Use DeMoivre's Theorem to find the following. Write your answer in
standard form.
Answers
Answer:
Option C.
Step-by-step explanation:
The given expression is
[tex]\left(\sqrt{2}cis \dfrac{9\pi}{20}\right)^5[/tex]
It can be rewritten as
[tex]\left(\sqrt{2}\cos \dfrac{9\pi}{20}+i\sin\dfrac{9\pi}{20}\right)^5[/tex]
According to De moivre's Theorem:
[tex][r(\cos\theta+i\sin \theta)]^n=r^n(\cos n\theta +i\sin n\theta)[/tex]
Using De moivre's Theorem, we get
[tex](\sqrt{2})^5\left(\cos \dfrac{9\pi\times 5}{20}+i\sin\dfrac{9\pi\times 5}{20}\right)[/tex]
[tex]=4\sqrt{2}\left(\cos \dfrac{9\pi}{4}+i\sin\dfrac{9\pi}{4}\right)[/tex]
[tex]=4\sqrt{2}\left(\cos (2\pi +\dfrac{\pi}{4})+i\sin (2\pi+\dfrac{\pi}{4})\right)[/tex]
[tex]=4\sqrt{2}\left(\cos \dfrac{\pi}{4}+i\sin \dfrac{\pi}{4}\right)[/tex]
[tex]=4\sqrt{2}\left(\dfrac{1}{\sqrt{2}}+i\dfrac{1}{\sqrt{2}}\right)[/tex]
[tex]=4+4i[/tex]
Therefore, the correct option is C.
Related Questions
can someone please help? i don’t know how to solve this.
Answers
Answer:
Hey there!
We have the absolute value of x is greater than two.
Thus, to solve absolute value inequalities, we want to break the inequality down.
We break this down to
x>2
x<-2
Thus, we don't need to simplify this any further, and have our answer.
Hope this helps :)
Answer:
x >= 2 or x <= -2
Step-by-step explanation:
|x|>= 2
Means the absolute value of x is greater than or equal to two.
When we deal with absolute value, we can equivalently break into two cases
+x >= 2 => x >= 2
-x >= 2 => x <= -2
On the number line it looks like what is shown in the attached figure.
Find 4 1/5 · 7 2/3 . a. 32 1/5 b. 32 3/5 c. 28 2/15 d. 28 3/10
Answers
Answer:
Hey there!
4 1/5 · 7 2/3= 32 1/5
Hope this helps :)
Answer:
32 1/5
Step-by-step explanation:
4 1/5 * 7 2/3
Change to improper fractions
( 5*4+1)/5 * ( 3*7+2)/3
21/5 * 23/3
Rewriting
21/3 * 23/5
7 * 23/5
161/5
Change back to a mixed number
5 goes into 161 32 times with 1 left over
32 1/5
ASAP!! can someone help answer these and include work, whoever answers most gets brainliest!! i underlined the questions
Answers
your work for the first one is perfect!!
the second one gives you the side lengths of x+2 and x-5. the formula for the area of a rectangle is l×w so you have to multiply (x+2)(x-5) and im sure you can figure that out but the answer to that is x^2-3x-10
you set up number 3 right, you just need to continue with the distribute property which would give you 4x^2-20x+25
number 4 is the same as the other ones and the answer is 36x^2-1
number 5 is just distributing the -z^3 into everything in the parenthesis giving you -5z^3-z^4+4z^5
I cant see all your work for number 6, but I can see that your term for c is 5, when it is actually -5. it looks like you didnt subtract it to make the quadratic equal to zero and to do that you would have to subtract 5.
tbh I cant read number 7
hope that helped :)
A backpack is on sale for 40% off the regular $50 price. What is the sale price?
$40
$30
$20
$10
Answers
Answer:
$30
Step-by-step explanation:
i think i did the right math
Answer:
Hey there!
40% of 50 dollars is 20
Thus, the price of the backpack is now 50-20, or 30 dollars.
Hope this helps :)
Susan placed $9900 in a savings account which compounds interest continuously at a rate of 1.9 %. How much will she have in the account after 6 years?
