Answer:
[tex]\left\{11, 21\right\} \times \left\{1, 1.5, 2\right\} = \left\{ (11, 1), (11, 1.5), (11, 2), (21, 1), (21, 1.5), (21, 2)\right\}[/tex]
Step-by-step explanation:
The Cartesian product between two discrete sets, is given by all possible ordered pairs originated with the combinations of the elements of the two sets, thus the requested Cartesian product is:
[tex]\left\{11, 21\right\} \times \left\{1, 1.5, 2\right\} = \left\{ (11, 1), (11, 1.5), (11, 2), (21, 1), (21, 1.5), (21, 2)\right\}[/tex]
[tex]A = (11, 1)\\B = (11, 1.5)\\C = (11, 2)\\D = (21, 1)\\E = (21, 1.5)\\F = (21, 2)\\[/tex]
You can see the attached file
Use the power series method to solve the given initial-value problem. (Format your final answer as an elementary function.)
(x − 1)y'' − xy' + y = 0, y(0) = −3, y'(0) = 2
[tex]y=\displaystyle\sum_{n\ge0}a_nx^n=a_0+\sum_{n\ge1}a_nx^n[/tex]
[tex]y'=\displaystyle\sum_{n\ge0}(n+1)a_{n+1}x^n=a_1+\sum_{n\ge0}(n+1)a_{n+1}x^n[/tex]
[tex]y''=\displaystyle\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n[/tex]
Notice that [tex]y(0)=-3=a_0[/tex], and [tex]y'(0)=2=a_1[/tex].
Substitute these series into the ODE:
[tex](x-1)y''-xy'+y=0[/tex]
[tex]\displaystyle\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^{n+1}-\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge0}(n+1)a_{n+1}x^{n+1}+\sum_{n\ge0}a_nx^n=0[/tex]
Shift the indices to get each series to include a [tex]x^n[/tex] term.
[tex]\displaystyle\sum_{n\ge1}n(n+1)a_{n+1}x^n-\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge1}na_nx^n+\sum_{n\ge0}a_nx^n=0[/tex]
Remove the first term from both series starting at [tex]n=0[/tex] to get all the series starting on the same index [tex]n=1[/tex]:
[tex]\displaystyle-2a_2+a_0+\sum_{n\ge1}n(n+1)a_{n+1}x^n-\sum_{n\ge1}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge1}na_nx^n+\sum_{n\ge1}a_nx^n=0[/tex]
[tex]\displaystyle-2a_2+a_0+\sum_{n\ge1}\bigg[n(n+1)a_{n+1}-(n+1)(n+2)a_{n+2}-(n-1)a_n\bigg]x^n=0[/tex]
The coefficients are given recursively by
[tex]\begin{cases}a_0=-3\\a_1=2\\\\a_n=\dfrac{(n-2)(n-1)a_{n-1}-(n-3)a_{n-2}}{n(n-1)}&\text{for }n>1\end{cases}[/tex]
Let's see if we can find a pattern to these coefficients.
[tex]a_2=\dfrac{a_0}2=-\dfrac32=-\dfrac3{2!}[/tex]
[tex]a_3=\dfrac{2a_2}{3\cdot2}=-\dfrac12=-\dfrac3{3!}[/tex]
[tex]a_4=\dfrac{2\cdot3a_3-a_2}{4\cdot3}=-\dfrac18=-\dfrac3{4!}[/tex]
[tex]a_5=\dfrac{3\cdot4a_4-2a_3}{5\cdot4}=-\dfrac1{40}=-\dfrac3{5!}[/tex]
[tex]a_6=\dfrac{4\cdot5a_5-3a_4}{6\cdot5}=-\dfrac1{240}=-\dfrac3{6!}[/tex]
and so on, suggesting that
[tex]a_n=-\dfrac3{n!}[/tex]
which is also consistent with [tex]a_0=3[/tex]. However,
[tex]a_1=2\neq-\dfrac3{1!}=-3[/tex]
but we can adjust for this easily:
[tex]y(x)=-3+2x-\dfrac3{2!}x^2-\dfrac3{3!}x^3-\dfrac3{4!}x^4+\cdots[/tex]
[tex]y(x)=5x-3-3x-\dfrac3{2!}x^2-\dfrac3{3!}x^3-\dfrac3{4!}x^4+\cdots[/tex]
Now all the terms following [tex]5x[/tex] resemble an exponential series:
[tex]y(x)=5x-3\displaystyle\sum_{n\ge0}\frac{x^n}{n!}[/tex]
[tex]\implies\boxed{y(x)=5x-3e^x}[/tex]
The given differential equation is a second-order homogeneous differential equation. The power series method may not straightforwardly work for this equation due to the x dependence in the coefficients. Even using advanced techniques like the Frobenius method, the solution cannot be expressed as an elementary function.
Explanation:The given differential equation (x − 1)y'' - xy' + y = 0 is an example of a second order homogeneous differential equation. To solve the equation using the power series method, let's assume a solution of the form y = ∑(from n=0 to ∞) c_n*x^n. Substituting this into the equation and comparing coefficients, we can find a relationship for the c_n's and thus the power series representation of the solution.
However, for this particular differential equation, the power series method is not straightforward because of the x dependence in the coefficients of y'' and y'. Therefore, the standard power series approach would not work, and we would need more advanced techniques like the Frobenius method which allows for non-constant coefficients.
