There exist two complex numbers c, say c1 and c2, so that 2+2i, 5+i, and c form the vertices of an equilateral triangle. Find the product c1c2 .enter image description here
So far, I have used the distance formula to get (a−2)2+(b−2)2=(a−5)2+(b−1)2 for c1(a,b). I expanded and simolified this to get b=3a−9. What do I do next?
Answers
The product of the two complex number c₁c₂ is thus 10 + 9i.
Let c₁ = a + ib and c₂ = c + id. Also z₁ represent 2 + 2i and z₂ represent 5 + i. The length of side of the equilateral triangle formed can be obtained by
|z₁ - z₂| = r = √((5 - 2)² + (1 - 2)² = √10
a² + b² - 4a - 4b + 8 = 10
a² + b² - 10a - 2b + 26 = 10
So 6a - 2b = 18 that is 3a - b = 9
Thus b = 3a - 9
Similarly c² + d² - 4c - 4d + 8 = 10
c² + d² - 10c - 2d + 26 = 10
So 3c - d = 9, that is d = 3c - 9
Midpoint of line joining c₁ and c₂ is same as that of z₁ and z₂, so
(c + a)/2 = ( 5 + 2)/ 2 and (d + b)/2 = (1 + 2)/2
So midpoint is 3.5 + 1.5i
Altitude of equilateral triangle of side a is √3a/2
That is √(a - 3.5)² + (b - 1.5)² = √3×√10/2
√(a - 3.5)² + (3a - 9 - 1.5)² = √30 /2
a² - 7a + 12.25 + 9a² - 63a + 110.25 = 30/4
On solving a = (7 ± √3)/ 2
So a = (7 - √3)/ 2, c = (7 + √3)/ 2
b = 3a - 9 = (3 - 3√3)/ 2, d = (3 + 3√3)/2
So c
c₁ = (7 - √3)/2 + i (3 - 3√3)/ 2
c₂ = (7 + √3)/2 + i (3 + 3√3)/ 2
So the product c₁c₂ is
= ((7 - √3)/ 2) * ((7 + √3)/ 2) - ((3 - √3)/ 2) * ((3 + √3)/ 2) + i (((7 - √3)/ 2) * ((3 + √3)/ 2) + ((7 + √3)/ 2) * ((3 - √3)/ 2))
= 10 + 9i
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Related Questions
Assume that the duration of human pregnancies can be described by a Normal model with mean 265 days and standard deviation 17 days.
a) What percentage of pregnancies should last between 269 and 282 days?
b) At least how many days should the longest 10% of all pregnancies last?
c) Suppose a certain obstetrician is currently providing prenatal care to 67 pregnant patients. Let y represent the mean length of their pregnancies. According to the
Central Limit Theorem, what is the distribution of this sample mean, y? Specify the model, mean, and standard deviation.
d) What's the probability that the mean duration of these patients' pregnancies will be less than 258 days?
Answers
a) The percentage of pregnancies should last between 269 and 282 days is 18%
b) The days should the longest 10% of all pregnancies last is 291 days.
c) According to the Central Limit Theorem, the distribution of this sample mean is 265. The model is a Normal model, and standard deviation is 17/(√(67)).
d) The probability that the mean duration of these patients' pregnancies will be less than 258 days is 0.03 or 3%
How do we determine the values?a) To find the percentage of pregnancies that should last between 269 and 282 days, we need to find the proportion of the area under the normal distribution curve that falls between those two values. This can be done using a standard normal table or calculator. Using a calculator, we find that the proportion of the area under the curve between 269 and 282 days is approximately 0.18 (or 18%).
b) To find the longest 10% of all pregnancies, we need to find the duration that corresponds to the 90th percentile. We can use the standard normal table or calculator to find that the duration that corresponds to the 90th percentile is approximately 291 days.
c) According to the Central Limit Theorem, the distribution of the sample mean, y, is a normal distribution with a mean of 265 days (the population mean) and a standard deviation of 17/(√(67)) days. So the model is a Normal model with mean 265 and standard deviation 17/(√(67))
d) To find the probability that the mean duration of these patients' pregnancies will be less than 258 days, we need to find the proportion of the area under the normal distribution curve that falls to the left of 258 days. Using a calculator, we can find that the proportion of the area under the curve to the left of 258 days is approximately 0.03 (or 3%).
