The complete question is;
The polynomial P(x) = 5x²(x − 1)³(x + 9) has degree ____. It has zeros 0, 1, and ____. The zero 0 has multiplicity ____ , and the zero 1 has multiplicity ____
Answer:
A) degree = 6
B) -9 is also a zero of the polynomial
C) 0 has multiplicity of 2
1 has multiplicity of 3
Step-by-step explanation:
A) To find the degree of the polynomial, we will first have to identify each term [term is for example (x - 1)³]. Thus, to find the degree of each term we will add the exponents.
The terms are;
5x², (x - 1)³, (x + 9)
The exponents are, 2, 3 and 1 respectively.
Thus, degree = 2 + 3 + 1 = 6
B) A zero of a polynomial is the value of x that causes the polynomial function to equal 0.
Since 0 and 1 are zeros, looking at the polynomial P(x) = 5x²(x − 1)³(x + 9), we can tell that when the term which when x = 0 makes the polynomial 0 is 5x².
Similarly, the term which when x = 1 makes the polynomial 0 is (x - 1)³
Thus,we are left with the term (x + 9)
So for the polynomial to be zero, (x + 9) = 0
Thus,x = -9
So -9 is a zero of the polynomial
C) The zero 0 is from the term 5x².
Thus,the multiplicity is the highest power of x which is 2.
The zero 1 is from the term (x - 1)³. Thus, the multiplicity is the highest power of x which is 3
A cola-dispensing machine is set to dispense 11 ounces of cola per cup, with a standard deviation of 1.0 ounce. The manufacturer of the machine would like to set the control limit in such a way that, for samples of 35, 5% of the sample means will be greater than the upper control limit, and 5% of the sample means will be less than the lower control limit.Required:a. At what value should the control limit be set?b. If the population mean shifts to 10.7, what is the probability that the change will be detected?c. If the population mean shifts to 11.7, what is the probability that the change will be detected?
Answer:
a. the control limits should be set at (10.72, 11.28)
b. [tex]\mathbf{P(10.72<x<11.28) = 0.4526}[/tex]
c. [tex]\mathbf{P(10.72<x<11.28) = 0.0065}[/tex]
Step-by-step explanation:
Given that:
population mean μ = 11
standard deviation [tex]\sigma[/tex] = 1.0
sample size n = 35
5% of the sample means will be greater than the upper control limit, and 5% of the sample means will be less than the lower control limit.
Therefore, level of significance ∝ = 0.05+0.05 = 0.10
Critical value for [tex]z_{1-\alpha/2} =z_{1-0.10 /2}[/tex]
[tex]\implies z_{1-0.05} = z_{0.95}[/tex]
Using the EXCEL FORMULA: = NORMSINV (0.95)
z = 1.64
The lower control limit and the upper control limit can be determined by using the respective formulas:
Lower control limit = [tex]\mathtt{\mu - z_{1-\alpha/2} \times \dfrac{\sigma}{\sqrt{n}}}[/tex]
Upper control limit = [tex]\mathtt{\mu + z_{1-\alpha/2} \times \dfrac{\sigma}{\sqrt{n}}}[/tex]
For the lower control limit = [tex]11-1.64 \times \dfrac{1.0}{\sqrt{35}}[/tex]
For the lower control limit = [tex]11-0.27721[/tex]
For the lower control limit = 10.72279
For the lower control limit [tex]\simeq[/tex] 10.72
For the upper control limit = [tex]11+1.64 \times \dfrac{1.0}{\sqrt{35}}[/tex]
For the upper control limit = 11 + 0.27721
For the upper control limit = 11.27721
For the upper control limit [tex]\simeq[/tex] 11.28
Therefore , the control limits should be set at (10.72, 11.28)
b. If the population mean shifts to 10.7, what is the probability that the change will be detected?