Unfortunately, even with the Frobenius method, the solution isn't an elementary function, meaning the solution cannot be expressed in terms of a finite combination of basic arithmetic operations, exponentials, logarithms, constants, and solutions to algebraic equations.
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The quantity demanded x for a certain brand of MP3 players is 100 units when the unit price p is set at $100. The quantity demanded is 1100 units when the unit price is $50. Find the demand equation.
P=
Answer:
x = 2100 - 20p
Step-by-step explanation:
Let the quantity demanded be 'x'
unit price be 'p'
thus, from the given relation in the question, we have
p (100) = $100
and,
p (1100) = $50
now, from the standard equation for the line
[tex]\frac{\textup{p - p(100)}}{\textup{x - 100}}[/tex] = [tex]\frac{\textup{p(1100) - p(100)}}{\textup{50 - 100}}[/tex]
or
[tex]\frac{\textup{p - 100}}{\textup{x - 100}}[/tex] = [tex]\frac{\textup{50 - 100}}{\textup{1100 - 100}}[/tex]
or
1000 × (p - 100) = - 50 × ( x - 100 )
or
20p - 2000 = - x + 100
or
x = 2100 - 20p
The demand equation for the MP3 players can be determined using the given data points. By setting up a system of linear equations and solving for the values of a and b, we can find the demand equation Qd = 1500 - 10P.
Explanation:The demand equation can be determined using the given information. We know that when the price is set at $100, the quantity demanded is 100 units, and when the price is $50, the quantity demanded is 1100 units.
We can set up a linear demand equation in the form Qd = a + bP, where Qd is the quantity demanded and P is the unit price. Using the two data points, we can solve for the values of a and b.
Substituting the first data point (100 units at $100) into the equation, we get 100 = a + b(100).Substituting the second data point (1100 units at $50) into the equation, we get 1100 = a + b(50).We now have a system of linear equations that we can solve to find the values of a and b.Solving the system of equations, we find that a = 1500 and b = -10.Therefore, the demand equation is Qd = 1500 - 10P.
1. The volume of a cube is increasing at a rate of 1200 cm/min at the moment when the lengths of the sides are 20cm. How fast are the lengths of the sides increasing at that [10] moment?
Answer:
[tex]1\,\,cm/min[/tex]
Step-by-step explanation:
Let V be the volume of cube and x be it's side .
We know that volume of cube is [tex]\left ( side \right )^{3}[/tex] i.e., [tex]x^3[/tex]
Given :
[tex]\frac{\mathrm{d} V}{\mathrm{d} t}=1200\,\,cm^3/min[/tex]
[tex]x=20\,\,cm[/tex]
To find : [tex]\frac{\mathrm{d} x}{\mathrm{d} t}[/tex]
Solution :
Consider equation [tex]V=x^3[/tex]
On differentiating both sides with respect to t , we get
[tex]\frac{\mathrm{d} V}{\mathrm{d} t}=3x^2\left ( \frac{\mathrm{d} x}{\mathrm{d} t} \right )\\1200=3(20)^2\left ( \frac{\mathrm{d} x}{\mathrm{d} t} \right )\\\frac{\mathrm{d} x}{\mathrm{d} t} =\frac{1200}{3(20)^2}=\frac{1200}{3\times 400}=\frac{1200}{1200}=1\,\,cm/min[/tex]
So,
Length of the side is increasing at the rate of [tex]1\,\,cm/min[/tex]
A nationwide study of American homeowners revealed that 65% have one or more lawn mowers. A lawn equipment manufacturer, located in Omaha, feels the estimate is too low for households in Omaha. Find the P-value for a test of the claim that the proportion with lawn mowers in Omaha is higher than 65%. Among 497 randomly selected homes in Omaha, 340 had one or more lawn mowers.
Answer:
0.0559
Step-by-step explanation:
Cant prove it but its right
To test the claim that more than 65% homeowners in Omaha possess a lawnmower, a one-tailed test of proportion was performed. After calculating a test statistic (z =1.52), it was determined that the P-value is 0.064, suggesting there is a 6.4% probability that a sample proportion this high could happen by chance given the null hypothesis is true.
Explanation:In this question, you're asked to calculate the P-value for a test of the claim that the proportion of homeowners with lawn mowers in Omaha is higher than 65%. The proportion from the sample size of Omaha is given as 340/497 = 0.68 or 68%. To test this claim, you would employ a one-tailed test of proportion. The null hypothesis (H0) is that the proportion is equal to 65%, while the alternative hypothesis (Ha) is that the proportion is greater than 65%.
To determine the P-value, you need to first calculate the test statistic (z) using the formula: z = (p - P) / sqrt [(P(1 - P)) / n], where p is the sample proportion, P is the population proportion, and n is the sample size. Substituting into the formula, z = (0.683 - 0.65) / sqrt [(0.65 * 0.35) / 497] = 1.52. The P-value is the probability of getting a z-score that is greater than or equal to 1.52.
Using a standard normal distribution table, or an online z-score calculator, you can find that P(Z > 1.52) = 0.064 (approximately). This is the P-value for the test, which indicates that if the null hypothesis is true, there is a 6.4% probability that a sample proportion this high could occur by chance.
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The finishing time for cyclists in a race are normally distributed with an unknown population mean and standard deviation. If a random sample of 25 cyclists is taken to estimate the mean finishing time, what t-score should be used to find a 98% confidence interval estimate for the population mean?
