Answer:
[tex]f(x) = x^4 + 9x^3 + 27x^2 + 27x[/tex]
Step-by-step explanation:
Zeros of a function:
Given a polynomial f(x), this polynomial has roots [tex]x_{1}, x_{2}, x_{n}[/tex] such that it can be written as: [tex]a(x - x_{1})*(x - x_{2})*...*(x-x_n)[/tex], in which a is the leading coefficient.
- 3 as a zero of multiplicity 3
So
[tex]f(x) = (x - (-3))^3 = (x + 3)^3 = x^3 + 9x^2 + 27x + 27[/tex]
0 as a zero of multiplicity 1.
So
[tex]f(x) = x(x^3 + 9x^2 + 27x + 27) = x^4 + 9x^3 + 27x^2 + 27x[/tex]
(Use 1 for the leading coefficient.)
Multiply the polynomial by 1, so it stays the same. The polynomial in expanded form is:
[tex]f(x) = x^4 + 9x^3 + 27x^2 + 27x[/tex]
The lifetimes of light bulbs of a particular type are normally distributed with a mean of 350 hours and a standard deviation of 6 hours. What percentage of the bulbs have lifetimes that lie within 1 standard deviation to either side of the mean?
Answer:
By the Empirical Rule, approximately 68% of the bulbs have lifetimes that lie within 1 standard deviation to either side of the mean.
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed random variable:
Approximately 68% of the measures are within 1 standard deviation of the mean.
Approximately 95% of the measures are within 2 standard deviations of the mean.
Approximately 99.7% of the measures are within 3 standard deviations of the mean.
What percentage of the bulbs have lifetimes that lie within 1 standard deviation to either side of the mean?
By the Empirical Rule, approximately 68% of the bulbs have lifetimes that lie within 1 standard deviation to either side of the mean.
If a tank holds 6000 gallons of water, which drains from the bottom of the tank in 50 minutes, then Toricelli's Law gives the volume V of water remaining in the tank after t minutes as
V=5000 (1-1/50*t)^2 0⤠t ⤠50.
1. Find the rate at which water is draining from the tank after the following amount of time. (Remember that the rate must be negative because the amount of water in the tank is decreasing.)
a. 5 min
b. 10 min
c. 20 min
d. 50 min
2. At what time is the water flowing out the fastest?
3. At what time is the water flowing out the slowest?
Answer: hello from the question the volume of tank = 6000 gallons while the value in the Torricelli's equation = 5000 hence I resolved your question using the Torricelli's law equation
answer:
1) a) -180 gallons/minute ,
b) -160 gallons/minute
c) -120 gallons/minute
d) 0
2) The water is flowing out fastest when t = 5 min
3) The water is flowing out slowest after t = 20 mins
Step-by-step explanation:
Volume of tank = 5000 gallons
Time to drain = 50 minutes
Volume of water remaining after t minutes by Torricelli's law
V = 5000 ( 1 - [tex]\frac{1}{50}t[/tex] )^2 ----- ( 1 )
1) Determine the rate at which water is draining from the tank
First step : differentiate equation 1 using the chain rule to determine the rate at which water is draining from the tank
V' = [tex]-10000[ ( 1 - \frac{1}{50}t ) (\frac{1}{50}) ][/tex]
a) After t = 5minutes
V' = - 10000[ ( 1 - 0.1 ) * ( 0.02 ) ]
= -180 gallons/minute
b) After t = 10 minutes
V' = - 10000[ ( 1 - 0.2 ) * ( 0.02 ) ]
= - 160 gallons/minute
c) After t = 20 minutes
V' = - 10000 [ ( 1 - 0.4 ) * ( 0.02 ) ]
= -120 gallons/minute
d) After t = 50 minutes
V' = - 10000 [ ( 1 - 1 ) * ( 0.02 ) ]
= 0 gallons/minute
2) The water is flowing out fastest when t = 5 min
3) The water is flowing out slowest after t = 20 mins because no water flows out after 50 minutes
For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. The library director in Owensboro, Kentucky feels this is not true, so she asked a local college statistic class to conduct a survey. The class randomly selected 100 patrons and found that 82 borrowed books. Did the class demonstrate that the percentage was higher in Owensboro, KY? Use α = 0.01 level of significance. What is the possible proportion of patrons that do borrow books from the Owensboro Library?