i.e
[tex]\mathtt{P(10.72<x<11.28) = P( \dfrac{X- \mu}{\dfrac{\sigma}{\sqrt{n}}}<z < \dfrac{X- \mu}{\dfrac{\sigma}{\sqrt{n}}}})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P( \dfrac{10.72- 10.7}{\dfrac{1.0}{\sqrt{35}}}<z < \dfrac{11.28- 10.7}{\dfrac{1.0}{\sqrt{35}}}})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P( \dfrac{0.02}{\dfrac{1.0}{5.916}}<z < \dfrac{0.58}{\dfrac{1.0}{5.916}}})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P(0.1183<z < 3.4313})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P(z< 3.4313) - P(z< 0.1183) }[/tex]
Using the EXCEL FORMULA: = NORMSDIST (3.4313) - NORMSDIST (0.118 ); we have:
[tex]\mathbf{P(10.72<x<11.28) = 0.4526}[/tex]
c If the population mean shifts to 11.7, what is the probability that the change will be detected?
i.e
[tex]\mathtt{P(10.72<x<11.28) = P( \dfrac{X- \mu}{\dfrac{\sigma}{\sqrt{n}}}<z < \dfrac{X- \mu}{\dfrac{\sigma}{\sqrt{n}}}})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P( \dfrac{10.72- 11.7}{\dfrac{1.0}{\sqrt{35}}}<z < \dfrac{11.28- 11.7}{\dfrac{1.0}{\sqrt{35}}}})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P( \dfrac{-0.98}{\dfrac{1.0}{5.916}}<z < \dfrac{-0.42}{\dfrac{1.0}{5.916}}})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P(-5.7978<z < -2.48472})[/tex]
[tex]\mathtt{P(10.72<x<11.28) = P(z< -2.48472) - P(z< -5.7978) }[/tex]
Using the EXCEL FORMULA: = NORMSDIST (-2.48472) - NORMSDIST (-5.7978); we have:
[tex]\mathbf{P(10.72<x<11.28) = 0.0065}[/tex]
Which of the following is not a real number?
Answer:
im pretty sure its the -3 one
Step-by-step explanation:
Answer:
The answer is A, Square root of -3 is not a real number.
Step-by-step explanation: You can take the square root of positive numbers, so we can eliminate choices C and D. We can take the square root of 0, which would equal 0, so B is incorrect. However, We cannot take the square root of negative numbers, so choice A is the answer for this question.
Let f(x)=x+8 and g(x)= x2-6x-7 find f(g2)
Answer:
-7.
Step-by-step explanation:
g(x) = x^2 - 6x - 7
g(2) = 2^2 - 6(2) - 7
= 4 - 12 - 7
= -8 - 7
= -15
f(x) = x + 8
f(-15) = (-15) + 8
= 8 - 15
= -7
Hope this helps!
3
BO
Evaluate the function f(x) = x2 + 4x + 1 at the given values of the independent variable and simplify.
a. f(6)
b. f(x +9)
c. f(-x)
Answer:
a) f(6)=(6)^2+4(6)+1=65
b)f (x+9)=(x+9)^2+4 (x+9)+1=(x^2+18x+81)+(4x+36)+1=x^2+22x+117
f (-x)=(-x)^2-4x+1
An expression is ???
Answer:
s-6
Step-by-step explanation:
difference means subtract
s-6
The quotient of 3 and the cube
of y+2
Answer:
[tex]\dfrac{3}{(y+2)^3}[/tex]
Step-by-step explanation:
Maybe you want this written using math symbols. It will be ...
[tex]\boxed{\dfrac{3}{(y+2)^3}}[/tex]
Jean Paul is an interior designer who is working with a difficult client. Part of his design requires that he put 11 colored vases in a row on a shelf. He has 3 identical blue vases, 2 identical green vases, 4 identical red vases, a purple vase and a yellow vase. He has put up 4 different arrangements of the vases that his client complained about. As he begins to put up the fifth arrangement, he wonders how many different arrangements he might have to go through before his client complains about all of them. How many different arrangements could Jean Paul make
Answer:
138600 arrangements
Step-by-step explanation:
Let n = 11
The different arrangements Jean Paul can make = n!/(4!)(3!)(2!)
Hence, 11!/(4!)(3!)(2!) = 1663200/12 = 138600
bon
Question 21
O pts
The recipe for a s'more is as follows:
1 graham cracker
chocolate bar
2 marshmallows
If you have 10 graham crackers, 7 marshmallows, and 5 chocolate bars, how many
complete s'mores can you make using this recipe?
