Solve the equation. (List your answers counterclockwise about the origin starting at the positive real axis. Express θ in radians.)
z^3 + 3 = -3i
Answers
Expressing the angles θ in radians, the solutions are: z1 ≈ 1.229 * \(e^{(-\pi i/12)\), z2 ≈ 1.229 * \(e^{(7\pi i/12)\) and z3 ≈ 1.229 * \(e^{(11\pi i/12)\). These solutions can be plotted counterclockwise about the origin starting at the positive real axis on the complex plane.
To solve the equation, we can rewrite it in exponential form using Euler's formula:
z³ + 3 = -3i
z³ = -3 - 3i
Now, let's convert -3 - 3i to polar form:
-3 - 3i = 3√2 * (-1/√2 - i/√2)
= 3√2 * \(e^{(-i\pi /4)\)
We can write z³ as r³ * \(e^{(i\theta3)\), where r is the magnitude of z and θ3 is the argument of z³.
So, we have:
r³ * e^(iθ3) = 3√2 * \(e^{(-i\pi /4)\)
Comparing the real and imaginary parts of both sides, we get:
r³ = 3√2
e^(iθ3) = \(e^{(-i\pi /4)\)
From the first equation, we can solve for r:
r = (3√2)¹/³
r ≈ 1.817
From the second equation, we know that θ3 = -π/4.
Now, let's find the three cube roots of r * \(e^{(i\theta)\):
z1 = r¹/³ * \(e^{(i\theta/3)\)
z1 ≈ 1.229 * \(e^{(-i\pi /12)\)
z2 = r¹/³ * \(e^{(i(\theta/3 + 2\pi /3))\)
z2 ≈ 1.229 * \(e^{(i7\pi /12)\)
z3 = r¹/³ * \(e^{(i(\theta/3 + 4\pi /3))\)
z3 ≈ 1.229 * \(e^{(i11\pi /12)\)
So, the solutions to the equation z³ + 3 = -3i are approximately:
z1 ≈ 1.229 * \(e^{(-i\pi /12)\)
z2 ≈ 1.229 * \(e^{(i7\pi /12)\)
z3 ≈ 1.229 * \(e^{(11\pi i/12)\)
Therefore, expressing the angles θ in radians, the solutions are: z1 ≈ 1.229 * \(e^{(-i\pi /12)\), z2 ≈ 1.229 * \(e^{(7\pi i/12)\)and z3 ≈ 1.229 * \(e^{(11\pi i/12)\). These solutions can be plotted counterclockwise about the origin starting at the positive real axis on the complex plane.
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Related Questions
Which line is perpendicular to 2x+3y=5
Answers
Answer:
2/3
Step-by-step explanation:
mx+b=y --> m=slope
if m is slope of a line.... -m is the slope of a line perp to the first
y=-2/3x+5/3
slope m=-2/3
slope perp to m -> -m = 2/3
Calculate the effective interest on £2000 at 3% interest
quarterly after 4 years.
Answers
The effective interest on £2000 at a 3% interest rate compounded quarterly over a period of 4 years is approximately £245.15.
To calculate the effective interest, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (including interest)
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of compounding periods per year
t = the number of years
In this case, the principal amount (P) is £2000, the annual interest rate (r) is 3% (or 0.03 as a decimal), the compounding is done quarterly (n = 4), and the investment period (t) is 4 years.
Plugging the values into the formula:
A = £2000(1 + 0.03/4)^(4*4)
= £2000(1 + 0.0075)^16
= £2000(1.0075)^16
≈ £2000(1.126825)
Calculating the future value:
A ≈ £2253.65
To find the effective interest, we subtract the principal amount from the future value:
Effective Interest = £2253.65 - £2000
≈ £253.65
Therefore, the effective interest on £2000 at a 3% interest rate compounded quarterly after 4 years is approximately £253.65.
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1. What are the 3 conditions for a function to be continuous at xa? 2. the below. Discuss the continuity of function defined by graph 3. Does the functionf(x) = { ***
Answers
The three conditions for a function to be continuous at a point x=a are:
a) The function is defined at x=a.
b) The limit of the function as x approaches a exists.
c) The limit of the function as x approaches a is equal to the value of the function at x=a.
The continuity of a function can be analyzed by observing its graph. However, as the graph is not provided, a specific discussion about its continuity cannot be made without further information. It is necessary to examine the behavior of the function around the point in question and determine if the three conditions for continuity are satisfied.