Answer:
The T-score is 2.49216
Step-by-step explanation:
A 98% confidence interval should be estimated for the end times of cyclists. Since the sample is small, a T-student distribution should be used, in such an estimate. The confidence interval is given by the expression:
[tex][\bar x -T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}, \bar x +T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}][/tex]
[tex]n = 25\\\alpha = 0.02\\T_{(n-1;\frac{\alpha}{2})}= T_{(24;0.01)} = 2.49216[/tex]
Then the T-score is 2.49216
Answer:
2.485
Step-by-step explanation:
Let p stand for "This statement is false." What can be said about the truth value of p. (Hint: Did we really assign a truth value to p? See Example 5 for a discussion of truth value assignment.)
Answer: P means "This statement is false"
then, P is a "function" of some statement,
if i write P( 3> 1932) this could be read as:
3>1932, this statement is false.
You could see that 3> 1932 is false, so P( 3>1932) is true.
Then you could se P(x) at something that is false if x is true, and true if x is false, so p is a negation.
1. Suppose that A , B and C are sets. Show that A \ (B U C) (A \ B) n (A \ C).
Step-by-step explanation:
We want to show that
[tex]=A \setminus (B \cup C) = (A\setminus B) \cap (A\setminus C)[/tex]
To prove it we just use the definition of [tex]X\setminus Y = X \cap Y^c[/tex]
So, we start from the left hand side:
[tex]=A \setminus (B \cup C) = A \cap (B \cup C)^c[/tex] (by definition)
[tex]=A \cap (B^c \cap C^c)[/tex] (by DeMorgan's laws)
[tex]=A \cap B^c \cap C^c[/tex] (since intersection is associative)
[tex]=A \cap B^c \cap A \cap C^c[/tex] (since intersecting once or twice A doesn't make any difference)
[tex]=(A \cap B^c) \cap (A \cap C^c)[/tex] (since again intersection is associative)
[tex]=(A\setminus B) \cap (A \setminus C)[/tex] (by definition)
And so we have reached our right hand side.
The augmented matrix of a consistent system of five equa- tions in seven unknowns has rank equal to three. How many parameters are needed to specify all solutions?
Answer:
4 parameters are necessary to specify all solutions and correspond to the number of free variables of the system.
Step-by-step explanation:
Remember that the number of free variables of a system is equal to m-rank(A) where m is the number of unknowns variables and A is the matrix of the system.
Since the system is consistent and the rank of the matrix is 3 then echelon form of the augmented matrix has two rows of zeros.
Then m-rank(A)=7-3=4.
A boy has 8 red marbles, 2 blue marbles 7 yellow and 3 green. what is the probability for selecting a red marble AND then selecting a green marble?
Answer: Our required probability is [tex]\dfrac{11}{20}[/tex]
Step-by-step explanation:
Since we have given that
Number of red marbles = 8
Number of blue marbles = 2
Number of yellow marbles = 7
Number of green marbles = 3
So, Total number of marbles = 8 + 2 + 7 + 3 = 20
So, Probability for selecting a red marble and then selecting a green marbles is given by
P(red) + P(green) is equal to
[tex]\dfrac{8}{20}+\dfrac{3}{20}\\\\=\dfrac{8+3}{20}\\\\=\dfrac{11}{20}[/tex]
Hence, our required probability is [tex]\dfrac{11}{20}[/tex]
Justin deposited $2,000 into an account 5 years ago. Simple interest was paid on the account. He has just withdrawn $2,876. What interest rate did he earn on the account?
Answer: [tex]8.76\%[/tex]
Step-by-step explanation:
The formula to find the final amount after getting simple interest :
[tex]A=P(1+rt)[/tex], where P is the principal amount , r is rate of interest ( in decimal )and t is time(years).
Given : Justin deposited $2,000 into an account 5 years ago.
i.e. P = $2,000 and t= 5 years
He has just withdrawn $2,876.
i.e. we assume that A = $2876
Now, Put all the values in the formula , we get
[tex](2876)=(2000)(1+r(5))\\\\\Rightarrow\ 1+5r=\dfrac{2876}{2000}\\\\\Rightarrow\ 1+5r=1.438\\\\\Righhtarrow\ 5r=0.438\\\\\Rightarrow\ r=\dfrac{0.438}{5}=0.0876[/tex]
In percent, [tex]r=0.0876\times100=8.76\%[/tex]
hence, He earned [tex]8.76\%[/tex] of interest on account.
Last year, Anthony's grandmother gave him 33 silver coins and 16 gold coins to start a coin collection. Now Anthony has six times as many coins in his collection. How many coins does Anthony have in his collection?
Answer:
59 coins
Step-by-step explanation:
Add 16 gold coins and 33 silver coinsMultiply by 6The number of coins in Anthony's collection is 294
Using the parameters given for our Calculation;
silver coins = 33gold coins = 16Total number of coins now :
33 + 16 = 49Six times as many coins can be calculated thus :
6(49) = 294Therefore, the number of coins in Anthony's collection is 294
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A store gave a customer an allowance of $10.50 on a dress that originally sold for $285.99. How much did the customer pay for the dress if the sales tax was 6.5%?