Therefore, the correct answers are as given above
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in a survey of 260 college students, the following data were obtained: 64 had taken a mathematics course, 94 had taken a computer science course, 58 had taken a business course, 28 had taken both a mathematics and a business course, 26 had taken both a mathematics and a computer science course, 22 had taken both a computer science and a business course, and 14 had taken all three types of courses.
Answers
There were 64 students who had taken a mathematics course. Of those 64, 28 had also taken a business course, and 26 had also taken a computer science course. This leaves 10 students who had taken only a mathematics course.
To calculate this, we start by subtracting the number of students who had taken all three courses (14) from the total number of students who had taken a mathematics course (64).
Then we subtract the number of students who had taken both a mathematics and a business course (28) and the number of students who had taken both a mathematics and a computer science course (26) from the result. This gives us 10 students who had taken only a mathematics course.
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For the following exercises, use a calculator to find the length of each side to four decimal places.
Answers
In the given right-angled triangle, the sides are 21.3617 and 22.8815.
The given figure is a right-angled triangle. Hence, sides can be determined by using trigonometric function and/or the Pythagorean theorem. It is known that angle A = 69 and side b = 8.2. The side b is an adjacent side to angle A. Using the trigonometric function of tangent
tan Ɵ = opposite side/adjacent side
tan A = a/b
a = b tan A = 8.2 tan 69 ≈ 21.3617
As the two sides a and b are known, the remaining side c can be determined by using Pythagorean theorem,
Hypotenuse^2 = Opposite^2 + Adjacent^2
c^2 = a^2 + b^2
c^2 = (21.3617)^2 + (8.2)^2
c^2 = 523.3635
c =√523.3635
c ≈ 22.8815
Note: The question is incomplete. The complete question probably is: Use a calculator to find the length of each side to four decimal places. (Assume triangle ABC where side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse.) side length b = 8.2 and angle A = 69 degrees.
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a car dealer is deciding what kinds of vehicles he should order from the factory. he looks at his sales report for the preceding period. construct a pareto chart to help him decide. choose the vertical scale so that the relative frequencies are represented.
Answers
The vertical scales for each type of vehicle are:
Economy, 0.2Sports, 0.5Family, 0.35Luxury, 0.1Truck, 0.3The rationale is to choose a vertical scale that represents the relative frequencies, or the proportion of each type of vehicle sold compared to the total number of vehicles sold. The calculation for each type of vehicle is as follows:
Economy: 30 / 150 = 0.2 Sports: 7.5 / 150 = 0.05Family: 52.5 / 150 = 0.35 Luxury: 15 / 150 = 0.1 Truck: 45 / 150 = 0.3Here, the total number of vehicles sold (150) is calculated by adding up the sales for each type of vehicle. The relative frequency for each type of vehicle is then calculated by dividing the sales for that type of vehicle by the total number of vehicles sold.
This question should be provided as:
A car dealer is deciding what kind of vehicles he should order from the factory. he looks at his sales report for the preceding period. choose the vertical scale so that the relative frequencies are represented:
Vehicle : economy, sports, family, luxury, truck.Sales : 30, 7.5, 52.5, 15, 45.Learn more about vertical axes here: brainly.com/question/11485924
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what is the slope of the line which passes through the origin (-3,5)
Answers
Answer:
Step-by-step explanation:
29. Candice and Leila arrived at a birthday party at 5:20 p.m. The party lasted for 3 hours. They then took 20 minutes to arrive home. What time did they arrive home?
