.5 · (3 cm) · (10 cm)
A = 15 cm²
Second, find the side lengths of the non-right triangle by Pythagorean theorem:
a = √[(10 cm)² + (6 cm)²]
a = 2√34 cm
b = √[(7 cm)² + (4 cm)²]
b = √65 cm
c = √[(3 cm)² + (10 cm)²]
c = √109 cm
Third, calculate the semiparameter of the non-right triangle:
s = 0.5 · (2√34 cm + √65 cm + √109 cm)
s = 15.082 cm
Fourth, calculate the area of the non-right triangle:
A = √[(15.082 cm) · (15.082 cm - 2√34 cm) · (15.082 cm - √65 cm) · (15.082 cm - √109 cm)]
A = 40.997 cm²
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Compute probability of rolling two dice that have a sum of 7. Write the probability as a simplified fraction.
Thank you!!!
Answers
The probability of rolling two dice that have a sum of 7 is: 1/6.
What is the probability?When rolling two dice each die will have 6 possible outcomes
Total number of possible outcomes= 6 * 6
Total number of possible outcomes = 36
Now let's roll a sum of 7 dice to determine the amount of favourable results. There are 6 various ways that the number 7 may be combined, and they are as follows:
(1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1)
The number of favorable outcomes is 6.
Let find the probability of rolling two dice that have a sum of 7 :
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 6 / 36
Probability = 1 / 6
Therefore the probability is 1/6.
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There were 470 people at a play. The admission price was $3 for adults and $1 for children.
The admission receipts were $910. How many adults and how many children attended?
Answers
Answer:
220 adults and 250 children
Step-by-step explanation:
we require to set up 2 equations and solve simultaneously.
let a be the number of adults and c the number of children , then
a + c = 470 → (1) based on number of people
3a + c = 910 → (2) based on ticket sales
subtract (1) from (2) term by term to eliminate c
(3a - a) + (c - c) = 910 - 470
2a + 0 = 440
2a = 440 ( divide both sides by 2 )
a = 220
substitute a = 220 into (1) and solve for c
220 + c = 470 ( subtract 220 from both sides )
c = 250
that is 220 adults and 250 children attended
Calculate the average Budget across the four quarters.
Next year, it is estimated that there will be an average
budget of £6,032 per quarter. How much more is this, as a
percentage?
Answers
Answer:
error
Step-by-step explanation:
I'm sorry, but I need more information to calculate the average budget across the four quarters. Can you please provide me with the values of the budget for each quarter? Once I have this information, I can calculate the average and then determine the percentage difference between the average and the estimated budget for next year.
A 4th of July sale at your local truck dealership is advertising big discounts on pickup trucks. The truck you've had your eye on is now listed at $23,000 which is 23% off the original price. What was the original price of the truck?
Answers
If the value of the truck is at $23,000 which is 23% off the original price. Then the original price of the truck will be $ 29,870.13.
What is the percentage?
The amount of something is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.
A 4th of July sale at your local truck dealership is advertising big discounts on pickup trucks.
The truck you've had your eye on is now listed at $23,000 which is 23% off the original price.
Then the original price of the truck will be given as
Let the original price of the truck will be x. Then we have
0.77 × x = $ 23,000
x = $ 29,87
Find the length of side x in simplest radical form
with a rational denominator.
30⁰
X
4
60°
Answers
The value of measure of x in the triangle is,
⇒ x = 4√3 units
We have to given that,
A triangle is shown in image.
Now, WE can formulate by trigonometry formula we get;
⇒ tan 30° = Opposite / Base
⇒ tan 30° = 4 / x
⇒ 1/√3 = 4/x
⇒ x = 4√3 units
Thus, The value of measure of x in the triangle is,
⇒ x = 4√3 units
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50 Points! Multiple choice algebra question. Photo attached. Thank you!
Answers
The data-set in this problem is classified as follows:
D) Negatively skewed.
How to classify the skewness of a data-set?Negative skew refers to a longer or fatter tail on the left side of the distribution, that is, the majority of the values are on the lower end of the distribution.
Positive skew refers to a longer or fatter tail on the right side of the distribution, that is, the majority of the values are on the right end of the distribution.
For a symmetric distribution, we have that the majority of the values are at the center of the distribution, with an equal proportion at both tails.
For this problem, we have that the larger percentage is less than $4.00, which is at the left tail of the distribution, hence the distribution is negatively skewed.
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Write the slope-intercept form of the equation of the line through the given point with the given slope
through: (4, 3), slope = undefined
20 POINTS‼️‼️
Answers
Answer:
x = 4
Step-by-step explanation:
a line with an undefined slope is a vertical line with equation
x = c ( c is the value of the x- coordinates the line passes through )
the line passes through the point (4, - 3 ) with x- coordinate 4 , then
x = 4 is the equation of the line
note this equation cannot be expressed in slope- intercept form
A bottle holds 750 milliliters of iced tea.