Answer:
$ 293.40
Step-by-step explanation:
Given,
The allowance given by the store = $ 10.50,
The original cost of the dress = $ 285.99,
So, the cost of the dress after allowance = 285.99 - 10.50
= $ 275.49,
Now, the sales tax = 6.5 %,
Hence, the final amount of the dress after tax = (100+6.5)% of 275.49
= 106.5% of 275.49
[tex]=\frac{106.5\times 275.49}{100}[/tex]
[tex]=\frac{29339.685}{100}[/tex]
= $ 293.39685
≈ $ 293.40
Final answer:
The customer paid $293.40 for the dress after subtracting the allowance and adding the sales tax, which was calculated and rounded to the nearest penny.
Explanation:
To calculate the final price the customer paid for the dress after receiving an allowance and including sales tax, we follow these steps:
Subtract the allowance from the original price of the dress: $285.99 - $10.50 = $275.49.Calculate the sales tax by converting the percentage to a decimal and multiplying it by the adjusted price: $275.49 × 0.065 = $17.90685.Round the sales tax to the nearest penny: $17.90685 rounded is $17.91.Add the rounded sales tax to the adjusted price to get the total amount paid: $275.49 + $17.91 = $293.40.Therefore, the customer paid $293.40 for the dress after the allowance and including the 6.5% sales tax.
Listed below are amounts (in millions of dollars) collected from parking meters by a security service company and other companies during similar time periods. Do the limited data listed here show evidence of stealing by the security service company's employees? Security Service Company: 1.4 1.6 1.7 1.6 1.5 1.6 1.6 1.5 1.4 1.8 Other Companies: 1.5 1.8 1.6 1.9 1.7 1.9 1.8 1.7 1.7 1.6 Find the coefficient of variation for each of the two samples, then compare the variation.
Answer:
[tex]CV=\frac{0.1252}{1.57}=0.07975[/tex]
[tex]CV=\frac{0.1317}{1.72}=0.07657[/tex]
The relative variability is almost equal in both samples a slight greater variability can be noticed in the first sample.
Step-by-step explanation:
The coefficient of variation of a sample is defined as the ratio between the mean standard deviation and the sample mean. And it represents the percentage relation of the variation of the data with respect to the average.
[tex]CV=\frac{S}{\bar X}[/tex]
In the case of the first sample you have:
[tex]CV=\frac{0.1252}{1.57}=0.07975[/tex]
In the case of the second sample you have:
[tex]CV=\frac{0.1317}{1.72}=0.07657[/tex]
The relative variability is almost equal in both samples a slight greater variability can be noticed in the first sample.
Dave received 120$ for his birthday from his parents and his friends. He went on a shopping spree and spent 2/5 of his money on a t-shirt and 1/6 on mountain bike magazines. How much money does he have left.
Answer: 52$
Step-by-step explanation:
The money Dave spent on a t-shirt is obtained multiplying 120$ by the fraction 2/5
120(2/5) = 48$
And the money he spent on mountain bike magazines is obtained multiplying 120$ by the fraction 1/6
120(1/6)= 20$
The money he has left is:
120$ - 48$ -20$ = 52$
Exercise 0.2.9 : Verify that x = C 1 e − t + C 2 e 2 t is a solution to x ′′ − x ′ − 2 x = 0 . Find C 1 and C 2 to solve for the initial conditions x (0) = 10 and x ′ (0) = 0 .
Answer:
Since 2 and -1 are eigenvalues of the differential equation,
[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]
is a solution to the differential equation
-----------------------------------------------------------
The solution to the initial value problem is:
[tex]x(t) = \frac{20}{3}e^{-t} + \frac{10}{3}e^{2t}[/tex]
Step-by-step explanation:
We have the following differential equation:
[tex]x'' - x' - 2x = 0[/tex]
The first step is finding the eigenvalues for this differential equation, that is, finding the roots of the following second order equation:
[tex]r^{2} - r - 2 = 0[/tex]
[tex]\bigtriangleup = (-1)^{2} -4*1*(-2) = 1 + 8 = 9[/tex]
[tex]r_{1} = \frac{-(-1) + \sqrt{\bigtriangleup}}{2*1} = \frac{1 + 3}{2} = 2[/tex]
[tex]r_{2} = \frac{-(-1) - \sqrt{\bigtriangleup}}{2*1} = \frac{1 - 3}{2} = -1[/tex]
Since 2 and -1 are eigenvalues of the differential equation,
[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]
is a solution to the differential equation.
Solution of the initial value problem:
[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]
[tex]x(0) = 10[/tex]
[tex]10 = c_{1}e^{-0} + c_{2}e^{2*0}[/tex]
[tex]c_{1} + c_{2} = 10[/tex]
---------------------
[tex]x'(t) = -c_{1}e^{-t} + 2c_{2}e^{2t}[/tex]
[tex]x'(0) = 0[/tex]
[tex]0 = -c_{1}e^{-0} + 2c_{2}e^{2*0}[/tex]
[tex]-c_{1} + 2c_{2} = 0[/tex]
[tex]c_{1} = 2c_{2}[/tex]
So, we have to solve the following system:
[tex]c_{1} + c_{2} = 10[/tex]
[tex]c_{1} = 2c_{2}[/tex]
[tex]2c_{2} + c_{2} = 10[/tex]
[tex]3c_{2} = 10[/tex]
[tex]c_{2} = \frac{10}{3}[/tex]
[tex]c_{1} = 2c_{2} = \frac{20}{3}[/tex]
The solution to the initial value problem is:
[tex]x(t) = \frac{20}{3}e^{-t} + \frac{10}{3}e^{2t}[/tex]
Twin Primes Conjecture A natural number is called a prime number if it has exactly two factors, 1 and itself. 1 is not a prime number because it has exactly one factor. If a number is not prime it is called composi -List the numbers 1 through 31 and circle the primes. What do you see? -You might notice that some pairs of prime numbers have exactly one composite number between them. Such pairs of prime numbers include 3&5, 11&13, 17&19, 29&31. These pair numbers are called twin primes. -Write your first impression regarding this question: Are there an infinite number of twin primes? Provide a justification for your thinking. -Do a little research on the twin prime conjecture and describe at least one interesting fact ti you find.