Answers
Explanation: 5+3=8
20+20=40
-12y - 16 i dont understand this please explain to me with a good explanation
Answers
Answer: -4(3y+4)
Step-by-step explanation:
First I found the Great common divisor (-4) and took that number and divided -12y and -16 with it getting 3y and 4 and then I set it up as if I was trying to get -12y and -16 so the best way to do that is to use the distributive property so I put 3y and 4 in parentheses like this (3y+4) then I added the -4 on to get the answer of -4(3y+4)
[tex]\huge\text{Hey there!}[/tex]
[tex]\mathsf{-12y - 16}[/tex]
[tex]\large\text{Let us look for the GCF for each of your numbers because it can}\\\large\text{make this equation somewhat easier to solve.}[/tex]
[tex]\mathsf{-12x \rightarrow -1, -2, -3, -4, -6, -12, \ \& \ -x}\\\\\mathsf{-16 \rightarrow -1, -2, -4, -8, \& -16}\\\\\large\text{Thus, your GCF for both of your numbers is: \boxed{- 4 }}\large\checkmark[/tex]
[tex]\large\text{So, let us put }\rm{-4}\large\text{ in front of your parentheses because we had to factor}\\\large\text{out the common like term they shared together.}[/tex]
[tex]\large\text{Now simplify it furthermore}[/tex]
[tex]\huge\text{This leaves your answer as:}[/tex]
[tex]\huge\boxed{\mathsf{-4(3y + 4)}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
How do you solve these step by step (circled problems)
Answers
To create a dot plot, we have to first choose an appropriate scale for the given number line, and then mark a dot over the given numbers.
What is a dot plot?Data points are represented as dots on a graph with an x- and y-axis in a dot plot, sometimes referred to as a strip plot or dot chart, which is a straightforward type of data visualization. These kinds of graphs are employed to visually represent particular data patterns or groups.
To create a dot plot, we have to first choose an appropriate scale for the given number line, and then mark a dot over the given numbers.
For the first question:
Scale on the number line from the first tick, 310 to 335.
For the second question:
Scale on the number line from the first tick, 18 to 31.
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Add or subtract. Write in simplest form. 6 1/3 + 1 2/3 + 5 5/9
Answers
6 1/3 + 1 2/3 + 5 5/9
The first step, what I like to do when I add/subtract mixed fractions is to break them apart, whole numbers and fractions separated:
(6 + 1 + 5) + (1/3 + 2/3 + 5/9)
Next step is to calculate the sum in the parentheses:
(12) + (14/9)
Now convert the improper fraction to a mixed fraction:
(12) + (1 5/9)
Now break the mixed fraction apart again:
(12) + (1) + (5/9)
Add the whole numbers:
(13) + (5/9)
And last step, add them together:
13 5/9
That’s your answer!
I hope this helped! This is the easiest method to learn and use (in my opinion), but is probably longer than some others.
Help me pls!!!!!!!! I NEED HELPPP
Answers
PLEASE HELP...PLEASE
Answers
Answer:
(4,8) (0,2) (-6,-7)
Step-by-step explanation:
Perpendicular means it sort of makes a cross with 90º angle from the already existing line. Imagine it starts at (4,8) coordinate and comes falling to the left, passing through p(-4,-4) coordinate, the points it will go through will be (4,8) (0,2) and (-6,-7) respectively.
A sure way to do this is to mark every point that's on the table below and see which ones form a perpendicular line with the original.
write a function void print int vector(int *ptr, int length) that prints the values in the vector of integers indicated by ptr of length length to the standard output. a vector is a 1-dimensional array. for example, the following c statement declares an integer vector of 10 elements: int avector[10]; each integer should be printed as a decimal number, one entry per line.
Answers
The function void print int vector(int *ptr, int length) prints the values in the vector of integers indicated by ptr of length length to the standard output.
The vector is a 1-dimensional array. The formula for this function is:
For i = 0 to length-1
Print ( ptr[i] )
End
The calculation for this function is as follows:
The function starts with a for loop, where the initial value of i is set to 0, and the condition for the for loop is i < length, and the increment value of i is 1. The body of the for loop prints the value of ptr[i], which is the value at the i-th position in the vector. This loop is repeated until the condition i < length is false, at which point the function terminates.
In summary, the function void print int vector(int *ptr, int length) prints the values in the vector of integers indicated by ptr of length length to the standard output. The formula for this function is a for loop, where the initial value of i is set to 0 and the condition for the for loop is i < length. The body of the for loop prints the value of ptr[i], and the loop is repeated until the condition i < length is false.
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Find domain and range of the relation: (-2,7), (-2,6), (-2,5), (-2,4).