Question 22
O pts
Answer:
3 s'mores
Step-by-step explanation:
If we center our attention on how many marshmallows you need per s'more which is 2 and you only have 7 you can only make 3 with one marshmallow remaining.
Marcelina uses a blend of white corn and yellow corn to make tortilla chips at her restaurant. She needs to buy 50kg of corn in total for her next order. White corn costs $0.30 per kilogram, yellow corn costs $0.15 per kilogram, and she wants to spend $12.00 in total. Here's a graph that shows a system of equations for this scenario where x is the amount of white corn she buys and y is the amount of yellow corn she buys.
Answer:
https://brainly.com/question/17155330
Step-by-step explanation:
Question:
Marcelina uses a blend of white corn and yellow corn to make tortilla chips at her restaurant. She needs to buy 50kg of corn in total for her next order. White corn costs $0.30 per kilogram, yellow corn costs $0.15 per kilogram, and she wants to spend $12 total.
What does point F represent in this context?
Answer:
Marcelina spends less that the intended amount of money and buy less than enough corn
Determine the length of chord BC. 1) 17.45 2) 30.96 3) 67.06 4) 33.53
Answer:
33.53
Step-by-step explanation:
OB is a radius of the circle, and OC is also a radius of the circle, so both are equal length. That makes ΔOBC an isosceles triangle.
If we cut ΔOBC in half, the angle formed is 125° / 2 = 62.5°.
Therefore:
sin 62.5 = (x/2) / 18.9
x = 37.8 sin 62.5
x ≈ 33.53
Answer:
33.5
Step-by-step explanation:
Given the following hypotheses: H0: μ = 490 H1: μ ≠ 490 A random sample of 15 observations is selected from a normal population. The sample mean was 495 and the sample standard deviation 9. Using the 0.01 significance level:
a.) State the decision rule.
b.) Compute the value of the test statistic.
c.) What is your decision regarding the null hypothesis?
Answer:
We conclude that the population mean is equal to 490.
Step-by-step explanation:
We are given that a random sample of 15 observations is selected from a normal population. The sample mean was 495 and the sample standard deviation 9.
Let [tex]\mu[/tex] = population mean.
So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] = 490 {means that the population mean is equal to 490}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu\neq[/tex] 490 {means that the population mean is different from 490}
The test statistics that will be used here is One-sample t-test statistics because we don't know about population standard deviation;
T.S. = [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_1_4[/tex]
where, [tex]\bar X[/tex] = sample mean = 495
s = sample standard deviation = 9
n = sample of observations = 15
So, the test statistics = [tex]\frac{495-490}{\frac{9}{\sqrt{15} } }[/tex] ~ [tex]t_1_4[/tex]
= 2.152
The value of t-test statistics is 2.152.
Now, at a 0.01 level of significance, the t table gives a critical value of -2.977 and 2.977 at 14 degrees of freedom for the two-tailed test.
Since the value of our test statistics lies within the range of critical values of t, so we have insufficient evidence to reject our null hypothesis as the test statistics will not fall in the rejection region.
Therefore, we conclude that the population mean is equal to 490.
PLEASE HELP- MATH
simplify the fraction
5bc/10b^2
[tex]\dfrac{5bc}{10b^2}=\dfrac{\not 5\cdot \not b\cdot c}{2\cdot \not 5\cdot \not b\cdot b}=\dfrac{c}{2b}[/tex]
Answer:
c / ( 2b)
Step-by-step explanation:
5bc/10b^2
Lets look at the numbers first
5/10 = 1/2
Then the variable b
b / b^2 = 1/b
Then the variable c
c/1 = c
Putting them back together
1/2 * 1/b * c/1
c/ 2b
donald is a taxi driver. for each ride in the taxi, the cost, c, is given by c = 500+130d, where c is in cents and d is the distance of the ride, in miles. what is the meaning of the value 500 in this equation? a) donald charges 500 cents per mile b) donald drives 500 customers per day c) donald charges at least 500 cents per taxi ride d) donald charges at most 500 cents per taxi ride
can u go to my page real quick and answer my question pls
Find the (a) mean, (b) median, (c) mode, and (d) midrange for the data and then (e) answer the given question. Listed below are the weights in pounds of 1111 players randomly selected from the roster of a championship sports team. Are the results likely to be representative of all players in that sport's league?