Answer:
See explanation below.
Step-by-step explanation:
The prime numbers are bold:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20 21 22 23 24 25 26 27 28 29 30 31
a) We can see that as we go higher, twin primes seem less frequent but even considering that, there is an infinite number of twin primes. If you go high enough you will still eventually find a prime that is separated from the next prime number by just one composite number.
b) I think it's interesting the amount of time that has been devoted to prove this conjecture and the amount of mathematicians who have been involved in this. One of the most interesting facts was that in 2004 a purported proof (by R. F. Arenstorf) of the conjecture was published but a serious error was found on it so the conjecture remains open.
A linens department received the following:
Answer:
b.
917 sheet sets.
Step-by-step explanation:
We are asked to find the total number of sheet sets received by linens department.
To find the total number of sheet sets received by linens department, we will add the number of dozens of each sheet.
[tex]\text{Total number of sheet sets received in dozens}=19\frac{1}{2}+33\frac{2}{3}+23\frac{1}{4}[/tex]
[tex]\text{Total number of sheet sets received in dozens}=\frac{39}{2}+\frac{101}{3}+\frac{93}{4}[/tex]
Make common denominator:
[tex]\text{Total number of sheet sets received in dozens}=\frac{39*6}{2*6}+\frac{101*4}{3*4}+\frac{93*3}{4*3}[/tex]
[tex]\text{Total number of sheet sets received in dozens}=\frac{234}{12}+\frac{404}{12}+\frac{279}{12}[/tex]
[tex]\text{Total number of sheet sets received in dozens}=\frac{234+404+279}{12}[/tex]
[tex]\text{Total number of sheet sets received in dozens}=\frac{917}{12}[/tex]
To find the number of sheets, we will multiply number of dozens of sheets by 12 as 1 dozen equals 12.
[tex]\text{Total number of sheet sets received}=\frac{917}{12}\times 12[/tex]
[tex]\text{Total number of sheet sets received}=917[/tex]
Therefore, the total number of sheet sets received is 917.
Researchers have created every possible "knockout" line in yeast. Each line has exactly one gene deleted and all the other genes present (Steinmetz et al. 2002). The growth rate - how fast the number of cells increases per hour - of each of these yeast lines has also been measured, expressed as a multiple of the growth rate of the wild type that has all the genes present. In other words, a growth rate greater than 1 means that a given knockout line grows faster than the wild type, whereas a growth rate less than 1 means it grows more slowly. Below is the growth rate of a random sample of knockout lines:
0.8, 0.98, 0.72, 1, 0.82, 0.63, 0.63, 0.75, 1.02, 0.97, 0.86
What is the standard deviation of growth rate this sample of yeast lines (answer to 3 decimals)?
Answer: 0.144
Step-by-step explanation:
Formula to find standard deviation: [tex]\sigma=\sqrt{\dfrac{\sum_{i=1}^n(x_i-\overline{x})^2}{n-1}}[/tex]
Given : The growth rate of a random sample of knockout lines:-
0.8, 0.98, 0.72, 1, 0.82, 0.63, 0.63, 0.75, 1.02, 0.97, 0.86
Here ,
[tex]\overline{x}=\dfrac{\sum_{i=1}^{10}x_i}{n}\\\\=\dfrac{0.8+0.98+0.72+1+0.82+0.63+0.63+ 0.75+1.02+ 0.97+ 0.86}{10}\\\\=\dfrac{9.18}{10}\approx0.83[/tex]
[tex]\sum_{i=1}^n(x_i-\overline{x})^2=(-0.03)^2+(0.15)^2+(-0.11)^2+(0.17)^2+(-0.01)^2+(-0.2)^2+(-0.2)^2+(-0.08)^2+(0.19)^2+(0.14)^2+(0.03)^2\\\\=0.2075[/tex]
Now, the standard deviation:
[tex]\sigma=\sqrt{\dfrac{0.2075}{10}}=0.144048602909\approx0.144[/tex]
Hence, the standard deviation of growth rate this sample of yeast lines =0.144
The standard deviation of the growth rates for the given sample of yeast knockout lines is 0.148 when rounded to three decimal places.
Explanation:To calculate the standard deviation of the growth rates of the yeast knockout lines, we first need to compute the mean (average) of the given data. Then, following the steps, we find the variance by calculating the difference between each value and the mean, squaring those differences, and finding their average. Finally, we take the square root of the variance to find the standard deviation.
