Answer:
(2x²+4x-24) in.
Step-by-step explanation:
.
100 POINTS AND BRAINLIEST FOR THIS WHOLE SEGMENT
a) Find zw, Write your answer in both polar form with ∈ [0, 2pi] and in complex form.
b) Find z^10. Write your answer in both polar form with ∈ [0, 2pi] and in complex form.
c) Find z/w. Write your answer in both polar form with ∈ [0, 2pi] and in complex form.
d) Find the three cube roots of z in complex form. Give answers correct to 4 decimal
places.
Answer:
See Below (Boxed Solutions).
Step-by-step explanation:
We are given the two complex numbers:
[tex]\displaystyle z = \sqrt{3} - i\text{ and } w = 6\left(\cos \frac{5\pi}{12} + i\sin \frac{5\pi}{12}\right)[/tex]
First, convert z to polar form. Recall that polar form of a complex number is:
[tex]z=r\left(\cos \theta + i\sin\theta\right)[/tex]
We will first find its modulus r, which is given by:
[tex]\displaystyle r = |z| = \sqrt{a^2+b^2}[/tex]
In this case, a = √3 and b = -1. Thus, the modulus is:
[tex]r = \sqrt{(\sqrt{3})^2 + (-1)^2} = 2[/tex]
Next, find the argument θ in [0, 2π). Recall that:
[tex]\displaystyle \tan \theta = \frac{b}{a}[/tex]
Therefore:
[tex]\displaystyle \theta = \arctan\frac{(-1)}{\sqrt{3}}[/tex]
Evaluate:
[tex]\displaystyle \theta = -\frac{\pi}{6}[/tex]
Since z must be in QIV, using reference angles, the argument will be:
[tex]\displaystyle \theta = \frac{11\pi}{6}[/tex]
Therefore, z in polar form is:
[tex]\displaystyle z=2\left(\cos \frac{11\pi}{6} + i \sin \frac{11\pi}{6}\right)[/tex]
Part A)
Recall that when multiplying two complex numbers z and w:
[tex]zw=r_1\cdot r_2 \left(\cos (\theta _1 + \theta _2) + i\sin(\theta_1 + \theta_2)\right)[/tex]
Therefore:
[tex]\displaystyle zw = (2)(6)\left(\cos\left(\frac{11\pi}{6} + \frac{5\pi}{12}\right) + i\sin\left(\frac{11\pi}{6} + \frac{5\pi}{12}\right)\right)[/tex]
Simplify. Hence, our polar form is:
[tex]\displaystyle\boxed{zw = 12\left(\cos\frac{9\pi}{4} + i\sin \frac{9\pi}{4}\right)}[/tex]
To find the complex form, evaluate:
[tex]\displaystyle zw = 12\cos \frac{9\pi}{4} + i\left(12\sin \frac{9\pi}{4}\right) =\boxed{ 6\sqrt{2} + 6i\sqrt{2}}[/tex]
Part B)
Recall that when raising a complex number to an exponent n:
[tex]\displaystyle z^n = r^n\left(\cos (n\cdot \theta) + i\sin (n\cdot \theta)\right)[/tex]
Therefore:
[tex]\displaystyle z^{10} = r^{10} \left(\cos (10\theta) + i\sin (10\theta)\right)[/tex]
Substitute:
[tex]\displaystyle z^{10} = (2)^{10} \left(\cos \left(10\left(\frac{11\pi}{6}\right)\right) + i\sin \left(10\left(\frac{11\pi}{6}\right)\right)\right)[/tex]
Simplify:
[tex]\displaystyle z^{10} = 1024\left(\cos\frac{55\pi}{3}+i\sin \frac{55\pi}{3}\right)[/tex]Simplify using coterminal angles. Thus, the polar form is:
[tex]\displaystyle \boxed{z^{10} = 1024\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)}[/tex]
And the complex form is:
[tex]\displaystyle z^{10} = 1024\cos \frac{\pi}{3} + i\left(1024\sin \frac{\pi}{3}\right) = \boxed{512+512i\sqrt{3}}[/tex]
Part C)
Recall that:
[tex]\displaystyle \frac{z}{w} = \frac{r_1}{r_2} \left(\cos (\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right)[/tex]
Therefore:
[tex]\displaystyle \frac{z}{w} = \frac{(2)}{(6)}\left(\cos \left(\frac{11\pi}{6} - \frac{5\pi}{12}\right) + i \sin \left(\frac{11\pi}{6} - \frac{5\pi}{12}\right)\right)[/tex]