Answers
Answer:
$11095.45
Step-by-step explanation:
This is a problem in continuous compounding, for which the formula is
P = Ae^rt, r being the annual interest rate as a decimal fraction and t being the time in years. Here we have:
P = $9900e^(0.019)(6) = $11095.45
Answer:
11095
Step-by-step explanation:
Rounding to the nearest dollar
How do I figure of the 7/3 I'm not sure how to add a fraction with a different denomination
Answers
Step-by-step explanation:
Hi there!!
Here, the thing is we should add them first yaa.
so, let's add them
[tex] = - 2 \frac{1}{6} + \frac{7}{3} [/tex]
by taking lcm we get their lcm is 6 and let's do it according to that,
[tex] = \frac{ - 12 \times 1}{6} + \frac{7 \times 2}{6} [/tex]
now, let's simplify them ok..
[tex] = \frac{ - 12 + 14}{6} [/tex]
so, we get 2/6 (i.e 1/3)
now, the number line must be greator than 1 and less than 1.
Hope it helps...
Suppose you deposit a principal amount of p dollars in a bank account that pays
compound interest. If the annual interest rate r (expressed as a decimal) and the bank
makes interest payments n times every year, the amount of money A you would have
after t years is given by:
A = P[1+ (r/n)]"
Find the account balance after 20 years if you started with a deposit of $1000, and the
bank was paying 4% interest compounded quarterly (4 times a year). Round your
answer to the nearest cent.
Answer:
Answers
Answer:
$2216.72
Step-by-step explanation:
Simply plug in your known variables into the formula:
a = 1000
r = 0.04 (4%)
n = 4
t = 20
A = 1000(1 + 0.04/4)⁴⁽²⁰⁾
A = 1000(1 + 0.01)⁸⁰
A = 1000(1.01)⁸⁰
A = 1000(2.21672)
A = 2216.72
The line L is shown on the grid. Find an equation for L. Give your answer in the form y=mx+c
Answers
Answer:
y=4x-5
Step-by-step explanation:
y=mx+b,
You can identify that the y-intercept is -5.
Using rise/run, there is a rise of +4 and a run of +1, 4/1=4.
Slope = 4
Y-intercept=-5
Slope-intercept form of equation of line is, y = mx + c
Equation of line L is, y = 5x - 5
From given figure, it is observed that line L is passing through point (0, -5) and (1, 0).
First we have to find slope (m) of line.
[tex]m=\frac{0-(-5)}{1-0} =5[/tex]
So, equation of line becomes,
[tex]y=5x+c[/tex]
Since, line passing through (1, 0). Therefore, substituting point (1, 0) in above equation of line.
We get, c = - 5
Thus, equation of line is, [tex]y=5x-5[/tex]
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Which equation could generate the curve in the graph below?
Answers
Answer:
The answer is option 1.
Step-by-step explanation:
In order to find the equation, you have to apply Discriminant Law, D = b² - 4ac. When D < 0, the equation has no real roots (no x-intercept). When D = 0, it has 2 and equal roots (1 x-intercept). When D > 0, it has 2 distinct roots (2 x-intercept).
Option 1,
[tex]y = 3 {x}^{2} - 2x + 1[/tex]
[tex]D = {( - 2)}^{2} - 4(3)(1)[/tex]
[tex]D = - 8[/tex]
[tex]D < 0[/tex]
Option 2,
[tex]y = 3 {x}^{2} - 6x + 3[/tex]
[tex]D = {( - 6)}^{2} - 4(3)(3)[/tex]
[tex]D = 0[/tex]
Option 3,
[tex]y = 3 {x}^{2} - 7x + 1[/tex]
[tex]D = {( - 7)}^{2} - 4(3)(1)[/tex]
[tex]D = 37[/tex]
[tex]D > 0[/tex]
Option 4,
[tex]y = 3 {x}^{2} - 4x - 2[/tex]
[tex]D = {( - 4)}^{2} - 4(3)( - 2)[/tex]
[tex]D = 40[/tex]
[tex]D > 0[/tex]
As you look at the graph, the curve does not meet x-axis so the discriminant must be less than 0.
The given curve in the graph is generated by the equation y =3x² - 2x +1. The correct option is A.
What is coordinate geometry?A coordinate plane is a 2D plane that is formed by the intersection of two perpendicular lines known as the x-axis and y-axis. A coordinate system in geometry is a method for determining the positions of the points by using one or more numbers or coordinates.
