Answer:
[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]
Step-by-step explanation:
[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]
We will prove this with the help of principal of mathematical induction.
For n = 1, [tex]x-1[/tex] is a factor [tex]x-1[/tex], which is true.
Let the given statement be true for n = k that is [tex]x-1[/tex] is a factor of [tex]x^k - 1[/tex].
Thus, [tex]x^k - 1[/tex] can be written equal to [tex]y(x-1)[/tex], where y is an integer.
Now, we will prove that the given statement is true for n = k+1
[tex]x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)[/tex]
Thus, [tex]x^k - 1[/tex] is divisible by [tex]x-1[/tex].
Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.
A batch of 20 semiconductor chips is inspected by choosing a sample of 3 chips. Assume 10 of the chips do not conform to customer requirements. Round your answers to the nearest integer. a. How many different samples are possible? b. How many samples of 3 contain exactly one nonconforming chip? c. How many samples of 3 contain at least one nonconforming chip?
The answers are:
a) 1,140 combinations.b) The probability is 0.395c) The probability is 0.895Finding the possible combinations of the sample.We know that if we have a set of N elements, the number of different groups of K elements that we can make, such that:
N ≥ K
Is given by:
[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]
a) If we have 20 semiconductors and we make groups of 3, then we can have:
[tex]C(20, 3) = \frac{20!}{(20 - 3)!*3!} = \frac{20*19*18}{3*2} = 1,140[/tex]
b) The probability that a randomly picked chip is nonconforming is given by the quotient between the number of nonconforming chips and the total number of chips.
So, if the first selected chip is the nonconforming one, the probability of selecting it is:
P = 10/20 = 1/2
The next two ones work properly, the probability is computed in the same way, but notice that now there are 10 proper chips and 19 chips in total.
Q = 10/19
And for the last one we have:
K = 9/18
The joint probability is:
P*Q*K = (1/2)*(9/19)*(8/18) = 0.132
But this is only for the case where the first one is the nonconforming, then we must take in account the possible permutations (there are 3 of these) then the probability is:
p = 3*0.132 = 0.395
c) The probability of getting at least one nonconforming chip is equal to the difference between 1 and the probability of not getting a nonconforming chip.
That probability is computed in the same way as above)
P = (10/20)*(9/19)*(8/18) = 0.105
Then we have:
1 - P = 1 - 0.105 = 0.895
So the probability of getting at least one nonconforming chip is 0.895.
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The total number of different samples is 1140, the number of samples with exactly one nonconforming chip is 450, and the number of samples with at least one nonconforming chip is 1020. The problem involves calculating combinations related to semiconductor chip samples.
In this problem, we are dealing with combinations and probabilities. Let's denote the total number of chips as 20, the number of nonconforming chips as 10, and the sample size as 3.
a. How many different samples are possible?
We use the combination formula to find the total number of different samples of 3 chips that can be drawn from 20 chips.
Combination formula: [tex]$C(n, k) = \dfrac{n!}{k! \cdot (n - k)!}$[/tex]Here, n = 20 and k = 3:[tex]\[C(20, 3) = \dfrac{20!}{3! \cdot (20 - 3)!} = 1140\][/tex]b. How many samples of 3 contain exactly one nonconforming chip?
To get samples with exactly one nonconforming chip, we need to choose 1 nonconforming chip from 10, and 2 conforming chips from the remaining 10.
Number of ways to choose 1 nonconforming chip: C(10, 1) = 10Number of ways to choose 2 conforming chips: C(10, 2) = 45Total number of samples with exactly one nonconforming chip: 10 * 45 = 450c. How many samples of 3 contain at least one nonconforming chip?
First, calculate the total number of samples that contain no nonconforming chips.