The function f(x) = { *** is not defined in the question. In order to discuss its continuity, the function needs to be provided or described. Without the specific form of the function, it is impossible to analyze its continuity. Different functions can exhibit different behaviors with respect to continuity, so additional information is required to determine whether or not the function is continuous at a particular point or interval.
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in pascal's triangle, each entry is the sum of the two entries above it. in which row of pascal's triangle do three consecutive entries occur that are in the ratio $3: 4: 5?$
Answers
The three consecutive entries in Pascal's Triangle that are in the ratio $3:4:5$ occur in the sixth row.
Let's start by writing out the first few rows of Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
We can see that the ratio between consecutive entries in each row is the same, so we can look for a row where the ratio is close to $3:4:5$. Let's divide each entry in the sixth row by the entry to its left:
1 / 5 = 0.2
5 / 10 = 0.5
10 / 10 = 1
10 / 5 = 2
5 / 1 = 5
We can see that there are three consecutive entries in this row that are close to the ratio $3:4:5$: the entries 10, 5, and 1 are in the ratio $10:5:1$ or $2:1:0.2$. Therefore, the ratio $3:4:5$ occur in the sixth row of Pascal's Triangle.
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need help with this problem
Answers
The equation of the line that passes through the point (-3, -5) and (-3, -3) will be; y = -5
What is the slope?The slope is the ratio of the vertical changes to the horizontal changes between two points of the line.
The slope of the line is calculated as follows:
m = Δy/Δx
Given that the line that passes through the points (-3, -5) and (-3, -3)
To find the slope of the line that passes through points (-3, -5) and (-3, -3)
Therefore, the slope of the line is;
m = Δy/Δx
m = (-3 + 5)/(-3 + 3)
m = 0
When we simplify it, then we get the equation;
y + 5 = 0
y = -5
Hence, the equation of the line is y = -5
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What is the magnitude of ?
V
(9,-4)
Answers
Answer:
The magnitude is sqrt((-4)^2 + (-9)^2) = 9.85. The angle is atan(-9/-4) = 180 deg + 66 deg = 246 deg = -114 deg.
Step-by-step explanation:
hope it help
Answer:
9.85
Step-by-step explanation:
|v|= √9²+(-4)²
=√81+16
=√97
|v|= 9.85
Cd is the perpendicular bisector of ab. Ad ≅ bd by definition of a bisector. ∠ cda and ∠ cdb are both right angles based on the definition of perpendicular lines. Cd ≅ cd by the reflexive property of congruence. Δ adc ≅ δ bdc by the _____. So ca ≅ cb because corresponding parts of congruent triangles are congruent.
Answers
Answer:
YesTheorem 8.3: If two angles are complementary to the same angle, then these two angles are congruent.
A and B are complementary, and C and B are complementary.
Two cards are drawn with replacement in succession. What is the probability that both cards drawn are Queen?1/16916/171/23/51
Answers
Given:
Two cards are drawn with replacementss in succession.
Required:
We need to find the probability that both cards are drawn is Queen.
Explanation:
The sample space = The number of cards.
\(n(S)=52\)Let A be the event of drawing a queen.
The favourable outcomes =The number of queen cards.
\(n(A)=4\)The probability of drawing the first card, which is queen.
\(P(A)=\frac{n(A)}{n(S)}=\frac{4}{52}=\frac{1}{13}\)The card is replaced.
Let B be the event of drawing the second card after replacement.
The probability of drawing the second card, which is queen is equal to the probability of drawing the first card, which is queen.
\(P(B)=P(A)=\frac{1}{13}\)The probability that both cards drawn are Queen.
\(P(both\text{ cards are queen\rparen=P\lparen A\rparen}\times P(B)=\frac{1}{13}\times\frac{1}{13}\)\(P(both\text{ cards are queen\rparen}=\frac{1}{169}\)Final answer:
\(P(both\text{ cards are queen\rparen}=\frac{1}{169}\)
can u help me with this and show work i only have 20 min and theres more
Answers
180 - 145 = 35
180 - 80 - 35 = 65
x = 65
Name the relationship: altérnate interior, corresponding, or altérnate exterior.
Which is the Answers?
A. Altérnate interior
B. Complementary
C. Altérnate exterior
D. Corresponding
Can you help me, I would appreciate it very much:)
Answers
Answer:
A. Alternate Interior
Step-by-step explanation:
A. Alternate interior
Step-by-step explanation:
Angles a and b are alternate interior angles,
Alternate interior angles are equal.