278 303 186 292 276 205 208 236 278 198 208
a. Find the mean.
The mean is ? pound(s).
(Type an integer or a decimal rounded to one decimal place asneeded.)
b. Find the median.
The median is ? pound(s).
(Type an integer or a decimal rounded to one decimal place asneeded.)
c. Find the mode.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The mode(s) is(are) ? pound(s).
(Type an integer or a decimal. Do not round. Use a comma to separate answers as needed.)
B. There is no mode.
d. Find the midrange.
The midrange is ? pound(s).
(Type an integer or a decimal rounded to one decimal place asneeded.)
e. Are the results likely to be representative of all players in that sport's league?
A. The results are not likely to be representative because the median is not equal to the mode.
B. The results are likely to be representative because a championship team is most likely representative of the entire league.
C. The results are not likely to be representative because the median is not equal to the mean.
D. The results are not likely to be representative because the championship team may not be representative of the entire league.
Answer:
Mean= 242.5 pounds
Median= 236 pounds
Mode= 208 and 278 pounds
Range=117 pounds
Mid-range= 58.5 pounds
B. The results are likely to be representative because a championship team is most likely representative of the entire league.
Step-by-step explanation:
278 303 186 292 276 205 208 236 278 198 208
Arranged in ascending order is
186 198 205 208 208 236 276 278 278 292 303
Mean = (186 +198+ 205+ 208 +208 +236 + 276+ 278 +278+ 292 +303)/11
Mean =2668/11
Mean= 242.5 pounds
Median = the middle number
Median= 236 pounds
Mode = highest occuring number(s)
Mode= 208 and 278 pounds
Range= highest number- smallest number
Range=303-186
Range=117 pounds
Mid-range= range/2
Mid-range= 117/2
Mid-range= 58.5 pounds
Fifteen chaperones went on a field trip with 225 students. Which fraction represents the number of chaperones to students on the field trip
Answer:
15/225
Step-by-step explanation:
The number of chaperones for the group of students can be represented with a ratio - 15:225 or 15/225.
Because there are 15 chaperones for the 225 students, you can state what the ratio does - for every 225 students, there are 15 chaperones.
However, 15/225 can be reduced to 1/15, so for every 15 students, there is 1 chaperone.
One of two small restaurants is chosen at random with equally likely probability, and then an employee is chosen at random from the chosen restaurant. Restaurant #1 has 10 full-timers and 6 part-timers. Restaurant #2 has 7 full-timers and 9 part-timers. What is the probability that Restaurant #1 was chosen at random, given that a full-time employee was chosen? Your answers should be rounded to 4 digits after the decimal.
Answer:
P(1 |F) = 10/17
Step-by-step explanation:
Let events
1 = restaurant 1
2 = restaurant 2
F = full-time worker chosen
P = part-time worken chosen
P(1 and F) = 1/2 * 10/16 = 5/16
P(2 and F) = 1/2 * 7/16 = 7/32
P( (1 or 2) and F ) = P(F) = 5/16+7/32 = 17/32
P(1 | F) Probability of choosing restaurant 1 given a full-time was chosen
= P(1 and F) / P(F)
= 5/16 / (17/32)
= 5/16 * 32/17
= 10 / 17
Which equation is equivalent to StartRoot x EndRoot + 11 = 15?
Answer:
D [tex]\sqrt{x} +11=15[/tex]
Step-by-step explanation:
Edge 2020
For the given expression √x + 11 = 15 the value of x will be equal to 16.
The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the expression √x + 11 =15. The expression will be solved as below,
√x + 11 =15
√x = 15 - 11
√x = 4
Squaring on both sides of the equation,
x = 4²
x = 16
To know more about an expression follow
https://brainly.com/question/25968875
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Determine if the process appears to be within statistical control. If not, state the reason why not.
a. It does not appear to be within statistical control because there is an upward shift.
b. It appears to be within statistical control.
c. It does not appear to be within statistical control because there is an upward trend.
d. It does not appear to be within statistical control because there is increasing variation.