Answer:
The p-value of the test is 0.0007 < 0.01, which means that the class demonstrates that the percentage was higher in Owensboro, KY.
The possible proportion of patrons that do borrow books from the Owensboro Library is 0.82.
Step-by-step explanation:
For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. Test if the proportion is higher in Owensboro, KY.
At the null hypothesis, we test if the proportion is of at most 0.67, that is:
[tex]H_0: p \leq 0.67[/tex]
At the alternative hypothesis, we test if the proportion is of more than 0.67, that is:
[tex]H_1: p > 0.67[/tex]
The test statistic is:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, [tex]\sigma[/tex] is the standard deviation and n is the size of the sample.
0.67 is tested at the null hypothesis:
This means that [tex]\mu = 0.67, \sigma = \sqrt{0.67*0.33}[/tex]
The class randomly selected 100 patrons and found that 82 borrowed books.
This means that [tex]n = 100, X = \frac{82}{100} = 0.82[/tex]
Value of the test statistic:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \frac{0.82 - 0.67}{\frac{\sqrt{0.67*0.33}}{\sqrt{100}}}[/tex]
[tex]z = 3.19[/tex]
P-value of the test and decision:
The p-value of the test is the probability of a finding a sample proportion of 0.82 or above, which is 1 subtracted by the p-value of z = 3.19.
Looking at the z-table, z = 3.19 has a p-value of 0.9993.
1 - 0.9993 = 0.0007
The p-value of the test is 0.0007 < 0.01, which means that the class demonstrates that the percentage was higher in Owensboro, KY.
What is the possible proportion of patrons that do borrow books from the Owensboro Library?
The sample proportion of 0.82.
Given the following data from a repeated-measures design study examining the effect of a treatment by measuring a group of 9 participants before and after they received treatment:
Participant Before After
A 8 7
B 7 5
C 6 6
D 7 6
E 9 7
F 8 5
G 5 4
H 9 4
I 7 4
a. Calculate the difference scores and MD.
b. Compute SS, sample variance, and estimated standard error.
c. Is there a significant treatment effect?
Answer:
MD = 2
SS = 18
SAMPLE VARIANCE = 2.25
STANDARD ERROR = 0.5
Step-by-step explanation:
Given :
A 8 7
B 7 5
C 6 6
D 7 6
E 9 7
F 8 5
G 5 4
H 9 4
I 7 4
Difference, d = Before - After
______ d
A 8 7 __ 1
B 7 5 __ 2
C 6 6 __ 0
D 7 6 __ 1
E 9 7 __ 2
F 8 5 __ 3
G 5 4 __ 1
H 9 4 __5
I 7 4 ___3
The mean of difference, MD ;
MD = Σd/ n = (1+2+0+1+2+3+1+5+3) / 9 = 18 / 9 = 2
The sum of square, SS ;
(1 - 2)^2 + (2 - 2)^2 + (0 - 2)^2 + (1 - 2)^2 + (2 - 2)^2 + (3 - 2)^2 + (1 - 2)^2 + (5 - 2)^2 + (3 - 2)^2 = 18
Sample variance, S² = SS/(N-1) = 18 / (9 - 1) = 18 / 8 = 2.25
Sample standard deviation, S = √Variance = √2.25 = 1.5
Standard Error, S.E = S / √n = 1.5 / √9 = 0.5
Test statistic : MD / S.E = 2 / 0.5 = 4
We test at α = 0.05 since no α - value is stated in the question.
Critical value at 0.05, df = 8 ;
Critical value = 2.306
Since; Test statistic > Critical value, then result is significant at α = 0.05
Please,look at this one.
9514 1404 393
Answer:
x = √2
Step-by-step explanation:
A graph indicates the only solution is near x=√2.
__
Square both sides, separate the radical and do it again.
[tex]\displaystyle(2-x)\sqrt{\frac{x+2}{x-1}}=\sqrt{x}+\sqrt{3x-4}\qquad\text{given}\\\\(2-x)^2\cdot\frac{x+2}{x-1}=x+(3x-4)+2\sqrt{x(3x-4)}\qquad\text{square}\\\\\frac{(2-x)^2(x+2)}{x-1}-4x+4=2\sqrt{x(3x-4)}\qquad\text{isolate radical}\\\\\left(\frac{(2-x)^2(x+2)-4(x-1)^2}{x-1}\right)^2=x(3x-4)\qquad\text{square}\\\\(x^3-6x^2+4x+4)^2=4(x-1)^2(3x^2-4x)\qquad\text{multiply by $(x-1)^2$}[/tex]
Now, we can put this polynomial equation into standard form and factor it.