Also,
a = b
How many solutions are there to the inequality x1 + x2 + x3 ≤ 11, where x1, x2, and x3 are nonnegative integers? [Hint: Introduce an auxiliary variable x4 such that x1 + x2 + x3 + x4 = 11.]
Answers
The number of nonnegative integer solutions to the inequality x1 + x2 + x3 ≤ 11 is C(14,3) = 364.
We can solve this inequality by introducing an auxiliary variable x4, such that x1 + x2 + x3 + x4 = 11. Here, x1, x2, x3, and x4 are all nonnegative integers.
We can interpret this equation as follows: imagine we have 11 identical objects and we want to distribute them among four boxes (x1, x2, x3, and x4). Each box can contain any number of objects, including zero. The number of solutions to this equation will give us the number of nonnegative integer solutions to the original inequality.
We can use a technique known as stars and bars to count the number of solutions to this equation. Imagine we represent the 11 objects as stars: ***********.
We can then place three bars to divide the stars into four groups, each group representing one of the variables x1, x2, x3, and x4. For example, if we place the first bar after the first star, the second bar after the third star, and the third bar after the fifth star, we get the following arrangement:
| ** | * | ****
This arrangement corresponds to the solution x1=1, x2=2, x3=1, and x4=7. Notice that the number of stars to the left of the first bar gives the value of x1, the number of stars between the first and second bars gives the value of x2, and so on.
We can place the bars in any order, so we need to count the number of ways to arrange three bars among 14 positions (11 stars and 3 bars). This is equivalent to choosing 3 positions out of 14 to place the bars, which can be done in C(14,3) ways.
Therefore, the number of nonnegative integer solutions to the inequality x1 + x2 + x3 ≤ 11 is C(14,3) = 364.
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15 (x-3)/5= 3 (2x-3) without the distributive property ( the / symbol is division)
Answers
Answer:
brooooooooooo I got x=0
I dont understand this please help
Answers
Answer: 190 miles
Solution:
=4 x 95 / 2
=(cross out the common factor)
=2 x 95 = 190
What is the area of triangle ABC?
Answers
So,the area is 36root3
Write the equation of the line in slope-intercept form (y=mx+b):
Answers
a = ( 0 , - 4 )
b = ( - 2 , - 2 )
\(m = - \frac{y(a) - y(b)}{x(a) - x(b)} \\ \)
\(m \: = \frac{ - 4 - ( - 2)}{0 - ( - 2)} \\ \)
\(m = \frac{ - 4 + 2}{2} \\ \)
\(m = \frac{ - 2}{2} \\ \)
\(m = - 1\)
Which is the better deal?
Cheerios
12 oz. for $ 1.99
36 oz. for $2.59
48 oz. for $3.99
60 oz. $4.59
Answers
Answer:
36 oz
Step-by-step explanation:
Get the other options down to 12oz units for their price and then you can compare which one is the cheapest for 12oz.
12oz = $1.99
36oz/3 = 12oz so $2.59/3 = $ .86
48oz/4= 12oz so $3.99/4 = $1.00
60oz/5= 12oz so $4.59/5 = $ .92
So looking at the breakdown, the 36oz one is the cheapest because for that price for a third of it comes out to .86 cents.
4z-10-9z please help me i need these
Answers
Answer:
-5z - 10
Step-by-step explanation:
Subtract 9z from 4z.
draw two normal curves that have the same mean but different standard deviations. describe the similarities and differences.
Answers
Two normal-curves with similar forms but distinct spreads will have the same mean but different standard deviations.
Since the centre of a normal distribution is determined by the mean, the centre of both curves on the horizontal axis will be the same. The spread of a normal distribution is, nevertheless, determined by its standard deviation. A smaller standard deviation indicates that the distribution is more tightly concentrated around the mean, whereas a bigger standard deviation indicates that the distribution is more dispersed.
Since extreme values are more likely to occur, the normal curve with the larger standard deviation will have more area under the curve in the tails. Extreme values are less likely to occur for the normal curve with the smaller standard deviation, which will have less area under the curve in the tails.
When compared to a normal curve with a smaller standard deviation, the larger standard deviation normal curve will visually appear wider and flatter. Both curves will have a bell-shaped curve with the highest point at the mean that is symmetrical around the mean.
Similarities:
The mean of both normal curves, which symbolises the distribution's centre, will be the same.
The mean will be the centre of symmetry for both normal curves.
The area under the curve for both normal curves will be equal to one.
Differences:
Greater standard deviation results in a wider normal curve than smaller standard deviation does.