Answer:
c. It does not appear to be within statistical control because there is an upward trend.
Step-by-step explanation:
Statistical process control is a method for quality control which employs statistical method to monitor and control process. It ensures operation efficiency and ensuring required specification to reduce wastes in production lines. Here the process variation is out of control because the statistical control has an upward trend.
the product of two consecutive positive integer is 306
Answer:
[tex]\Large \boxed{\sf 17 \ and \ 18}[/tex]
Step-by-step explanation:
The product means multiplication.
There are two positive consecutive integers.
Let the first positive consecutive integer be x.
Let the second positive consecutive integer be x+1.
[tex](x) \times (x+1) =306[/tex]
Solve for x.
Expand brackets.
[tex]x^2 +x =306[/tex]
Subtract 306 from both sides.
[tex]x^2 +x -306=306-306[/tex]
[tex]x^2 +x -306=0[/tex]
Factor left side of the equation.
[tex](x-17)(x+18)=0[/tex]
Set factors equal to 0.
[tex]x-17=0[/tex]
[tex]x=17[/tex]
[tex]x+18=0[/tex]
[tex]x=-18[/tex]
The value of x cannot be negative.
Substitute x=17 for the second consecutive positive integer.
[tex](17)+1[/tex]
[tex]18[/tex]
The two integers are 17 and 18.
The product of two consecutive positive integers is 306.
We need to find the integers
solution : Let two consecutive numbers are x and (x + 1)
A/C to question,
product of x and (x + 1) = 306
⇒x(x + 1) = 306
⇒x² + x - 306 = 0
⇒ x² + 18x - 17x - 306 = 0
⇒x(x + 18) - 17(x + 18) = 0
⇒(x + 18)(x - 17) = 0⇒ x = 17 and -18
so x = 17 and (x +1) = 18
Therefore the numbers are 17 and 18.
Hope it helped u if yes mark me BRAINLIEST
TYSM!
if the current time is 10:35 how long until it turns 3:15
Answer:
10:35-3:15
5 hoursThe average score of all golfers for a particular course has a mean of 70and a standard deviation of 5.Suppose 100golfers played the course today. Find the probability that the average score of the 100golfers exceeded 71.Round to four decimal places.
Answer:
0.9773
Step-by-step explanation:
Here, we start by calculating the z-scores statistic
Mathematically;
z-score = (x-mean)/SD/√n
From the question, we have;
x = 71, mean = 70, SD = 5 and n = 100
Plugging these values in the equation above, we have;
z-score = (71-70)/5/√100 = 1/5/10 = 1/0.5 = 2
So the probability we want to calculate is that;
P(z > 2)
This is obtainable from the standard normal distribution table
P(z > 2) = 0.97725 which is 0.9773 to 4 decimal places
A regression was run to determine if there is a relationship between hours of TV watched per day (x) and number of situps a person can do (y). The results of the regression were:
1. y=a+bx
2. b=-0.669
3. a=27.41
4. r2=0.760384
5. r=-0.872
Use this to predict the number of situps a person who watches 3.5 hours of TV can do. Round to one decimal place.
Answer: The correct answer is 19 sit ups.
Step-by-step explanation: Given that the regression equation to find a a relationship between hours of TV watched per day (x) and number of situps a person can do (y) was done.
The result was
y = ax+b
Correlation coefficient = 0.865
To predict the number of situps a person who watches 3 hours of TV
y = -1.23(3)+22.738
= 19.048
Approximately 19 situps.
what is the least number to be added to 1500 to make it a perfect square?