I fluid ounce 30 milliliters
Approximately how many fluid ounces of iced tea does the bottle hold, rounded to the nearest whole number?
Answers
The volume of the bottle is 25 fluid ounce of iced tea.
Calculating the number of fluid ounces of iced tea the bottle holdfrom the question, we have the following parameters that can be used in our computation:
Volume of bottle = 750 milliliters of iced tea.
Also, we have
1 fluid ounce = 30 milliliters
This means that
1/30 fluid ounce = 1 milliliters
So, we have
Volume of bottle = 750 * 1/30 fluid ounce of iced tea.
Evaluate
Volume of bottle = 25 fluid ounce of iced tea.
Hence, the volume of bottle is 25 fluid ounce of iced tea.
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a car traveling at 30m/s comes to a stop in 5 seconds. What is the accerlation of this car
Answers
Answer: The acceleration of the car is -6 m/s²
Step-by-step explanation:
The acceleration of the car can be found by using the given below equation:
Acceleration = (final velocity - initial velocity) / time
Given:
Initial velocity (u) = 30 m/s
Final velocity (v) = 0 m/s
Time (t) = 5 seconds
Putting the values in the equation:
acceleration = (0 - 30) m/s / 5 s
acceleration = -30 m/s / 5 s
acceleration = -6 m/s²
Therefore, the acceleration of the car is -6 m/s² (negative sign indicates deceleration).
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Charlie the trainer has two solo workout plans that he offers his clients: Plan A and plan B. Each client does either one or the other (not both). On Wednesday there were 5 clients who did plan A and 6 who did plan B. On Thursday there were 3 clients who did plan A and 2 who did plan B. Charlie trained his Wednesday clients for a total of 7 hours and his Thursday clients for a total of 3 hours. How long does each of the workout plans last?
Answers
Plan A lasts for 0.5 hours, and Plan B lasts for 0.75 hours based on the given information and the solution to the System of equations.
To determine the duration of each workout plan, let's assume that both Plan A and Plan B have a fixed duration. Let's denote the duration of Plan A as "x" hours and the duration of Plan B as "y" hours.
From the given information, we can form the following equations:
On Wednesday:
5 clients did Plan A, so the total duration for Plan A is 5x hours.
6 clients did Plan B, so the total duration for Plan B is 6y hours.
The total training duration on Wednesday is 7 hours, so we have the equation: 5x + 6y = 7 ... Equation (1)
On Thursday:
3 clients did Plan A, so the total duration for Plan A is 3x hours.
2 clients did Plan B, so the total duration for Plan B is 2y hours.
The total training duration on Thursday is 3 hours, so we have the equation: 3x + 2y = 3 ... Equation (2)
We now have a system of equations (Equation 1 and Equation 2) that we can solve to find the values of x and y.
Multiplying Equation (1) by 2 and Equation (2) by 5 to eliminate the y variable, we get:
10x + 12y = 14 ... Equation (3)
15x + 10y = 15 ... Equation (4)
Multiplying Equation (3) by 3 and Equation (4) by 2 to create a new system of equations:
30x + 36y = 42 ... Equation (5)
30x + 20y = 30 ... Equation (6)
Subtracting Equation (6) from Equation (5), we can eliminate the x variable:
(30x + 36y) - (30x + 20y) = 42 - 30
16y = 12
y = 12/16
y = 0.75
Substituting the value of y into Equation (2), we can solve for x:
3x + 2(0.75) = 3
3x + 1.5 = 3
3x = 3 - 1.5
3x = 1.5
x = 1.5/3
x = 0.5
Therefore, the duration of Plan A is 0.5 hours, and the duration of Plan B is 0.75 hours.
Plan A lasts for 0.5 hours, and Plan B lasts for 0.75 hours based on the given information and the solution to the system of equations.
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Part 3: Room 2 You will be tiling the floor below with 12 inch by 12 inch tiles. You will need to figure 20% overage for this job.
7. What is the area of the floor?
8. How many tiles will you need to purchase? I
Answers
Based on the information, we can infer that the area of the floor is 130 square inches. Besides, we can infer that 2 tiles are needed to cover the entire floor area.
How to calculate the floor area?To calculate the area of the floor we must divide the figure into fragments to find the area more easily. First of all we can find the area of the left zone that measures 10 base by 9 height.
10 * 9 = 90
For the right part the area would be 10 * 4. So:
10 * 4 = 4040 + 90 = 130
According to the above, the area of the place would be 130. On the other hand, we can infer that 2 tiles are needed to cover the entire floor area.
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Solve for x. Round to the nearest tenth of a
degree, if necessary.
X
73
83
W
to
+
Answers
Answer:
Set your calculator to Degree mode.
[tex]x = {sin}^{ - 1} \frac{73}{83} = 61.6 \: degrees[/tex]
Angle x measures about 61.6°.
in a certain country people own a total of about 356 million fish, cats, and dogs as pets. The number of fish owned is 12 million more than the total number of cats and dogs owned, and 12 million more cats are owned than dogs. how many of each type of pet do people in this country own?