[tex]x^6 -12x^5+32x^4-76x^2+48x+16=0\\\\(x-2)^2(x^2-2)(x^2-8x-2)=0\qquad\text{factor it}\\\\x\in \{2,\pm\sqrt{2},4\pm3\sqrt{2}\}[/tex]
The original equation requires that we restrict the domain of possible solutions. In order for the radicals to be non-negative, we must have x ≥ 4/3. In order for the left side of the equation to be non-negative, we must have x ≤ 2. So, the only potential solutions will be in the interval [4/3, 2].
The only values in the above list that match this requirement are {√2, 2}. We know that the right side of the equation cannot be zero, so the value x=2 is also an extraneous solution.
The only solution is x = √2.
_____
Additional comment
For solving higher-degree polynomials, I like to use a graphing calculator to help me find the roots. The second attachment shows the roots of the 6th-degree polynomial. They can help us factor the equation. (There are also various machine solvers available that will show factors and roots.)
Mr. Allway’s math class surveyed all the seventh-grade students to find out their favorite sport. The following circle graph shows a breakdown of the survey findings.
Find the number of degrees represented by Basketball.
108°
101°
11°
140°
Answer:
101 is the answer of the question
Answer:
101 degrees
Step-by-step explanation:
First you add all the percentages
39 + 28 + 30 + 3 = 100%
To find the number of degrees of basketball you multiply 28% by 360 because it’s a circle.
28/100 * 360 = 10,080/100 = 100.8 ~ 101
The average breaking strength of a certain brand of steel cable is 2000 pounds, with a standard deviation of 100 pounds. A sample of 20 cables is selected and tested. Find the sample mean that will cut off the upper 95% of all samples of size 20 taken from the population. Assume the variable is normally distributed.
Answer:
The sample mean that will cut off the upper 95% of all samples of size 20 taken from the population is of 1963.2 pounds.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The average breaking strength of a certain brand of steel cable is 2000 pounds, with a standard deviation of 100 pounds.
This means that [tex]\mu = 2000, \sigma = 100[/tex]
A sample of 20 cables is selected and tested.
This means that [tex]n = 20, s = \frac{100}{\sqrt{20}} = 22.361[/tex]
Find the sample mean that will cut off the upper 95% of all samples of size 20 taken from the population.
This is the 100 - 95 = 5th percentile, which is X when Z has a p-value of 0.05, so X when Z = -1.645. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-1.645 = \frac{X - 2000}{22.361}[/tex]
[tex]X - 2000 = -1.645*22.361[/tex]
[tex]X = 1963.2[/tex]
The sample mean that will cut off the upper 95% of all samples of size 20 taken from the population is of 1963.2 pounds.
Which of the following rational functions is graphed below?
o
A. F(x) = 1/2x
B. AX) = 1/x-2
C. F(x) = 1/x+2
Answer:
Option B.
Step-by-step explanation:
We can see that we have an asymptote at x = 2
Remember that in a rational function, the asymptote is at the x-value such that the denominator is equal to zero.
So, the denominator is something like:
(x + a)
we have that the denominator is zero when x = 2
Then:
(2 + a) = 0
solving that for a, we get:
a = -2
Then the denominator of the rational function is:
(x - 2)
For the given options, the only one with this denominator is option B, then the correct option is B.
Answer:
B. f(x) = 1/x-2
Step-by-step explanation:
Math is ez bro.
find all the missing measurement
Answer:
Hello,
|FD|=15
Step-by-step explanation:
Since the triangles are similar, the bissects are also.
k*35=21 ==> k=21/35
k*25=|FD|
|FD|=(21/35)*25=15
How many numbers multiple of 3 are in the range [2,2000]?
Answer: so there are 666 multiples of 3 between 2 and 2000.
Step-by-step explanation:
the smallest number = 3 which is 3*1. The largest number is = 1998 = 3*666
multiples of 3 between {2,2000} = 666-1+1 = 666
When a teacher in a private school points out to her high school principal that since there are empty seats in all classrooms, the cost of additional students is really zero, she is using the
Answer:
The answer is "Principal of marginal analysis".