Simplify. Hence, our polar form is:
[tex]\displaystyle\boxed{ \frac{z}{w} = \frac{1}{3} \left(\cos \frac{17\pi}{12} + i \sin \frac{17\pi}{12}\right)}[/tex]
And the complex form is:
[tex]\displaystyle \begin{aligned} \frac{z}{w} &= \frac{1}{3} \cos\frac{5\pi}{12} + i \left(\frac{1}{3} \sin \frac{5\pi}{12}\right)\right)\\ \\ &=\frac{1}{3}\left(\frac{\sqrt{2}-\sqrt{6}}{4}\right) + i\left(\frac{1}{3}\left(- \frac{\sqrt{6} + \sqrt{2}}{4}\right)\right) \\ \\ &= \boxed{\frac{\sqrt{2} - \sqrt{6}}{12} -\frac{\sqrt{6}+\sqrt{2}}{12}i}\end{aligned}[/tex]
Part D)
Let a be a cube root of z. Then by definition:
[tex]\displaystyle a^3 = z = 2\left(\cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6}\right)[/tex]
From the property in Part B, we know that:
[tex]\displaystyle a^3 = r^3\left(\cos (3\theta) + i\sin(3\theta)\right)[/tex]
Therefore:
[tex]\displaystyle r^3\left(\cos (3\theta) + i\sin (3\theta)\right) = 2\left(\cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6}\right)[/tex]
If two complex numbers are equal, their modulus and arguments must be equivalent. Thus:
[tex]\displaystyle r^3 = 2\text{ and } 3\theta = \frac{11\pi}{6}[/tex]
The first equation can be easily solved:
[tex]r=\sqrt[3]{2}[/tex]
For the second equation, 3θ must equal 11π/6 and any other rotation. In other words:
[tex]\displaystyle 3\theta = \frac{11\pi}{6} + 2\pi n\text{ where } n\in \mathbb{Z}[/tex]
Solve for the argument:
[tex]\displaystyle \theta = \frac{11\pi}{18} + \frac{2n\pi}{3} \text{ where } n \in \mathbb{Z}[/tex]
There are three distinct solutions within [0, 2π):
[tex]\displaystyle \theta = \frac{11\pi}{18} , \frac{23\pi}{18}\text{ and } \frac{35\pi}{18}[/tex]
Hence, the three roots are:
[tex]\displaystyle a_1 = \sqrt[3]{2} \left(\cos\frac{11\pi}{18}+ \sin \frac{11\pi}{18}\right) \\ \\ \\ a_2 = \sqrt[3]{2} \left(\cos \frac{23\pi}{18} + i\sin\frac{23\pi}{18}\right) \\ \\ \\ a_3 = \sqrt[3]{2} \left(\cos \frac{35\pi}{18} + i\sin \frac{35\pi}{18}\right)[/tex]
Or, approximately:
[tex]\displaystyle\boxed{ a _ 1\approx -0.4309 + 1.1839i,} \\ \\ \boxed{a_2 \approx -0.8099-0.9652i,} \\ \\ \boxed{a_3\approx 1.2408-0.2188i}[/tex]
Hi please answer ASAP please and thank you
Answer:
1 1/4
Step-by-step explanation:
2 3/4 - 1 1/2
3 3/4 - 1 2/4
1 1/4
To purchase a car costing $10,000, the buyer bor-
rowed part of the money from the bank at 9% sim-
ple interest and the rest from her mother-in-law at
12% simple interest. If her total interest for the year
was $1080, how much did she borrow from the
bank?
Answer:
She borrowed 4000 from bank.
Step-by-step explanation:
Let 'y' be the amount borrowed from bank. Then 10000-y is the amount borrowed from her mother-in-law.
Let x= interest amount gained by bank . Then 1080- x = interest gained by mother-in-law
I1= interest rate by bank= 9%
I2= interest rate by mother-in-law=12%
Time(T) = 1 year
Now, By Simple Interest formula:
x=PTR/100
Or, x=(y*1*9)/100
Or,100x=9y
or,9y-100x=0...........................Equation (i)
Again 1080-x= ((10000-y)*1*12)/100
Or, 108000-100x=120000-12y
Or, 12y-100x=12000.................Equation(ii)
Solving equation (i) and (ii), we get
y= 4000, which the amount borrowed from bank.