In order to find the equation, you have to apply Discriminant Law, D = b² - 4ac. When D < 0, the equation has no real roots (no x-intercept). When D = 0, it has 2 and equal roots (1 x-intercept). When D > 0, it has 2 distinct roots (2 x-intercepts).
Option 1,
y =3x² - 2x +1
D = b² - 4ac
D = ( -2 )² - ( 4 x 3 x 1 )
D = 4 - 12
D = -6
D < 0
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In the circle below, O is the center, MP is a diameter, and m angle POL = 85º. Find the measure of LNP.
Answers
Answer:170
Step-by-step explanation:
The measure of angle LNP is 275 degrees.
What is the inscribed angle theorem?The inscribed angle theorem says that an inscribed angle is half the intercepted arc measure.
In the circle below, O is the center, MP is the diameter, and mangle POL = 85º.
The measure of angle LNP is determined in the following steps given below.
[tex]\rm \angle LNP=360-\angle POL\\\\ \angle LNP=360-85\\\\ \angle LNP=275[/tex]
Hence, the measure of angle LNP is 275 degrees.
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What is the Domain and Range *DON'T FORGET PARENTHESES AND BRACKETS*
Answers
Answer:
D= {-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6}
R={-6,-5,-4,-3,-2,-1,1,2,5)
Step-by-step explanation:
Find the indicated probability. Round to the nearest thousandth.
A study conducted at a certain college shows that 55% of the school's graduates find a job in their chosen field within a year after graduation. Find
the probability that among 7 randomly selected graduates, at least one finds a job in his or her chosen field within a year of graduating.
0.985
0.996
0.550
0.143
Answers
Answer:
[tex]P(At\ least\ 1) = 0.985[/tex]
Step-by-step explanation:
Given
Proportion = 55%
Required
Probability that at least one out of 7 selected finds a job
Let the proportion of students that finds job be represented with p
[tex]p = 55\%[/tex]
Convert to decimal
[tex]p = 0.55[/tex]
Let the proportion of students that do not find job be represented with q
Such that;
[tex]p + q = 1[/tex]
Make q the subject of formula
[tex]q = 1 - p[/tex]
[tex]q = 1 - 0.55[/tex]
[tex]q = 0.45[/tex]
In probability; opposite probabilities add up to 1;
In this case;
Probability of none getting a job + Probability of at least 1 getting a job = 1
Represent Probability of none getting a job with P(none)
Represent Probability of at least 1 getting a job with P(At least 1)
So;
[tex]P(none) + P(At\ least\ 1) = 1[/tex]
Solving for the probability of none getting a job using binomial expansion
[tex](p + q)^n = ^nC_0p^nq^0 + ^nC_1p^{n-1}q^1 +.....+^nC_np^0q^n[/tex]
Where [tex]^nC_r = \frac{n!}{(n-r)!r!}[/tex] and n = 7; i.e. total number of graduates
For none to get a job, means 0 graduate got a job;
So, we set r to 0 (r = 0)
The probability becomes
[tex]P(none) = ^nC_0p^nq^0[/tex]
Substitute 7 for n
[tex]P(none) = \frac{7!}{(7-0)!0!} * p^7 * q^0[/tex]
[tex]P(none) = \frac{7!}{7!0!} * p^7 * q^0[/tex]
[tex]P(none) = \frac{7!}{7! * 1} * p^7 * q^0[/tex]
[tex]P(none) = 1 * p^7 * q^0[/tex]
Substitute [tex]p = 0.55[/tex] and [tex]q = 0.45[/tex]
[tex]P(none) = 1 * 0.55^7 * 0,45^0[/tex]
[tex]P(none) = 0.01522435234[/tex]
Recall that
[tex]P(none) + P(At\ least\ 1) = 1[/tex]
Substitute [tex]P(none) = 0.01522435234[/tex]
[tex]0.01522435234+ P(At\ least\ 1) = 1[/tex]
Make P(At least 1) the subject of formula
[tex]P(At\ least\ 1) = 1 - 0.01522435234[/tex]
[tex]P(At\ least\ 1) = 0.98477564766[/tex]
[tex]P(At\ least\ 1) = 0.985[/tex] (Approximated)
HELP ASAP!