Then determine whether the relation of a function (yes or no).
Answers
The domain and range of the relation are { -2 } and { 4, 5, 6, 7 } respectively. The relation is not a function.
Is the relation a function? What are the domain and range?A function is simply a relationship that maps one input to one output, that is, each x-value can only have one y-value.
Given the relation in the question;
(-2,7), (-2,6), (-2,5), (-2,4)
For a relation to be a function, every x-value must have only one y-value mapped to it.
-2 appeared twice and its corresponding y-values are different.
Hence, the relation is Not a Function.
The Domains are the x-values and the Range are the y-values,
Domain: { -2 }
Range: { 4, 5, 6, 7 }
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7.) +9 – +16 = N
8.) +85 – +12 = N
9.) –6 –2 = N
10.) +13 – +12 = N Someone please need it right now
Answers
Arithmetic operations on the directed numbers gives;
7) -7
8) +73
9) - 8
10) + 1
What are directed numbers?Any given number which has either a positive or negative sign is a directed number. Thus all numbers are directed in one way or the other. Some examples are: 8, +2, - 8.35, 10, 520 etc.
Performing an arithmetic operations on the given question, we have;
7) +9 - + 16 = +9 -(+16)
applying the rule of signs
+9 - (+16) = + 9 - 16
= -7
8) +85 - + 12 = +85 - (+12)
applying the rule of signs
+85 - (+12) = +85 - 12
= +73
9) -6 - 2
applying the rule of signs
-6 -2 = -8
10) +13 - + 12 = +13 - (+12)
applying the rule of signs
+13 - (+12) = +13 - 12
= +1
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Note: Please make sure to properly format your answers. All dollar figures in the answers need to include the dollar sign and any amount over 1,000 should include the comma
($2,354.67). All percentage values in the answers need to include a percentage sign (96). For all items without specific rounding instructions, round your answers to two decimal places,
show both decimal places (5.06).
Use the graph of a piecewise function with three equations.
25,000
20,000
$21,197-
15,000
10,000
$5,448-5,000
$1,045-
0
20,000 40.000 60,000 80,000
$39,800
$10,450
Taxable Income
a. A taxpayer has a taxable income of $39,800. What is her tax?
b.
A taxpayer will owe $21,197. What is his taxable income?
100,000
$102,800
Answers
a. Taxpayer having a taxable income of $39,800 has $5,448 as the income tax.
b. The taxable income of taxpayer owing $21,197 is $102,800.
What is Income Tax?
Income tax is the tax which is levied on all individuals and businesses. The individual income taxes are calculated on the basis of the income received by the individual. Typically, it is categorized as a direct tax because the burden is thought to fall on the people who pay it. Net profits, calculated as the excess of receipts over permitted costs, are subject to corporate income tax.
We are given graph of a piecewise function with three equations which is attached below.
From the graph, it can be observed that the taxpayer who has the taxable income of $39,800 has to pay income tax of $5,448.
Also, a taxpayer owing $21,197 means that he has to pay income tax of $21,197.
His taxable income is $102,800 as shown in the graph below.
Hence, taxpayer having a taxable income of $39,800 has $5,448 as the income tax and the taxable income of taxpayer owing $21,197 is $102,800.
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non-decreasing function over r is continuous at any point except on a countable set using g delta set
Answers
A discontinuity the left and right limits still need to exist but are different.
Since bounded, growing sequences of numbers have limits, a non-decreasing function can only have leap discontinuities, which means that at
Every leap indicates that one interval is missing from the function's range, and these intervals are all disconnected (since the function is non-decreasing, the skipped intervals march up the y-axis).
Since there are only a finite number of discontinuous intervals, there can only be a finite number of discontinuity sites (to see this last bit, just pick a rational number in each interval).
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An aquarium tank can hold 6300 liters of water. There are two pipes that can be used to fill the tank. The first pipe alone can fill the tank in 84 minutes. The second pipe can fill the tank in 42 minutes by itself. When both pipes are working together, how long does it take them to fill the tank?
Answers
It will take 28 minutes for them to fill the tank.