Answer:
21
Step-by-step explanation:
√1500 ≈ 38.7
round that up to 39 and square it:
39² = 1521
A blue die and a red die are thrown. B is the event that the blue comes up with a 6. E is the event that both dice come up even. Write the sizes of the sets |E ∩ B| and |B|a. |E ∩ B| = ___b. |B| = ____
Answer:
Size of |E n B| = 2
Size of |B| = 1
Step-by-step explanation:
I'll assume both die are 6 sides
Given
Blue die and Red Die
Required
Sizes of sets
- [tex]|E\ n\ B|[/tex]
- [tex]|B|[/tex]
The question stated the following;
B = Event that blue die comes up with 6
E = Event that both dice come even
So first; we'll list out the sample space of both events
[tex]B = \{6\}[/tex]
[tex]E = \{2,4,6\}[/tex]
Calculating the size of |E n B|
[tex]|E n B| = \{2,4,6\}\ n\ \{6\}[/tex]
[tex]|E n B| = \{2,4,6\}[/tex]
The size = 3 because it contains 3 possible outcomes
Calculating the size of |B|
[tex]B = \{6\}[/tex]
The size = 1 because it contains 1 possible outcome
Q1) Two balls are randomly selected without replacement from a box containing three black balls numbered 1, 2, 3 and two white balls numbered 4 and 5. Assuming that all outcomes are equally likely. Find out the probabilities of following events. a) Probability that the color of second ball is white. b) Probability that the color of second ball is black. c) Probability that both balls are black. d) Probability that both balls are white.
[tex]|\Omega|=5\cdot4=20[/tex]
a)
[tex]|A|=3\cdot2+2\cdot1=8\\\\P(A)=\dfrac{8}{20}=\dfrac{2}{5}[/tex]
b)
[tex]|A|=3\cdot2+2\cdot3=12\\\\P(A)=\dfrac{12}{20}=\dfrac{3}{5}[/tex]
c)
[tex]|A|=3\cdot2=6\\\\P(A)=\dfrac{6}{20}=\dfrac{3}{10}[/tex]
d)
[tex]|A|=2\cdot1=2\\\\P(A)=\dfrac{2}{20}=\dfrac{1}{10}[/tex]
g The intersection of events A and B is the event that occurs when: a. either A or B occurs but not both b. neither A nor B occur c. both A and B occur d. All of these choices are true. a. b. c. d.
Answer:
c. both A and B
Step-by-step explanation:
Given that there are two events A and B.
To find:
Intersection of the two sets represents which of the following events:
a. either A or B occurs but not both
b. neither A nor B occur
c. both A and B occur
d. All of these choices are true. a. b. c. d
Solution:
First of all, let us learn about the concept of intersection.
Intersection of two events means the common part in the two events.
Explanation using set theory:
Let set P contains the outcomes of roll of a dice.
P = {1, 2, 3, 4, 5, 6}
And set Q contains the set of even numbers less than 10.
Q = {2, 4, 6, 8}
Common elements are {2, 4, 6}
So, intersection of P and Q:
[tex]P \cap Q[/tex] = {2, 4, 6}
Explanation using Venn diagram:
Please refer to the image attached in the answer area.
The shaded region is the intersection of the two sets P and Q.
When we apply the above concept in events, we can clearly say from the above explanation that the intersection of two events A and B is the event that occurs when both A and B occur.
So, correct answer is:
c. both A and B
Answer:
C.
Step-by-step explanation:
Evaluate fx, fy, fz at the given point a) f (x, y z) = x³yz² at the point (1, 2, 3) b) f (x, y, z) = x² - 2xy + 3yxz² at the point (3, 1, -2)
Answer:
a) (fx, fy, fz) = (54, 9, 12)b) (fx, fy, fz) = (16, 30, 9)Step-by-step explanation:
a) The partial derivatives of f(x, y, z) = x³yz² are ...
fx = 3x²yz²fy = x³z²fz = 2x³yzAt the given point, these are ...
fx(1, 2, 3) = 3(1²)(2)(3²) = 54fy(1, 2, 3) = (1³)(3²) = 9fz(1, 2, 3) = 2(1³)(2)(3) = 12__
b) The partial derivatives of f(x, y, z) = x² -2xy +3xyz² are ...
fx = 2x -2y +3yz²fy = -2x +3xz²fz = 3xyAt the given point, these are ...