The data will be more dispersed over the normal curve with a higher standard deviation, whereas the data will be more firmly grouped along the normal curve with a smaller standard deviation.
The peak of the normal curve with a higher standard deviation will be flatter than the peak of the normal curve with a lower standard deviation.
Overall, the two normal curves are comparable in that they both have the same mean, but they differ mostly in terms of their standard deviation, which has an impact on both their spread and the possibility of extreme values.
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can someone please help me, will give brainliest
Answers
Answer:
(d) 1 mile = 10 blocks
Step-by-step explanation:
The ballpark is 8 blocks west and 15 blocks north of Jeff's home. The Pythagorean theorem tells us the straight-line distance is ...
d² = 8² +15² = 64 +225 = 289
d = √289 = 17 . . . blocks
The problem tells us this represents 1.7 miles, so the proportion is ...
blocks/miles = 17 blocks/(1.7 miles) = 10 blocks/mile
The map scale is 1 mile = 10 blocks.
A beauty supply store surveyed 1,000 customers to determine which of 3 brands of concealer they like the best. 335 customers chose Brand A, 402 customers chose Brand B, while 263 customers chose Brand C. What percent of customers chose Brand B? Round off your answer to the nearest tenths place.
Answers
40.2 percent of the customers at the beauty supply store choose brand B.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
The percent of customers that choose brand B = (number of brand B customers / total number of customers) * 100%
The percent of customers that choose brand B = (402/1000) * 100% = 40.2%
40.2 percent of the customers at the beauty supply store choose brand B.
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Can someone PLEASE help me out??!!!!!!!
Answers
Answer:
\(\huge\boxed{66.42 \textdegree}\)
Step-by-step explanation:
In order to solve for this missing angle, we can use trigonometry.
With angle B, we only have two sides: the adjacent of the angle and the hypotenuse of the angle. The adjacent side is the one laying directly next to the angle that isn't the hypotenuse.
Using SOH CAH TOA, we can see that the correct trig function for this would be cos because of CAH (Cosine = Adjacent over Hypotenuse)
Since we know the length of the sides but we don't know the angle, we are going to use inverse trig.
The adjacent side over our hypotenuse will be \(\frac{2}{5}\) since 2 is the length of the adjacent and 5 is the length of the hypotenuse.
We know have the equation \(cos^{-1}(B) = \frac{2}{5}\).
We now find what \(\frac{2}{5}\) is (0.4) and find the inverse cosine of it. We can do this by plugging it into a calculator.
When we do this, we get the number \(66.42182152\) etc. Rounding to the nearest hundredth gets us 66.42.
Hope this helped!
Parametrize the portion of the cone z- V8x2 + 8y2 with 0 s zs V8. (Your instructors prefer angle bracket notation>for vectors.)
Answers
The parametric equations for the portion of the cone are: r(t) = t, θ(t) = t, z(t) = 2√2t where t is a parameter that ranges from 0 to √8.
To parametrize the portion of the cone z - √(8x^2 + 8y^2) with 0 ≤ z ≤ √8, we can use cylindrical coordinates. Let's denote the parameters as r, θ, and z.
We know that x = rcosθ, y = rsinθ, and z = z.
Substituting these values into the equation of the cone, we have:
z - √(8(rcosθ)^2 + 8(rsinθ)^2) = 0
Simplifying the expression inside the square root, we get:
z - √(8r^2(cos^2θ + sin^2θ)) = 0
z - √(8r^2) = 0
z - 2√2r = 0
From this equation, we can express z in terms of r as:
z = 2√2r
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approximately how much do i save per bottel of juice if i buy a pack of 8 rather than a pack of 6
Answers
Answer: Your question is missing information but this is what you do to answer your question step by step
B = Cost of 8 pack / 8 bottles = $ Cost of 8 pack/ 8 =$ ____ per bottle
A = Cost of 6 pack/ 6 bottles = $ Cost of 6 pack/ 6= $ ____ per bottle
then do a simple subtraction A -B = savings
Step-by-step explanation: Information is incomplete
if an 8 pack cost $16
If the 6 pack cost $15
All you do is a simple subtraction after calculations
Cost of 8 pack / 8 bottles = $16/8 =$2 per bottle
Cost of 6 pack/ 6 bottles = $15/ 6= $2.50 per bottle
in my example
You would save $ 0.50 per bottle by purchasing the 8 pack
Help me please it’s due tomorrow morning
Answers
The value of the given inequality is x≥16.