Calculate the mean (average) of the sample data.Subtract the mean from each data point and square the result.Find the average of these squared differences, which gives us the variance.Take the square root of the variance to get the standard deviation.Here are the calculations using the given growth rates:
Mean (average) = (0.8 + 0.98 + 0.72 + 1 + 0.82 + 0.63 + 0.63 + 0.75 + 1.02 + 0.97 + 0.86) / 11 = 0.836
[tex]Variance = [(0.8 - 0.836)^2 + (0.98 - 0.836)^2 + (0.72 - 0.836)^2 + (1 - 0.836)^2 + (0.82 - 0.836)^2 + (0.63 - 0.836)^2 + (0.63 - 0.836)^2 + (0.75 - 0.836)^2 + (1.02 - 0.836)^2 + (0.97 - 0.836)^2 + (0.86 - 0.836)^2] / (11 - 1)[/tex]Variance = 0.021918
Standard Deviation = sqrt(variance) = sqrt(0.021918) ≈ 0.148
Therefore, the standard deviation of the sample growth rates, rounded to three decimal places, is 0.148.
a,b,c,d are integers and GCD(a,b)=1. if c divides a and d divides b, prove that GCD(c,d) = 1.
Answer:
One proof can be as follows:
Step-by-step explanation:
We have that [tex]g.c.d(a,b)=1[/tex] and [tex]a=cp, b=dq[/tex] for some integers [tex]p, q[/tex], since [tex]c[/tex] divides [tex]a[/tex] and [tex]d[/tex] divides [tex]b[/tex]. By the Bezout identity two numbers [tex]a,b[/tex] are relatively primes if and only if there exists integers [tex]x,y[/tex] such that
[tex]ax+by=1[/tex]
Then, we can write
[tex]1=ax+by=(cp)x+(dq)y=c(px)+d(qy)=cx'+dy'[/tex]
Then [tex]c[/tex] and [tex]d[/tex] are relatively primes, that is to say,
[tex]g.c.d(c,d)=1[/tex]
How many grams of Total Sugars in one serving of this food can be attributed to naturally occurring sugar (i.e., NOT added sugars)?
22g
2g
12g
10g
Answer:
Taking into account the information presented in the image attached for this food.
Naturally sugar are
2 g
Step-by-step explanation:
Information presented regarding to the composition of this food is attached.
One serving has a weight of 55 g.
Total carbohydrates in one serving are 37 g. This value includes fiber and sugars.
If we detail sugars, it has 12 g of total sugars (St), where 10 g are added sugars (Sa) and all the rest Natural sugars (Sn).
Expressing sugars as an equation
St = Sa + Sn
Isolating Sn value
Sn = St - Sa
Sn = 12 g - 10 g
Sn = 2 g
Finally, Natural sugars (Sn) is 2 g.
All 22 grams of total sugars per serving in the food are naturally occurring, as the label specifies there are 0g of added sugars.
Explanation:To determine the grams of naturally occurring sugar in a serving of food, we need to examine the nutritional information provided.
According to the label, it includes 0g added sugars, which implies that all of the sugars listed are naturally occurring.
Therefore, if there are 22g total sugars in one serving of the food, and none of these are added sugars, then all 22 grams can be attributed to naturally occurring sugars.
Let A fa, b,c. B [a, b, d), and C tb, d,e. Find the union of A and B, and then the union of this with C. Also, find the union of the B and C and then the union of this with A. Try to formulate what you have observed.
Answer: A ∪ B = {a, b, c, d}
(A ∪ B) ∪ C = = {a, b, c, d, e}
B ∪ C = = {a, b, d, e}
(B ∪ C) ∪ A = = {a, b, c, d, e}
Step-by-step explanation:
A = {a, b, c} B = {a, b, d} C = {b, d, e}
Union means "to join" so combine the sets to form a union.
A ∪ B = {A & B}
= {a, b, c & a, b, d}
= {a, b, c, d} because we do not need to list a & b twice
(A ∪ B) ∪ C = {(A ∪ B) & C)
= {a, b, c, d & b, d, e}
= {a, b, c, d, e} because we do not need to list b & d twice
B ∪ C = {A & B}
= {a, b, d & b, d, e}
= {a, b, d, e} because we do not need to list b & d twice
(B ∪ C) ∪ A = {(B ∪ C) & A)
= {a, b, d, e & a, b, c}
= {a, b, c, d, e} because we do not need to list a & b twice
Final answer:
We utilized set theory to find the union of sets A, B, and C in different orders and observed that the associative property of union holds true, meaning the order of union operations does not change the outcome which is {a, b, c, d, e}.
Explanation:
To answer this student's question, we can apply set theory concepts, specifically the concepts of the union and associativity.
Given sets A = {a, b, c}, B = {a, b, d}, and C = {b, d, e}, the union of these sets can be found as follows:
The union of A and B is A ∪ B = {a, b, c, d} - the set containing all elements from both A and B.The union of A ∪ B with C is (A ∪ B) ∪ C = {a, b, c, d, e} - the set containing all elements from A, B, and C.Similarly, the union of B and C is B ∪ C = {a, b, d, e}.The union of B ∪ C with A is (B ∪ C) ∪ A = {a, b, c, d, e}.From these operations, we observe that regardless of the order in which we take the union of the three sets, the result is the same. This demonstrates the associative property of union in set theory, where the order in which unions are performed does not affect the final outcome. We can generalize this as A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C.
Which nutrient category on the Nutrition Facts panel does NOT have a Daily Value or % Daily Value ascribed to it?
Dietary Fiber
Protein
Total Fat
Sodium
Answer:
Protein
Step-by-step explanation:
According to U.S. Food and Drug Administration (FDA) information, the current scientific evidence indicates that protein intake is not a matter of public concern, this is why the FDA does not request to add a % Daily Value on the Nutrition Facts panel, unless if the product is made for protein such as 'high protein' products or if this food is meant for use by childrend under 4 years old.