Identify the outlier in the data set and then find the mean and median with and without the outlier.
4, 14, 5, 2, 16, 138, 11
If necessary, round your answers to the nearest tenth.
outlier = 4
mean with outlier = 27.1
median with outlier = 11
mean without outlier = 41.6
median without outlier = 12.5
outlier = 4
mean with outlier = 11
median with outlier = 27.1
mean without outlier = 8
median without outlier = 8.7
outlier = 138
mean with outlier = 27.1
median with outlier = 11
mean without outlier = 8.7
median without outlier = 8
outlier = 138
mean with outlier = 8.7
median with outlier = 8
mean without outlier = 27.1
median without outlier = 11
Answers
Answer:
C. outlier = 138
mean with outlier = 27.1
median with outlier = 11
mean without outlier = 8.7
median without outlier = 8
Step-by-step explanation:
Hey there!
Well if the outlier is 138 then we can cross out choices A and B.
So let's find the mean with and without the outlier.
With - 27.1
Without - 8.7
Median)
With - 11
Without - 8
By using this information we can conclude that the answer is C.
Hope this helps :)
What is the measure of the angle between the minute and the hour hands, when they show 4:20 PM?
Answers
Answer:
Won't they be on the same place, so 360° i think
Answer:
10 degrees
Step-by-step explanation:
15 POINTS!!!!! How is a linear function affected when a constant is multiplied by y?
Answers
Answer:
a constant multiplied by y can affect the linear, because it determines a dependent variables.
Step-by-step explanation:
a linear relationship is a statistical term used to described a straight line relationship between two variables.
1. A man decided to buy new furniture from Futuristie Purniture for $2600, Futuriste Vurniture
gave the man two choices pay the entire amount in one payment with cash, or pay $1000 as a dews
payment and 1120 per month for two full years in the finanong plan. If the man chooses the
financing plan, how much more would he payl
Answers
Answer:
Also, there seem to be many typos in this question, I am trying my best to decipher them all :) next time maybe double check the wording thx!
Step-by-step explanation:
the question doesn't make sense because if you were to pay $1120 per month, for 2 years, you would go over the amount that you bought. I also am not sure what a dews payment is. Try wording it differently/check you spelling.
When you simplify,
10!/(10 - 3)!-3
is equal to:?
Answers
Answer:
120
Step-by-step explanation:
expand and calculate
10x9x8x7!/(10-3)!x3!
10x9x8x7!/7!x3!
reduce
10x9x8x7!/7!x3!
10x9x8/3!
Calculate the product
10x9x8/3!
Now Cacluate the factorial
720/3!
720/6
reduce
120
Answer: 120
Step-by-step explanation:
Select the correct answer from each drop-down menu. In the figure, angle D measures 31° and angle A measures 27°. The measurement of angle B is °. The measurement of angle F is °.
Answers
Answer:
∠B = 63°, ∠F = 59°
Step-by-step explanation:
Given that angle D measures 31° and angle A measures 27°.
From the diagram attached, ∠ E = 90°, ∠G = 90°.
Also ∠C + ∠G = 180° (sum of angles on a straight line)
∠C + 90 = 180
∠C = 180 - 90 = 90°
To find the measure of ∠B and ∠F, we use the triangles ABC and DEF, remember that the sum of all interior angles of a triangle is 180°. In triangle ABC:
∠A + ∠B + ∠C = 180° (angles in a triangle).
27 + ∠B + 90 = 180
∠B + 117 = 180
∠B = 180 - 117
∠B = 63°
Also in triangle DEF:
∠D + ∠E + ∠F = 180° (angles in a triangle).
31 + 90 + ∠F = 180
∠F + 121 = 180
∠F = 180 - 121
∠F = 59°
What value, written as a decimal, should Lena use as the common ratio? Lena is asked to write an explicit formula for the graphed geometric sequence. On a coordinate plane, 3 points are plotted. The points are (1, 1), (2, 2.5), (3, 6.25).
Answers
Answer:
a(n) = 1(2.5)^(n - 1)
Step-by-step explanation:
The geometric sequence is 1, 2.5, 6.25, 15.625, and so on; the first term is 1. The common ratio is 2.5; each new term is 2.5 times the previous term.