Number of ways to choose 3 conforming chips from 10: C(10, 3) = 120Using the complement rule:
Number of samples with at least one nonconforming chip: Total samples - Samples with no nonconforming chips
1140 - 120 = 1020Find the point 3x - 7x₂ =5. of the lines x, - 5x2 = 1 and of intersection
Answer:
([tex]\frac{9}{4},\frac{1}{4})[/tex]
Step-by-step explanation:
We are given that two lines
[tex]x_1-7x_2=5[/tex] and [tex]x_1-5x_2=1[/tex]
We have to find the intersection point of two lines
Let [tex]3x_1-7x_2=5[/tex] (equation 1)
[tex]x_1-5x_2=1[/tex] (Equation 2)
Multiply equation 2 by 3 then subtract from equation 1
[tex]-7x_2+15x_2=5-3[/tex]
[tex]8x_2=2[/tex]
[tex]x_2=\frac{2}{8}=\frac{1}{4}[/tex]
Substitute [tex] x_2=\frac{1}{4}[/tex] in the equation 1
Then, we get
[tex]3x_1-7\frac{1}{4}=5[/tex]
[tex]3x_1-\frac{7}{4}=5[/tex]
[tex]3x_1=5+\frac{7}{4}=\frac{20+7}{4}=\frac{27}{4}[/tex]
[tex]x_1=\frac{27}{4\times 3}=\frac{9}{4}[/tex]
Hence, the intersection point of two given lines is ([tex]\frac{9}{4},\frac{1}{4})[/tex]
Show that the set of even whole numbers, E, is equivalent to the set of odd whole numbers, O, by carefully describing a one-to-one correspondence between the sets
Answer:
E and O are equivalents.
Step-by-step explanation:
We will define the one-to-one correspondence this way
E O
0 1
2 3
4 5
6 7
8 9
.................
2n 2n - 1
This way, every 2n even number will have its correspondence with a 2n-1 odd number.
Write down the explicit solution for each of the following: a) x’=t–sin(t); x(0)=1
b) x’+2x=4; x(0)=5
c) x’’+4x=0; x(0)=0; x’(0)=1
Answer:
a) x=(t^2)/2+cos(t), b) x=2+3e^(-2t), c) x=(1/2)sin(2t)
Step-by-step explanation:
Let's solve by separating variables:
[tex]x'=\frac{dx}{dt}[/tex]
a) x’=t–sin(t), x(0)=1
[tex]dx=(t-sint)dt[/tex]
Apply integral both sides:
[tex]\int {} \, dx=\int {(t-sint)} \, dt\\\\x=\frac{t^2}{2}+cost +k[/tex]
where k is a constant due to integration. With x(0)=1, substitute:
[tex]1=0+cos0+k\\\\1=1+k\\k=0[/tex]
Finally:
[tex]x=\frac{t^2}{2} +cos(t)[/tex]
b) x’+2x=4; x(0)=5
[tex]dx=(4-2x)dt\\\\\frac{dx}{4-2x}=dt \\\\\int {\frac{dx}{4-2x}}= \int {dt}\\[/tex]
Completing the integral:
[tex]-\frac{1}{2} \int{\frac{(-2)dx}{4-2x}}= \int {dt}[/tex]
Solving the operator:
[tex]-\frac{1}{2}ln(4-2x)=t+k[/tex]
Using algebra, it becomes explicit:
[tex]x=2+ke^{-2t}[/tex]
With x(0)=5, substitute:
[tex]5=2+ke^{-2(0)}=2+k(1)\\\\k=3[/tex]
Finally:
[tex]x=2+3e^{-2t}[/tex]
c) x’’+4x=0; x(0)=0; x’(0)=1
Let [tex]x=e^{mt}[/tex] be the solution for the equation, then:
[tex]x'=me^{mt}\\x''=m^{2}e^{mt}[/tex]
Substituting these equations in c)
[tex]m^{2}e^{mt}+4(e^{mt})=0\\\\m^{2}+4=0\\\\m^{2}=-4\\\\m=2i[/tex]
This becomes the solution m=α±βi where α=0 and β=2
[tex]x=e^{\alpha t}[Asin\beta t+Bcos\beta t]\\\\x=e^{0}[Asin((2)t)+Bcos((2)t)]\\\\x=Asin((2)t)+Bcos((2)t)[/tex]
Where A and B are constants. With x(0)=0; x’(0)=1:
[tex]x=Asin(2t)+Bcos(2t)\\\\x'=2Acos(2t)-2Bsin(2t)\\\\0=Asin(2(0))+Bcos(2(0))\\\\0=0+B(1)\\\\B=0\\\\1=2Acos(2(0))\\\\1=2A\\\\A=\frac{1}{2}[/tex]
Finally:
[tex]x=\frac{1}{2} sin(2t)[/tex]
Ana and Bella are sisters.They are responsible for keeping the
family's driveway clear of snow.If they work together they can
complete the shoveling in 30 minutes.