Pls i need help
The continuous growth rate of wind energy per year is ?%
Answer:
Do you need it in percentage, a graph or just normal annual calculation?
Which applies the power of a power rule properly to simplify this expression?
Answer:
D.
Step-by-step explanation:
[7^(-8)]^(-4) =
A power raised to a power: multiply the exponents.
= 7^[(-8)*(4)]
= 7^(32)
Answer: D.
Answer:
7^32, the fourth option
Step-by-step explanation:
For the power rule, if you have a number to a power, for example [tex]x^{a}[/tex] and raise it to another power so it becomes, [tex](x^{a} )^{b}[/tex], We simplify by multiplying the powers together. So the simplified answer would be [tex]x^{(a*b)}[/tex].
Julie assembles shelves for a department store and gets paid $3.25 per shelf. She can assemble 5 per hour and works 8 hours per day. Determine Julie’s gross pay for 1 week
Pay per shelf = $3.25
No of shelfs per hour = 5
Total hours per day = 8
Total days to find pay of = 7
= 3.25×5×8×7
= 910
Therefore she is paid $910 after 1 week.
Must click thanks and mark brainliest
−30=5(x+1)
what is x?
[tex]\\ \rm\Rrightarrow -30=5(x+1)[/tex]
[tex]\\ \rm\Rrightarrow -30=5x+5[/tex]
[tex]\\ \rm\Rrightarrow 5x=-30-5[/tex]
[tex]\\ \rm\Rrightarrow 5x=-35[/tex]
[tex]\\ \rm\Rrightarrow x=\dfrac{-35}{-5}[/tex]
[tex]\\ \rm\Rrightarrow x=7[/tex]
Answer:
x = -7
Step-by-step explanation:
-30 = 5 (x -1 )
5 ( x + 1 ) =-30
5 (x + 1 ) = - 30
5 5
x + 1 = -6
x + 1 -1 = -6 -1
x = - 7
can someone explain step by step how to get the answer?
Answer: x³+8x²+11x-20
Step-by-step explanation:
To find which polynomial has the roots of -5, -4, and 1, we want to first put them into an equation.
-5 is the same as x+5=0
-4 is the same as x+4=0
1 is the same as x-1=0
Now that we have the factors, we can multiply them together.
(x+5)(x+4)(x-1) [FOIL]
(x²+4x+5x+20)(x-1) [combine like terms]
(x²+9x+20)(x-1) [FOIL]
x³-x²+9x²-9x+20x-20 [combine like terms]
x³+8x²+11x-20
Therefore, x³+8x²+11x-20 is the correct polynomial with those roots.
Solve for x.
6(4x+2)= 3(8x+4)
help help help
help
help help
Answer:
R= (-9, -10) S=(-1,-10) T=(-1, -8) U=(-9, -8)
Step-by-step explanation:
There isn't really an explanation it's just reading the points on the graph. Hope this helps!! :)
by selling an article sonu makes a profit of 20%. if the cp decreased by 10% and sp also increased by 10%,calculate her profit percentage
Answer:
37.8 %
Step-by-step explanation:
Let CP = 100
[tex]SP =\frac{100+profit}{100}*CP\\\\=\frac{120}{100}*100[/tex]
SP = 120
New CP:
CP decreased by 10%
[tex]Decreased \ amount=\frac{10}{100}*CP\\\\=\frac{10}{100}*100[/tex]
= 10
New CP = 100- 10 = 90
New SP:
SP increased by 10%
Increase amount = [tex]\frac{10}{100}*old \ SP[/tex]
[tex]= \frac{10}{100}*120\\\\= 12[/tex]
New SP = 120 + 12 = 132
Profit = new SP - new CP
= 132 - 90 = 42
Profit percentage = [tex]\frac{Profit}{CP}*100[/tex]
[tex]= \frac{42}{90}*100\\[/tex]
= 46.67%
Step-by-step explanation:
Here your ans..
HOPE IT HELPS YOU.....