Answers
People in this country own approximately 111 million dogs, 123 million cats, and 258 million fish as pets.
Let's solve the problem step by step to determine how many of each type of pet people in this country own.
Let's assume the number of dogs as D.
According to the information given, the number of cats would be D + 12 million since there are 12 million more cats than dogs.
The number of fish would be the sum of the number of cats and dogs plus 12 million, as there are 12 million more fish than the total number of cats and dogs combined.
So, the equation representing the total number of pets is:
D + (D + 12 million) + (D + 12 million) = 356 million
Simplifying the equation, we have:
3D + 24 million = 356 million
Subtracting 24 million from both sides, we get:
3D = 332 million
Dividing both sides by 3, we find:
D = 110.67 million
Since the number of pets cannot be in fractions, we can round D to the nearest whole number. In this case, D ≈ 111 million.
Now, we can calculate the number of cats:
Cats = D + 12 million
= 111 million + 12 million
= 123 million
And the number of fish:
Fish = (D + Cats) + 12 million
= (111 million + 123 million) + 12 million
= 246 million + 12 million
= 258 million
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100 Points! Algebra question. Graph the function. Photo attached. Thank you!
Answers
The cost of each pound of grass seed is $6.
A graph of the solution for the cost of each pound of grass seed is shown below.
How to determine the cost of each pound of grass seed?In order to determine the cost of each pound of grass seed, we would assign a variable to the cost of each pound of grass seed and then translate the word problem into an algebraic linear equation.
Let the variable x represent the cost of each pound of grass seed.
Since Lori bought 3.6 pounds of grass seed and she was charged $22.90, including a tax of $1.30, an algebraic linear equation that best represent this situation is given by;
3.6x + 1.30 = 22.90
3.6x = 22.90 - 1.30
3.6x = 21.60
x = 21.60/3.6
x = $6.
In conclusion, we would sue an online graphing calculator to plot the solution for the cost of each pound of grass seed as shown in the image attached below.
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x varies inversely as y. y = 10 when x = 3.
Find the equation that relates x and y:
Then find x when y = 15:;
Answers
Answer:
Solution is in the attached photo.
Step-by-step explanation:
This question tests on the concept on deriving equations and substitution of values.
50 Points! Multiple choice algebra question. Photo attached. Thank you!
Answers
The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is, 3 / 5
We have to given that;
the terminal side of theta in standard position contains the point (6,-8)
Hence, For given condition we get;
The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is,
Here, x = 6, y = - 8
Hence, We get;
r = √x² + y²
r = √6² + 8²
r = √36 + 64
r = √100
r = 10
So, The exact value of cos theta if the terminal side of theta in standard position contains the point (6,-8) is,
cos θ = x / r
cos θ = 6 / 10
cos θ = 3 / 5
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please help! thank uu ~ :)
Answers
Certain isn’t the answer because he still has a 5/100 of not winning.
Unlikely would be less than 50
50-50 would mean he bought 50 tickets, but he didn’t. He bought 95.
Given:
AC
≅
BD
and ∠CAB≅∠DBA.
Prove: △ABC≅△BAD.
Answers
Triangle ABC is congruent to triangle BAD because all their corresponding angles are equal.
What are congruent triangles?Congruent triangles are triangles with equal corresponding angles and equal corresponding sides. This means that the angles will be equal and their corresponding sides will be equal.
Since AC is equal to BD , then the corresponding angles facing the two lines will be equal and angle A is equal to angle B
Therefore angle C will be equal to angle D, therefore we can say triangle ABC is congruent to triangle BAD.
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Please help me):00000000000
Answers
The mean monthly rainfall is 3.245 inches.
How to find the mean of the monthly rainfall?For a set of N numbers {x₁, x₂, ...}, the mean of these N numbers is:
M = {x₁ + x₂ + ...}/N
Here we want to find the mean of the rainfall in inches, then we just need to add the 12 values in the table and then divide by 12, we will get:
M = (2.22+1.51+1.86+2.06+3.48+4.57+ 3.27 + 5.40 + 5.45 + 4.34+2.64+ 2.14)/12
M = 3.245
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Express in polar form 6√3 + 6i
Answers
Answer:
Step-by-step explanation:
To express 6√3 + 6i in polar form, we first need to find the magnitude and argument of the complex number.
The magnitude (or modulus) of the complex number is given by the formula:
|z| = √(a^2 + b^2)
where a and b are the real and imaginary parts of the complex number, respectively.
In this case, a = 6√3 and b = 6, so:
|6√3 + 6i| = √((6√3)^2 + 6^2) = √(108 + 36) = √144 = 12
The argument (or angle) of the complex number is given by the formula:
arg(z) = tan^-1(b/a)
In this case, arg(z) = tan^-1(6/6√3) = tan^-1(1/√3) = π/6
Therefore, the polar form of 6√3 + 6i is:
12(cos(π/6) + i sin(π/6))
or
12e^(iπ/6)