A connection in mathematics that compares two numbers or other mathematical expressions unequally is known as an inequality. [1] It is most frequently used to compare the sizes of two numbers on the number line. To indicate various sorts of inequalities, a variety of notations are used:
A less than symbol (a b) indicates that an is less than b.
A bigger value than b is indicated by the notation a > b.
In either scenario, a and b are not equal. In these relationships, an is strictly less than or strictly greater than b, which is known as a strict inequality[1]. Comparability is not included.
Two kinds of inequality relations are looser than strict inequalities:
We have inequality
x-4≥12
add 4 on both sides
x-4+4≥12+4
x≥16
Hence,
The value of the given inequality is x≥16.
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The table shows y as a function of x. Suppose a point is added to this table. Which choice gives a point that preserves the function?
Responses
A (−3, −8)(−3, −8)
B (−6, 3)(−6, 3)
C (−5, −2)(−5, −2)
D (9, 8)
Answers
The point that preserves the function is option A(-3,-8).
What do you mean by function?
A function in mathematics from a set X to a set Y assigns exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
According to the given question,
We have four option for the answer.
Note: if f is a function then each number of the domain has an unique image.
Since (regarding the table) the image of 9 is 6 then the answer D is wrong
The same apply on answers B and C .
Therefore, the right answer is A(-3,-8).
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You are planning to build a walkway that surrounds the area of a rectangular garden. The width of the walkway around the
garden is the same on every side. Which of the following polynomials represents the total area of the garden and
walkway?
Answers
Answer:
B. (2x + 10)(2x + 9)
Step-by-step explanation:
Length (L) of the garden and walkway = x + 10 + x = (2x + 10) ft
Width (W) of the garden and walkway = x + 9 + x = (2x + 9)
Area of the garden and walkway = area of rectangle = L*W
= (2x + 10)(2x + 9)
The polynomial representing the total area of the garden and walkway = (2x + 10)(2x + 9)
a box with a square base and open top must have a volume of 512000 cm3 . find the dimensions of the box that minimize the amount of material used.
Answers
The box measures 100.79 cm by 100.79 cm by 50.40 cm in size, which minimizes the quantity of material used.
Given,
Let the side of the square base be x
h be the height of the box
Volume V = x²h
512000 = x²h
h = 512000/x² ..... 1
Surface area = x² + 2xh + 2xh
Surface area S = x² + 4xh ...... 2
Substitute 1 into 2;
From 2;
S = x² + 4xh
S = x² + 4x(512000/x²)
S = x² + 2048000/x
To minimize the amount of material used;
dS/dx = 0
dS/dx = 2x - 2048000/x²
0 = 2x - 2048000/x²
0 = 2x³ - 2048000
2x³ = 2048000
x³ = 1024000
x = ∛1024000
x = 100.79 cm
Since Volume = x²h
512000 = 100.79²h
h = 512000/10158.62
h = 50.40 cm
Hence the dimensions of the box that minimize the amount of material used is 100.79 cm by 100.79 cm by 50.40 cm
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Please help me with this :((
Answers
Answer:x=5 or F
Step-by-step explanation:(8x+6)/(5x-2)=2, because it is the midsegment, cross multiply, and you get 2x=10, so x=5
Based on the 2010 census,the population of Georgia was 9.6x10 6 people . Which state had a larger population?
Answers
Answer:
new york
Step-by-step explanation:
1.9 x 70 > 7.1 x 50
An algorithm is a calculation that determines how long it will take to solve a problem. True or False?
Answers
The given statement "An algorithm is a calculation that determines how long it will take to solve a problem." is False.
An algorithm is not just a calculation that determines how long it will take to solve a problem. An algorithm is a step-by-step set of instructions or a process used to solve a problem or perform a specific task. It is a systematic approach that allows a computer or human to break down a problem into smaller, manageable parts and reach a solution effectively.
Algorithms are the foundation of computer programming and can be applied in various fields such as mathematics, data processing, and problem-solving. They can be simple, like finding the largest number in a list, or complex, like solving a Rubik's Cube.
Efficiency is a key factor in evaluating algorithms. The time and resources required for an algorithm to solve a problem can vary greatly depending on the method used. However, the primary purpose of an algorithm is to provide a clear and concise procedure to reach a solution, rather than just estimating the time needed to solve a problem.
In summary, an algorithm is a well-defined process designed to perform a specific task or solve a problem, rather than just calculating the time required to do so. Its effectiveness depends on its efficiency, accuracy, and the simplicity of the steps involved.
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