PLEASE MARK ME BRAINLIST.....What is (9.3x10^34)
(3.1x10^17) in scientific notation?
Answer:
3x10^17
Step-by-step explanation:
(9.3/3.1) * 10^(34-17) = 3^17
law of indices, x^m/x^n =x^m-n
a rectangular postage stamp has a length of 3/2 inches and a width of 3/4 inch. what is the area of the stamp in square inches?
Answer:
9/8 or 1.125
Step-by-step explanation:
We want to find the area of a rectangular postage stamp
The area of a rectangle can be found by multiplying the length by the width
Given length: 3/2
Given width: 3/4
Area = 3/2 * 3/4 = 9/8 or 1.125
The area of a 2D form is the amount of space within its perimeter. The area of the stamp in square inches is 1 1/8 inches².
What is an area?The area of a 2D form is the amount of space within its perimeter. It is measured in square units such as cm², m², and so on. To find the area of a square formula or another quadrilateral, multiply its length by its width.
Given that a rectangular postage stamp has a length of 3/2 inches and a width of 3/4 inch. Therefore, the area of the stamp in square inches is,
Area of the stamp = Length × Width
= 3/2 inches × 3/4 inches
= 9/8 inches²
= 1 1/8 inches²
Hence, the area of the stamp in square inches is 1 1/8 inches².
Learn more about the Area here:
https://brainly.com/question/1631786
#SPJ2
How do we derive the sum rule in differentiation? (ie. (u+v)' = u' + v')
It follows from the definition of the derivative and basic properties of arithmetic. Let f(x) and g(x) be functions. Their derivatives, if the following limits exist, are
[tex]\displaystyle f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}h\text{ and }g'(x)\lim_{h\to0}\frac{g(x+h)-g(x)}h[/tex]
The derivative of f(x) + g(x) is then
[tex]\displaystyle \big(f(x)+g(x)\big)' = \lim_{h\to0}\big(f(x)+g(x)\big) \\\\ \big(f(x)+g(x)\big)' = \lim_{h\to0}\frac{\big(f(x+h)+g(x+h)\big)-\big(f(x)+g(x)\big)}h \\\\ \big(f(x)+g(x)\big)' = \lim_{h\to0}\frac{\big(f(x+h)-f(x)\big)+\big(g(x+h)-g(x)\big)}h \\\\ \big(f(x)+g(x)\big)' = \lim_{h\to0}\frac{f(x+h)-f(x)}h+\lim_{h\to0}\frac{g(x+h)-g(x)}h \\\\ \big(f(x)+g(x)\big)' = f'(x) + g'(x)[/tex]
PLEASE HELP Find the value of the following expression (2 ^8 • 3 ^-5 • 6 ^0) ^-2 • (3 ^-2 over 2 ^3 ) ^4 •2 ^28
the answer is very simple if you don't understand ask to your teacher
Find the equation of the linear function represented by the table below in slope-intercept form.
Answer:
y = 3x+1
Step-by-step explanation:
First find the slope
m = ( y2-y1)/(x2-x1)
= (13-4)/(4-1)
= 9/3
= 3
Slope intercept form is
y = mx+b where m is the slope and b is the y intercept
y = 3x+b
Using a point from the table
10 = 3(3)+b
10 =9+b
10-9 =b
1=b
y = 3x+1
Please help me solve this problem guys
Answer:
17%
Step-by-step explanation:
Again, as the amount of years increase, the population of bees gets multiplied by 0.83. We can rewrite this to 83%, and then again rewrite this to 100%-17%. We can see now that the population of bees decreases by 17% each year.
Ray is making his reward winning lemonade recipe for a party he is comparison shopping for lemons at super pioneer supermarket he can buy 4 lemons for 1.60 ray visits keyfood and found 3 lemons cost 1.80 use the table below to compare the values
Answer:
classified info jk juss use a mf calculater
Step-by-step explanation:
Select the correct answer from each drop-down menu.