Step-by-step explanation:
To determine unless the benefits of even an aggressive resource would outweigh its costs, and therefore increase utility, individuals and businesses can use a valuation model to compare the risks versus the benefits of more activities, like whether to create or consuming more. It's the amount during which net value is greater than or equal to marginal cost that's the optimal quantity in this situation. The amount where the marginal social cost curve and consumer surplus line connect.
Help me with this math problem !!!
Answer:
multiply the numerator together and denominator together
A simple random sample of 27 filtered 100-mm cigarettes is obtained from a normally distributed population, and the tar content of each cigarette is measured. The sample has a standard deviation of 0.20 mg. Use a 0.05 significance level to test the claim that the tar content of filtered 100-mm cigarettes has a standard deviation different from 0.30 mg, which is the standard deviation for unfiltered king-size cigarettes. Complete parts (a) through (d) below. a. What are the null and alternative hypotheses?
Answer:
The null hypothesis is [tex]H_0: \sigma = 0.3[/tex]
The alternative hypothesis is [tex]H_1: \sigma \neq 0.3[/tex]
Step-by-step explanation:
Test if the tar content of filtered 100-mm cigarettes has a standard deviation different from 0.30 mg.
At the null hypothesis, we test if the standard deviation is of 0.3, that is:
[tex]H_0: \sigma = 0.3[/tex]
At the alternative hypothesis, we test if the standard deviation is different of 0.3, that is:
[tex]H_1: \sigma \neq 0.3[/tex]
What is the distance from point N to LM in the figure below?
N
8.4
8.1
7.8
O
O A. 3.11
B. 0.8
C. 8.1
D. 2.18
E. 7.8
F. 8.4
Answer:
the answer to your question is 7.8 (E)
The distance from point N to LM is 7.8, 8.1 and 8.4 unit.
What is perpendicular?Perpendicular lines are those that cross at a straight angle to one another. Examples include the opposite sides of a rectangle and the steps of a straight staircase. the icon used to represent two parallel lines.
Perpendicular lines are two separate lines that cross one other at a right angle, or a 90° angle.
Given:
In ΔNOM
The Perpendicular distance is 7.8 unit
and, Hypotenuse distance id 8.1 unit
Now, In ΔNOL
The Perpendicular distance is 7.8 unit
and, Hypotenuse distance id 8.4 unit
Learn more about perpendicular here:
https://brainly.com/question/29268451
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Convert 37.5% to a fraction. (Reduce your answer to lowest terms.)
Answer:
3/8
Step-by-step explanation:
The value of 37.5% reduced to the lowest terms is 15/40.
What is a fraction?A fraction is written in the form p/q, where q ≠ 0. Fractions are of two types they are proper fractions and improper fractions. Fractions can also be converted into percentages by multiplying them by 100.
We know e can convert percentages into decimals and fractions by dividing the percentage value by hundred.
Given we have to convert 37.5% as a fraction which is,
= 37.5/100.
= 375/1000.
= 75/200.
= 15/40.
learn more about percentages here :
https://brainly.com/question/24159063
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The correlation coefficient, r, between the ages of employees, x, and the number of sick days taken per year, y, equals 0.81.
Complete the statement based on the information provided.
The value of r is
✔ positive
and is relatively close to
✔ 1
, so the variables are
✔ closely
associated. It appears that, as the age of an employee increases, the number of sick days taken
✔ increases
.
Answer:
✔ positive
✔ 1
✔ closely
✔ increases
ED2021
At a hockey game, a vender sold a combined total of 192 sodas and hot dogs. The number of sodas sold was two times the number of hot dogs sold. Find the
number of sodas and the number of hot dogs sold.
Answer:
they sold 64hotdogs and 128 sodas
Step-by-step explanation:
2x+x=192 3x=192 x=64
Find the Sample size for 99% confidence level with a margin of error of 4% and p unknown.
Answer:
za/2: Divide the confidence level by two, and look that area up in the z-table: .95 / 2 = 0.475. ...
E (margin of error): Divide the given width by 2. 6% / 2. ...
: use the given percentage. 41% = 0.41. ...
: subtract. from 1.
Graph 5 and it’s opposite on the number line
Answer:
See pic below.
Step-by-step explanation:
The opposite of 5 is -5.