The explicit formula desired is thus a(n) = 1(2.5)^(n - 1)
Answer:
ITS CCCCCCC (2.5)
which number line represents the solution to [x-5]=1
Answers
Answer:
Step-by-step explanation:
If there are illustrations to given problems, please share those illustrations. Thank you.
The only value of x for which [x-5]=1 is true is 6: [6-5]=1
Steps to solve:
x - 5 = 1
~Add 5 to both sides
x - 5 + 5 = 1 + 5
~Simplify
x = 6
Best of Luck!
I wish to have $10,000 at the end of 8 years in a bank offering a simple interest rate of 7.5% per year. How much should I deposit in the bank? Please ANSWER CORRECTLY! I will give brainliest for CORRECT answer
Answers
Answer:
$6250
Step-by-step explanation:
6250x.075x8=3750
6250+3750=10,000
Golden Corral charges $11 for a buffet plus $1 for each drink. Western Sizzlin charges $9 for a buffet plus $2 for each drink. Which restaurant has the best deal? (I made up these prices!!) 1. Write 2 equations to represent the 2 restaurants. Predict if these equations will be linear of nonlinear (how do you know?). If they are linear will the lines point upward or downward (how do you know?).
Answers
Answer:
see explanation
Step-by-step explanation:
GC = 11B + 1D
WS = 9B + 2D
Linear equations will point downward
Western Sizzlin is the best deal per person because in total it would be $11 and Golden Corral would be $12.
The lengths of a rectangular garden, the inner rectangle, is 9ft more than its width. For the garden. Let x = width, x + 9 = length. It is surrounded by a brick walkway 4ft wide. Suppose the total area of the walkway is 400ft^2.
a. Write a polynomial to represent the length of PQ
b. What are the dimensions of the garden?
Answers
Answer:
[tex]PQ = x + 13[/tex]
The dimension of the garden is 52.5ft by 43.5ft
Step-by-step explanation:
Given
The shape above
Required
Determine PQ
Determine dimension of the garden
Calculating Length PQ
Represent the length of the inner rectangle as L
Represent the width of the inner rectangle as W
[tex]W = x[/tex]
[tex]L = x + 9[/tex]
The distance between the inner rectangle and the outer triangle is 4ft
This implies that;
[tex]PQ = L + 4[/tex]
Substitute [tex]x + 9[/tex] for [tex]L[/tex]
[tex]PQ = x + 9 + 4[/tex]
[tex]PQ = x + 13[/tex]
Also;
[tex]QR = W + 4[/tex]
Substitute [tex]x[/tex] for [tex]W[/tex]
[tex]QR = x + 4[/tex]
Calculating The Dimension of The Garden
First, we need to determine the Area of the inner rectangle
[tex]Area_1 = L * W[/tex]
Recall that [tex]W = x[/tex] and [tex]L = x + 9[/tex]
So;
[tex]Area_1 = x * (x + 9)[/tex]
For the bigger rectangle
[tex]Area_2 = PQ * QR[/tex]
Recall that [tex]PQ = x + 13[/tex] and [tex]QR = x + 4[/tex]
So;
[tex]Area_2 = (x + 13)(x + 4)[/tex]
Given that the Area of the walkway is [tex]400ft^2[/tex]
This implies that
[tex]Area_2 = Area_1 + 400[/tex]
Substitute [tex]Area_1 = x * (x + 9)[/tex] and [tex]Area_2 = (x + 13)(x + 4)[/tex]
[tex](x + 13)(x + 4) = x * (x + 9) + 400[/tex]
Open All Brackets
[tex]x^2 + 13x + 4x + 52 = x^2 + 9x + 400[/tex]
[tex]x^2 + 17x + 52 = x^2 + 9x + 400[/tex]
Collect Like Terms
[tex]x^2 - x^2 + 17x - 9x = 400- 52[/tex]
[tex]8x = 348[/tex]
Divide both sides by 8
[tex]\frac{8x}{8} = \frac{348}{8}[/tex]
[tex]x = \frac{348}{8}[/tex]
[tex]x =43.5[/tex]
Since the dimensions of the garden is [tex]W = x[/tex] and [tex]L = x + 9[/tex]
Substitute 43.5 for x in both cases
[tex]L = 43.5 + 9 = 52.5[/tex]
[tex]W = 43.5[/tex]
Hence, the dimension of the garden is 52.5ft by 43.5ft
- (1 /-11 )( 77+1100)
Answers
-(1/-11)(77+1100)=107
Answer:
107
Step-by-step explanation:
-(1/-11)(77+1100)
1/11(1177)
=107
PLEASE HELP!!!! I WILL GIVE BRAINLIEST AND LOTS OF POINTS!!! Please provide an explanation and answer only if you know the answer for sure. In the following diagram, BC is parallel to DE. What is the measure of angle х? **Angles are not necessarily drawn to scale.**
Answers
Answer:
x = 29
Step-by-step explanation:
If the lines are parallel
< CBA = < DAB due to the fact that they are alternate interior angles
42 = DAB
We also know that
DAB + BAC + CAE = 180 since they form a straight line
Substituting the given values
42+ 109 + x = 180
Combine like terms
151 + x = 180
151+x-151 = 180-151
x = 29
At the age of 8, Roger received an inheritance of $5,000. His father invested the money in an account bearing 6% annual interest. How long will it take for this inheritance to double in value? Round your answer to 2 decimal places.