What is work done?Work done is defined as the amount of force needed to move and object a certain distance. In essence, it is a measure of an energy transferred to or from an object which allows it to be moved.
Given that, an aquarium tank can hold 6300 liters of water. There are two pipes that can be used to fill the tank. The first pipe alone can fill the tank in 84 minutes. The second pipe can fill the tank in 42 minutes by itself.
Work done by first pipe in 1 minute = 1/84
Work done by second pipe in 1 minute = 1/42
The time taken by both the pipes together =
1 / 84 + 1 / 42 = 3 / 84
= 1/28
Therefore, time taken by both the pipe = 28 minutes.
Hence, it will take 28 minutes for them to fill the tank.
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in four football games, antoine carried the ball for 98, 146, 84, and 105 yards. how many total yards did antoine have for the four games?
Answers
Answer: she had 433 yards
Step-by-step explanation:
Drag the files to the correct boxes to complete the pairs make each system of linear equations with the correct number of solutions 
Answers
The number of solutions for given three system of linear equations.
We can simply solve this numerical problem by using the following process.
3x - y = 4 ==> a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2
What is meant by numerical?
The phrase "numerical" refers to something that involves numbers.Numerical expression is made up of the two words numerical, which refers to numbers, and expression, which refers to a statement. Consequently, it is a statement that uses numbers.A numerical solution entails speculating about the answer and determining if the issue is sufficiently resolved to end.This formula has numerous complicated terms, making it difficult to recall. Additionally, finding a solution takes time. There are instances when a system has access to a lot of data, and the data can also be analyzed to find a solution. But because there is so much data, finding a solution to these issues takes time.Based on the formula is:
Let the system of equations be a₁x + b₁y = c₁ and a₂x + b₂y = c₂ then
Condition for One solution: a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂
Condition for No Solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Condition for infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂
Now in the given question:
a.) 3x - y = 4
6x - 2y = 8
⇒ a₁/a₂=3/6 = 1/2 ; b₁/b₂ = -1/-2 = 1/2 ; c₁/c₂ = 4/8 = 1/2
⇒ a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2
The given system of equations is coincident lines, hence has infinitely many solutions.
b.) -3x + y = 7
2x - 4y = -8
⇒ a₁/a₂= -3/2 ; b₁/b₂ = 1/-4 ; c₁/c₂ = 7/-8
⇒ a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂
The given system of equations is intersecting lines, hence having a single solution.
c.) y = -4x - 5
y = -4x +1
⇒ a₁/a₂= -4/-4 = 1 ; b₁/b₂ = 1/1 = 1 ; c₁/c₂ = -5/1 = -5
⇒ a₁/a₂ = b₁/b₂ ≠ c₁/c₂
The given system of equations is parallel lines, hence having no solution.
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3 > y-2 graph the solution
Answers
Answer:
12345678901234567890
this week lee got a promotion at work that came with a 5% pay increase. if now his monthly salary is 1732.5, how much was he making before the raise
Answers
Answer:
1650
Step-by-step explanation:
1732.5 / (1+5%) = 1732.5 / 1.05 = 1650
Select the correct answer.
Simplify the following expression.
A.
B.
C.
D.
Answers
4ab^2
It’s spindle focus and try simplifying it
solve the equation: x2+7x=0
Answer: X=
Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s). For example, if you get
and
as your answers, then enter 4,-2/3 in the box.
Answers
The solution to the given equation x² + 7x = 0 is x = 0, -7.
What is a quadratic equation?The quadratic equation is defined as a function containing the highest power of a variable is two.
The quadratic equation is given in the question, as follows:
x² + 7x = 0
We have to determine the solution to the given equation.
As per the question, we have
x² + 7x = 0
x(x + 7) = 0
Equating factors to zero, we get
x = 0 and (x + 7) = 0
x = 0 and x = -7
Thus, the solution to the given equation is x = 0, -7.
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The figure below is a square find the length of size x in simplest radical form with a rational denominator
Answers
x² + x² = 4
2x² = 4
x² = 2
x=√2
hope it helps
What is the distributive property of 32+44?