Answer:Protein
Step-by-step explanation: took quiz :)
Simplify the following expressions
X2 X-2/3
X1/4/ X-5/2
(4/5)x-2/5 y3/2 / (2/3) x3/5y1/2
(2/3)x2/3y2/3 / (1/3)x-1/3y-1/3
(z2/3 x2/3y-2/3 ) y + (z2/3 x-1/3y-1/3 ) x
(4x3/5y3 z2 ) 1/3
Step-by-step explanation:
For each case we have the next step by step solution.
[tex]x^2(\dfrac{x-2}{3})=\dfrac{x^3-2x^2}{3}[/tex][tex]\dfrac{x^{1/4}}{x-\dfrac{5}{2}}=\dfrac{x^{1/4}}{\dfrac{2x}{2}-\dfrac{5}{2}}=\dfrac{x^{1/4}}{\dfrac{2x-5}{2}}=\dfrac{2x^{1/4}}{{2x-5}}[/tex][tex]\dfrac{\dfrac{4}{5}x-\dfrac{2}{5}y^{3/2}}{\dfrac{\dfrac{2}{3}x^3}{5y^{1/2}}}=\dfrac{\dfrac{4x}{5}-\dfrac{2y^{3/2}}{5}}{\dfrac{\dfrac{2x^3}{3}}{5y^{1/2}}}=\dfrac{\dfrac{4x-2y^{3/2}}{5}}{\dfrac{2x^3}{15y^{1/2}}}={\dfrac{(4x-2y^{3/2})\cdot 15y^{1/2}}{5\cdot 2x^3}}[/tex] [tex]{\dfrac{(4x-2y^{3/2})\cdot 15y^{1/2}}{5\cdot 2x^3}}={\dfrac{(60xy^{1/2}-30y^{3/2}y^{1/2})}{10x^3}}={\dfrac{(60xy^{1/2}-30y^{4/2})}{10x^3}}={\dfrac{(60xy^{1/2}-30y^{2})}{10x^3}}[/tex][tex]\dfrac{\dfrac{\dfrac{2}{3}x^2}{3y^{2/3}}}{\dfrac{1}{3}x-\dfrac{1}{3}y-\dfrac{1}{3}}=\dfrac{\dfrac{\dfrac{2x^2}{3}}{3y^{2/3}}}{\dfrac{x}{3}-\dfrac{y}{3}-\dfrac{1}{3}}=\dfrac{\dfrac{2x^2}{9y^{2/3}}}{\dfrac{x-y-1}{3}}=\dfrac{2x^2\cdot 3}{(x-y-1)\cdot 9y^{2/3}}}=\dfrac{6x^2}{(9xy^{2/3}-9yy^{2/3}-9y^{2/3})}}=\dfrac{6x^2}{(9xy^{2/3}-9y^{5/3}-9y^{2/3})}}[/tex][tex](z^{2/3}x^{2/3}y+\dfrac{2}{3})y+(z^{2/3}x-\dfrac{1}{3}y-\dfrac{1}{3})x=(z^{2/3}x^{2/3}y^2+\dfrac{2}{3}y)+(z^{2/3}x^2-\dfrac{1}{3}yx-\dfrac{x}{3})=z^{2/3}x^{2/3}y^2+z^{2/3}x^2-\dfrac{1}{3}yx+\dfrac{2}{3}y-\dfrac{x}{3}[/tex][tex](\dfrac{4x^3}{5y^2}z^2)^{1/3}=\dfrac{(4x^3)^{1/3}}{(5y^2)^{1/3}}(z^2)^{1/3}=\dfrac{4^{1/3}x}{5^{1/3}y^{2/3}}z^{2/3}[/tex]
Calculate (a) the number of milligrams of metoclopramide HCl in each milliliter of the prescription:
Metoclopramide HCl 10 g
Methylparaben 50 mg
Propylparaben 20 mg
NaCl 800 mg
Purifed water qs ad 100 mL
Answer:
There are 100 milligrams of metoclopramide HCl in each milliliter of the prescription
Step-by-step explanation:
When the prescription says Purified water qs ad 100 mL means that if we were to make this, we should add the quantities given and then, fill it up with water until we have 100 mL of solution, being the key words qs ad, meaning sufficient quantity to get the amount of mixture given.
Then, knowing there is 10 grams of metoclopramide HCl per 100 mL of prescription, that means there is (1 gram = 1000 milligrams) 10000 milligrams of metoclopramide HCl per 100 mL of prescription. That is a concentration given in a mass/volume way.
Knowing the concentration, we can calculate it per mL instead of per 100 mL
[tex]Concentration_{metoclopramide HCL}= \frac{10000mg}{100mL} =100 \frac{mg}{mL}[/tex]
A man in a maze makes three consecutive displacements. His first displacement is 6.70 m westward, and the second is 11.0 m northward. At the end of his third displacement he is back to where he started. Use the graphical method to find the magnitude and direction of his third displacement.
Answer:
The man had a displacement of 12.88 m southeastward
Step-by-step explanation:
The path of man forms a right triangle. The first two magnitudes given in the problem form the legs and the displacement that we must calculate forms the hypotenuse of the triangle. To do this we will use the equation of the pythagorean theorem.