Which is an example of using an open-ended question to uncover a problem? O a) "Do you have a problem you'd like addressed today?" b) "Is there a problem? C) "What seems to be the problem?" O d) "Can I help you?"
Answer:
(D)
Step-by-step explanation:
i think dat is an open ended question
Find the missing ? Step by step need it
Answer:
30 degrees
Step-by-step explanation:
Use the Tan formula.
write
the following numbers using Roman numerals 20
Step-by-step explanation:
xx is the Roman number of 20
Solve x/4 > 2 Question 10 options: x ≥ 8 x < –8 x > 8 x ≤ –8
Answer:
x > 8
Step-by-step explanation:
You can start y multiplying both sides by 4 to cancel out the division by 4:
x/4 > 2
*4 *4
x > 8
Answer:
x > 8
Step-by-step explanation:
x/4 > 2
=> x > 2 × 4
=> x > 8
Solve the equation by factoring: 5x^2 - x = 0
Answer:
Step-by-step explanation:
x = 0, 1/5
Which equation does the graph of the systems of equations solve?
two linear functions intersecting at 3, negative 2
−one thirdx + 3 = x − 1
one thirdx − 3 = −x + 1
−one thirdx + 3 = −x − 1
one thirdx + 3 = x − 1
Answer:
-1/3x+3 = x-1
Step-by-step explanation:
The solution is (3,-2)
Check and see if the point solves the equation
-1/3x+3 = x-1
-1/3(3) +3 = 3-1
-1+3 = 3-1
2=2 yes
Answer:
C
Step-by-step explanation:
write your answer in simplest radical form
Answer:
[tex]9\sqrt{3}[/tex]
Step-by-step explanation:
This is a 30-60-90 triangle.
It's good to remember this. The side length opposite to the 60 degree angle is always the base multiplied by [tex]\sqrt{3}[/tex]
Answer:
9√3.
Step-by-step explanation:
tan 60 = √3
So w/9 =√3
w = 9√3
Volunteering: The General Social Survey asked 1309 people whether they performed any volunteer work during the past year. A total of 518 people said they did. Part: 0 / 30 of 3 Parts Complete Part 1 of 3 (a) Find a point estimate for the population proportion of people who performed volunteer work in the past year. Round the answer to at least three decimal places. The point estimate for the proportion of people who performed volunteer work in the past year is .
Answer:
The point estimate for the population proportion of people who performed volunteer work in the past year is 0.396.
Step-by-step explanation:
Point estimate of a proportion:
Proportion is the number of desired outcomes divided by the number of total outcomes.
518 out of 1309 people performed volunteer work:
This means that:
[tex]p = \frac{518}{1309} = 0.396[/tex]
The point estimate for the population proportion of people who performed volunteer work in the past year is 0.396.
PLS HELP !! Is the following a fair sampling of the contents of the jar? Why?
Pour a 2” layer of lentils into a jar. Then pour a 2” layer of kidney beans into the jar. Then pour a 2” layer of pinto beans into the jar. Stir the contents of the jar well. Then pull out a handful of beans.
Consider points a, b, and c in the graph. Determine which of these points is relative maxima on the interval x = –1 and x = 2 in the graph.
Given:
The graph of a function is given.
To find:
The point that is the relative maxima on the interval x = –1 and x = 2 in the graph.
Solution:
Relative maxima: It is the maximum point of a function over a short interval.
From the given graph it is clear that the graph of the function over the interval x = –1 and x = 2 has a relative maxima at (0,0).
Clearly, (0,0) is represented by point a.
So, the point a is the relative maxima on the interval x = –1 and x = 2 in the graph.
Therefore, the correct option is A.
Emma went out shopping with her father and bought a dress that cost $40.00. In class, she
learned to find the sales tax by multiplying by .08 (the sales tax in her state is 8%). Emma
found the tax, and then added the tax to the original amount. Emma's mother suggested that
she should just multiply the cost of the dress by 1.08 and that this method would give her the
final answer with the tax included. Emma was confused. Who is right? Work it out both ways
and explain your thinking.
Answer:
Both ways are correct
If you multiply the cost by 8% and add, you will still get 108% as your total.
Answer:
Both ways are correct
Step-by-step explanation:
Father's way is ( 40 × 0.08 ) + 40 = $43.2
Mothers way is 40 × 1.08 = $43.2