Answers
Answer:
16.66 years
Step-by-step explanation:
100(the percentage that it needs to increase) divided by 6 (the percentage of intrest) You could do 1.0/0.06 but i find it eaiser to do it this way
It will take 33.33 years to double his money.
We have roger who inherited $5000 from his father who invested the money in an account bearing 6% annual interest.
We have to find out how long it would take for this inheritance to double in value.
What is the formula to calculate the Simple Interest on the given amount of principal money ?The formula to calculate the Simple Interest on a given amount of principal money (x) at the rate of (r%) for a given amount of time (t) is -
[tex]SI =\frac{x \times r \times t}{100}[/tex]
In the question given : Simple Interest = 2 x Principal = 2[tex]x[/tex]
Hence, the equation becomes -
[tex]2x =\frac{x\times r\times t}{100} \\\\t = \frac{200}{r} \\\\t = \frac{200}{6} \\[/tex]
t = 33.33 years
Hence, it will take 33.33 years to double his money.
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What is the slope of the line ?
Answers
Answer:
The slope is 1
Step-by-step explanation:
there's actually a trick you can do to find the slope. if you go from where the y-intercept is and go over to the x- intercept it will give you the slope hope this helps you
Which of the following theorems verifies that WXY=PME?
A. HA
B. HL
C. LA
D. AA
Answers
HL = hypotenuse leg
We see that WX = PM are the congruent legs (they are the same length due to the similar single tickmark). Also, XY = ME is the congruent pair of hypotenuses. The HL rule only works for right triangles, as the term "hypotenuse" would not apply for a non-right triangle.
The theorem used is HL.
What is theorem?
A theorem is a statement which has been proved true by a special kind of logical argument called a rigorous proof.
Given:
In both triangles
XW = MP (given)
XY = ME (given)
As, hypotenuse of both triangle is equal and leg of one right triangle are congruent to the hypotenuse.
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What is the value of 5 to the power of 4 over 5 to the power of 6? 1 over 25 1 over 5 5 25
Answers
Answer:
1/25
Step-by-step explanation:
[tex]\frac{5^{4} }{5^{6} } =\frac{1}{5^{2} } =\frac{1}{25}[/tex]
suppose that all of the points on the circular dartboard shown are equally likely to be hit by a dart. If the radius of the shaded center circle is 2 and the radius of the entire dartboard is 4, what is the probability of throwing a dart and hitting the white part of the board? Round your answer to the nearest whole number.
Answers
Answer:
P(white region) = 0.75 or 75%
Step-by-step explanation:
Find the area of the shaded region:
A = πr²
A = π(2)²
A = 12.56
Find the area of the entire dartboard:
A = πr²
A = π(4)²
A = 50.24
Find the probability that the dart hits the blue region:
P(blue region) = Area of blue region/ Total Area
P(blue region) = 12.56 / 50.24
P(blue region) = 0.25
As total probability is equal to 1, find the probability of throwing a dart that hits the white part of the board by substracting 0.25 from 1
P(white region) = 1 - P(blue region)
P(white region) = 1 - 0.25
P(white region) = 0.75