H = magnitude of displacement
[tex]H^2 = \sqrt{L_1^2 + L_2^2} = \sqrt{6.70^2 + 11.0^2} = \sqrt{165.89} =12.88 m[/tex]
using the graphic method, we will realize that the displacement is oriented towards the southeast
Suppose the manager of a shoe store wants to determine the current percentage of customers who are males. How many customers should the manager survey in order to be 99% confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of customers who are males?
Answer:
The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units
Step-by-step explanation:
In calculating the sample size to estimate a population proportion in which there is no information on an initial sample proportion, the principle of maximum uncertainty is assumed and a ratio [tex]P = 1/2[/tex] is assumed. The expression to calculate the size is:
[tex] n=\frac{z_{\alpha /2}^2}{4 \epsilon^2} [/tex]
With
Z value (for 0.005) [tex] Z _ {\alpha / 2} = 1.64485 [/tex]
Significance level [tex] \alpha = 0.01 [/tex]
Estimation error [tex] \epsilon = 0.03 [/tex]
[tex] n=\frac{(1.64485)^2}{(4)(0.03)^2} = 751.5398[/tex]
The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units
To determine the percentage of male customers with a margin of error of 3% at a 99% confidence level, the manager would need to survey approximately 1847 customers.
Explanation:The question is related to the concept of statistics, and more specifically to the idea of a confidence interval
for a population proportion. When you want to be very sure about your estimates, you use a high level of confidence. The standard formula for calculating the sample size needed in order to get a certain margin of error at a certain confidence level is n = [Z^2 * P * (1-P)] / E^2. In this formula, n is the sample size, Z is the z-score associated with your desired level of confidence, P is the preliminary estimate of the population proportion, and E is the desired margin of error. If the manager doesn't have a precursory idea of what the proportion of male customers is, it's standard to use P = 0.5. The Z score for a 99% confidence interval is approximately 2.576. Substituting these values, the manager would need a sample size of approximately 1846 customers. For more accuracy, it's better to round up to the next nearest whole number, so the minimum sample size required would be 1847.
Learn more about confidence intervals here:https://brainly.com/question/32278466
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If the probability that a bird will lay an egg is 75%, the probability that the egg will hatch is 50%, and the probability that the chick will be eaten by a snake before it fledges is 20%, what is the probability that a parent will have progeny that survive to adulthood?
(A) 30%
(B) 14.5%
(C) 7.5%
(D) 2%
Answer: (A) 30%
Step-by-step explanation:
Given : The probability that a bird will lay an egg =0.75
The probability that the egg will hatch =0.50
Now, the probability that the egg will lay an egg and hatch = [tex]0.75\times0.50[/tex]
Also,The probability that the chick will be eaten by a snake before it fledges =0.20
Then, the probability that the chick will not be eaten by a snake before it fledges : 1-0.20=0.80
Now, the probability that a parent will have progeny that survive to adulthood will be :-
[tex]0.75\times0.50\times0.80=0.30=30\%[/tex]
Hence, the probability that a parent will have progeny that survive to adulthood = 30%
This year, Druehl, Inc., will produce 57,600 hot water heaters at its plant in Delaware, in order to meet expected global demand. To accomplish this, each laborer at the plant will work 160 hours per month. If the labor productivity at the plant is 0.15 hot water heaters per labor-hour, how many laborers are employed at the plant?
Answer:
2400 laborers
Step-by-step explanation:
Let N be the amount of laborers employed at the plant. The amount of heaters produced by all workers in an hour is:
heaters = 0.15heaters/hour * N
In a month the total amunt of heaters will be:
heaters = 0.15 * N * 160
Since this has to be 57600 to meet the expected demand:
57600 = 0.15 * N * 160 Solving for N we get:
N = 57600 / (0.15 * 160) = 2400 laborers
DECISION SCIENCE Assume more restrictions. The books pass through two departments: Graphics and Printing, 1. before they can be sold, X requires 3 hours in Graphics and 2 hours in printing. Y requires 1 hour in graphics and 1 hour in printing. There are 21 hours available in graphics and 19 hours in printing respectively. Solve for x and y.
Answer:
Step-by-step explanation:
The given information can be tabulated as follows:
Graphics G Printing P Total
X 3 2 5
Y 1 1 2
Available 21 19
We have the constraints as
[tex]3x+y\leq 21\\2x+y\leq 19\\x\leq 2\\y\leq 15[/tex]
Thus we have solutions as
[tex]0\leq x\leq 2\\0\leq y\leq 15[/tex]
Prove that the following two sets are the same. S1 = {a + bx : a, b ∈ R} = all polynomials which can expressed as a linear combination of 1 and x; S2 = {ax + b(2 + x) : a, b ∈ R} = all polynomials which can expressed as a linear combination of x and 2 + x.
Answer with Step-by-step explanation:
We are given that twos sets
[tex]S_1[/tex]={a+bx:[tex]a,b\in R[/tex]}=All polynomials which can expressed as a linear combination of 1 and x.
[tex]S_2[/tex]={ax+b(2+x):[tex]a,b\in R[/tex]}=All polynomials which can be expressed as a linear combination of x and 2+x.
We have to prove that given two sets are same.
[tex]S_2[/tex]={ax+2b+bx}={(a+b)x+2b}={cx+d}
[tex]S_2[/tex]={cx+d}=All polynomials which can be expressed as a linear combination of 1 and x.
Because a+b=c=Constant
2b= Constant=d
Hence, the two sets are same .
slope-intercept formula y=mx+b