A company makes cylindrical vases. The capacity, in cubic centimeters, of a cylindrical vase the company produces is given by the
function C() = 6.2873 + 28.26x2, where x is the radius, in centimeters. The area of the circular base of a vase, in square
centimeters, is given by the function A () = 3.14.2
To find the height of the vase, divide
represents the height of the vase.
the expressions modeling functions C(x) and A(z). The expression
Answer:
divide, 2x+9
Step-by-step explanation:
got it right
In a certain class, a teacher distributed a few candies and a few bars among the students such that each student got an equal number of candies and an equal number of bars and no candies or bars remained undistributed. How many students were there in the class
Answer:
C BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
Step-by-step explanation:
In a certain class, a teacher distributed a few candies and a few bars among the students such that each student got an equal number of candies and an equal number of bars and no candies or bars remained undistributed. How many students were there in the class?
(1) The teacher distributed 180 candies and 40 bars.
(2) The total number of items received by each student was less than 20.
A Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient to answer the question asked.
B Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient to answer the question asked.
C BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D EACH statement ALONE is sufficient to answer the question asked.
E Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
180 candies and 40 bars
The highest common factor of 180 candies and 40 bars = 20
A survey was conducted by asking 120 students in a town how they traveled to school.
The following pie chart shows the result of the survey
Car 30%
Cycle 25%
Walk 10%
Bus ?
What are the number of students that travel to school by bus
Answer:
42
Step-by-step explanation:
30+25+10=65%
bus=35%
35/100×120=42
BUS=42
Find the volume of the cylinder please
ASAP
Answer:
33ft^3
Step-by-step explanation:
radius is half the diameter, half of 2=1 and 1^2=1
3(1)(11)=33
Answer: V = 33 ft³
Step-by-step explanation:
π = 3
r = (1/2)d = (1/2) (2) = 1 ft
h = 11 ft
Given Formula
V = π r² h
Substitute values into the formula
V = (3) (1)² (11)
Simplify exponents
V = (3) (1) (11)
Simplify by multiplication
V = 33 ft³
Hope this helps!! :)
Please let me know if you have any questions
find the measure of the indicated angle to the nearest degree
Answer:
43
Step-by-step explanation:
First you divide the opposite side from the angle by the hypotenuse.
33/44
then you take the inverse sine of 33/44, resulting in
43.43°, which rounds to 43°
Please help I’ll mark as brainlist
Answer:
Ekta and Preyal
Step-by-step explanation:
The weekly wages of employees of Volta gold are normally distributed about a mean of$1250 with a standard deviation of $120. Find the probability of an employee having a weekly wage lying 1) between $1320 and $970 2) over $1290 3) under $1400
Answer:
1) 0.7104 = 71%
2) 0.6615 = 66%
3) 0.8944 = 89%
Step-by-step explanation:
1)
Z(low)=-2.333 0.009815329
Z(upper)=0.583 0.720165536
2)
Z(low)=0.333 0.63055866
Z(upper)=8322.908 1
3)
Z(low)=-10.417 0
Z(upper)=1.25 0.894350226
what expression represents 19 more than 6 times a number, n
1. 19n+6
2. 6n-19
3.6n+19
4.19n-6
Answer:
6n+19
Step-by-step explanation:
6 times a number, n
6n
19 more
6n+19
Answer:
6n + 19
Step-by-step explanation:
Since the expression is 6 times n
We need to have 6 x n or simply just 6n
Since the expression is 19 more than 6n
We need to have 6n + 19
Classify the triangle as acute, right, or obtuse and as equilateral, isosceles, or scalene.
9514 1404 393
Answer:
(d) Right, scalene
Step-by-step explanation:
The little square in the upper left corner tells you that is a right angle. Any triangle with a right angle is a right triangle. This one is scalene, because the sides are all different lengths.
__
Additional comment
An obtuse triangle cannot be equilateral, and vice versa.
An equilateral triangle has all sides the same length, and all angles the same measure: 60°. It is an acute triangle.
On two investments totaling $9,500, Peter lost 3% on one and earned 7% on the other. If his net annual receipts were $169, how much was each investment?
Answer:
23$ was each investment
Step-by-step explanation:
[tex]\sqrt{x} x^{2}[/tex] +3
help me pls??????? :)
Answer:4 in each bad 2 left over
Step-by-step explanation:
Answer:
4 in each bag and 2 left over
Step-by-step explanation:
divide 14 by 3
3 goes into 14, 4 times
14 - 12 = 2
4 in each bag and then 2 left over
A child is picking 3 days of the 7 days of the week to have Jello for lunch. What is the size of the sample space in this experiment?