fx(3, 1, -2) = 2(3) -2(1) +3(1)(-2)² = 16fy(3, 1, -2) = -2(3) +3(3)(-2)² = 30fz(3, 1, -2) = 3(3)(1) = 9Express the quotient of z1 and z2 in standard form given that [tex]z_{1} = 6[cos(\frac{2\pi }{5}) + isin(\frac{2\pi }{5})][/tex] and [tex]z_{2} = 2\sqrt{2} [cos(\frac{-\pi }{2}) + isin(\frac{-\pi }{2})][/tex]
Answer:
Solution : - 2.017 + 0.656i
Step-by-step explanation:
The quotient of the two expressions would be the following,
[tex]6\left[\cos \left(\frac{2\pi }{5}\right)+i\sin \left(\frac{2\pi \:}{5}\right)\right][/tex] ÷ [tex]2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi \:}{2}\right)\right][/tex]
So if we want to determine this expression in standard complex form, we can first convert it into trigonometric form, then apply trivial identities. Either that, or we can straight away apply the following identities and substitute,
( 1 ) cos(x) = sin(π / 2 - x)
( 2 ) sin(x) = cos(π / 2 - x)
If cos(x) = sin(π / 2 - x), then cos(2π / 5) = sin(π / 2 - 2π / 5) = sin(π / 10). Respectively sin(2π / 5) = cos(π / 2 - 2π / 5) = cos(π / 10). Let's simplify sin(π / 10) and cos(π / 10) with two more identities,
( 1 ) [tex]\cos \left(\frac{x}{2}\right)=\sqrt{\frac{1+\cos \left(x\right)}{2}}[/tex]
( 2 ) [tex]\sin \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos \left(x\right)}{2}}[/tex]
These two identities makes sin(π / 10) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], and cos(π / 10) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex].
Therefore cos(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], and sin(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex]. Substitute,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi \:}{2}\right)\right][/tex]
Remember that cos(- π / 2) = 0, and sin(- π / 2) = - 1. Substituting those values,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[0-i\right][/tex]
And now simplify this expression to receive our answer,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[0-i\right][/tex] = [tex]-\frac{3\sqrt{5+\sqrt{5}}}{4}+\frac{3\sqrt{3-\sqrt{5}}}{4}i[/tex],
[tex]-\frac{3\sqrt{5+\sqrt{5}}}{4}[/tex] = [tex]-2.01749\dots[/tex] and [tex]\:\frac{3\sqrt{3-\sqrt{5}}}{4}[/tex] = [tex]0.65552\dots[/tex]
= [tex]-2.01749+0.65552i[/tex]
As you can see our solution is option c. - 2.01749 was rounded to - 2.017, and 0.65552 was rounded to 0.656.
Which sequence has a common ratio of 2? a{20, 40, 80, 160, 320, 640, …} b{20, 10, 5, 2.5, 1.25, 0.625, …} c{20, 15, 10, 5, 0, -5, …} d{20, 4, 0.80, 0.16, 0.032, 0.0064, …}
Answer:
A
Step-by-step explanation:
40/20=2
80/40=2
Therefore the common ration is 2
The correct sequence which has a common ratio of 2 is,
⇒ {20, 40, 80, 160, 320, 640, …}
What is Geometric sequence?An sequence has the ratio of every two successive terms is a constant, is called a Geometric sequence.
Given that;
The common ratio of sequence is,
⇒ 2
Now, By option 1;
The sequence is,
⇒ {20, 40, 80, 160, 320, 640, …}
Hence, Common ratio = 40 / 20
= 2
And, 80 / 40 = 2
Thus, The correct sequence which has a common ratio of 2 is,
⇒ {20, 40, 80, 160, 320, 640, …}
Learn more about the geometric sequence visit:
https://brainly.com/question/25461416
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Question (1)
(i) Explain Pythagorean theorem in detail.
(ii) What is "Hippacus of Croton"?
Answer:
● The pythagorian theorem
The pythagorian theorem is used to find a missing side of a right triangle.
It states that the square of the hypotenus of a right triangle is equal to the sum of the squares of the two other sides.
Let a be the hypotenus, b and c are the othet sides:
☆☆☆☆☆ a^2 = b^2 + c^2☆☆☆☆☆
There are more than 350 way to prove this theorem.
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● Hippasus of Croton was a member of the highly-secretive school og Pythagoras in Croton. He is credited in history as the first person to prove the existence of irrational numbers.