Answers
Answer:
66
Step-by-step explanation:
32+44=66
32
+42
____
66
peter's international barbershop is a popular haircutting and styling saloon near the campus of the brooklyn college. one barber is available to work full time and spend an average of 4 minutes on each customer. customers arrive all day long at an average rate of 5 minutes arrivals tend to follow the poisson distribution, and service times are exponentially distributed. explain your results. [5 points]
Answers
If we assume that the average rate of arrivals is 5 minutes, then this means that the Poisson distribution will have a mean of 5 customers/minutesThis means that the probability that a customer arrives in the next minute is 0.2 (1/5).
The exponential distribution will have a mean of 4 minutes, meaning that the probability that the customer is served in the next minute is 0.25 (1/4).
Therefore, the expected number of customers that the barber can serve in an hour is 15 (5*4). The expected number of customers served in a given minute is 0.2 (1/5) * 0.25 (1/4) = 0.05. This means that the barber can expect to serve an average of 3 customers per hour.
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It is said that one of the keys to becoming money savvy is to learn how to separate "needs" from "wants". Please explain why this is true. (EXTRA POINTS UR OWN WORDS!!!!!
Answers
Answer:
absolutely
Step-by-step explanation:
Needs are for survival in life like food, rent or mortgage, utilities, and other expenses. Transportation costs, insurance coverage, and any clothing and tools you need for work are included in this part of your budget. A want includes expenses that you can comfortably live without and is not essential for survival.
You plan to drive north from city A to town B and then continue North to city C. The distance between City A and town B is 39mi and the distance between town B and city C is 99mi.
A. Assuming that you follow a straight driving path, after how many miles of driving will you reach the midpoint between city A and city C?
B. If you drive an average of 46/mi/h how long will it take you to drive from city A to city C?
Answers
Here we are given that the distance between city A and B is 39miles and that between city B and C is 99miles . in order to find after how many miles we would reach the midpoint of city A and C , divide the total distance between A and C by 2 as ,
Total distance between A and C =[tex] d_{AB}+d_{BC}[/tex] = 39mi. + 99mi. = 138 miles
Divide this distance by 2 ,
138/2 = 69miles [ answer to Part A ]
_______________
Now the speed is 46miles/hr . In order to find time , use relation,
Speed * time = distance
So that,
time = distance/speed
time = 138/46
time = 3hrs [ answer to Part B ]
and we are done!
Noah had 151 stamps. He had has 42 more stamps than Jonas. How many stamps did Jonas have? How many stamps must Noah give to Jonas so that they each have the same number of stamps?
Answers
=109
No. of stamps that Noah should give to Jona to get equal no. of stamps = 151 -31=130 and 109+31=130 So now they get an equal share.10) A boat travels 30 km up a river in the same time it takes to travel 50 km down the same river. If the current of water is 5km/h. What is the speed of the boat in still water? Be sure to show your work to earn full credit. Here's a chart to get you started: Travel Rate Time Distance
Answers
As a result, when a boat goes 30 km up a river in the same amount of time, its speed in still water is 20 km/h.
what is equation ?The numeral is said to be in this simplest form when divided into its smallest equivalent fraction. How to do to find the most basic form. Find common factors in the denominator and numerators. Verify a fractional integer to verify if it passes as a prime number. A statement having two collinear and an equally sign in the middle is called an equation in mathematics. The general form of any equation contains the degrees of all the variables in descending order. The normal form of a linear function is an x + b = 0. The general form of a linear equation with two variables is an x + b y + c = 0. (or something similar). Equations can be categorised as identities or conditional equations. An identity holds true regardless of the value given to the variables.
given
Let the boat's speed in calm water be x km/h.
Boat speed downstream is equal to x plus 5.
Upstream boat speed is equal to x - 5
Distance x time speed
Time is therefore determined by speed and distance.
Boat travel time is equal to 30/km upstream (x – 5)
50/(x + 5) is the time it takes a boat to travel 50 kilometres downstream.
Since the boat takes the same amount of time in both scenarios, 30 (x + 5) = 50 (x -5), 30 x +150 = 50 (x -250), 20 (x = 400), and 20 (x = 20).
As a result, when a boat goes 30 km up a river in the same amount of time, its speed in still water is 20 km/h.
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