One day Ana had to do the job alone.It took her 40 minutes.How long
would it have taken Bella to shovel the same amont alone?You can
assume that each girl always shovels at a constant speed.
Answer:
Bella can complete work alone in 120 minutes
Step-by-step explanation:
Ana and Bella together can complete the shoveling in 30 minutes.
They can do a part of work in 1 minute = [tex]\frac{1}{30}[/tex]
Ana can do work alone in 40 minutes
She can do part of work in 1 minute = [tex]\frac{1}{40}[/tex]
So, Bella can do a part of work in 1 minute = [tex]\frac{1}{30}-\frac{1}{40}[/tex]
= [tex]\frac{1}{120}[/tex]
Bella can do [tex]\frac{1}{120}[/tex] part of work in 1 minute
Bella can do whole work in minutes = 120
Hence Bella can complete work alone in 120 minutes
A particular group of men have heights with a mean of 174 cm and a standard deviation of 6 cm. Earl had a height of 192 cm. a. What is the positive difference between Earl's height and the mean? b. How many standard deviations is that [the difference found in part (a)]? c. Convert Earl's height to a z score. d. If we consider "usual" heights to be those that convert to z scores between minus2 and 2, is Earl's height usual or unusual? a. The positive difference between Earl's height and the mean is nothing cm.
Answer:
a) 18 cm
b) 18
c) 3
The Earl's height is unusual because the z score does not lies in the given range of usual i.e -2 and 2
Step-by-step explanation:
Given:
Mean height, μ = 174 cm
Standard deviation = 6 cm
height of Earl, x = 192 cm
a) The positive difference between Earl height and the mean = x - μ
= 192 - 174 = 18 cm
b) standard deviations is 18
c) Now,
the z score is calculated as:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
or
[tex]z=\frac{192-174}{6}[/tex]
or
z = 3
The Earl's height is unusual because the z score does not lies in the given range of usual i.e -2 and 2
Final answer:
Earl's height is 18 cm above the mean, which is 3 standard deviations from the mean, resulting in a z-score of 3. Considering the usual z-score range of -2 to 2, Earl's height is hence considered unusual.
Explanation:
The problem at hand involves understanding the concepts of statistics, particularly regarding mean, standard deviation, and z-scores in the context of normal distribution.
a. Positive Difference Between Earl's Height and the Mean
The positive difference between Earl's height and the mean is simply calculated by subtracting the mean from Earl's height. If Earl's height is 192 cm and the mean height is 174 cm, the difference is:
192 cm - 174 cm = 18 cm
b. Standard Deviations from the Mean
The number of standard deviations from the mean is found by dividing the difference by the standard deviation. Since the standard deviation is 6 cm,
18 cm / 6 cm = 3 standard deviations
c. Z-score Conversion
The z-score is calculated using the formula:
z = (Earl's height - mean) / standard deviation
z = (192 cm - 174 cm) / 6 cm = 3
d. Usuality of Earl's Height
Since "usual" heights translate to z-scores between -2 and 2, a z-score of 3 indicates that Earl's height is unusual.
An inspector for the U.S. Food and Drug Administration obtains all vitamin pills produced in an hour at the Health Supply Company. She thoroughly mixes them, then scoops a sample of 10 pills that are to be tested for the exact amount of vitamin content. Does this sampling plan result in a random sample? Explain.
Answer:
No. See explanation below.
Step-by-step explanation:
A random sample is a type of sample in which every item has the same probability of being selected. Equally, after we take one item, the remaining items keep having the same probability of being selected.
In this problem, the vitamin pills are firstly obtained from the pills produced in an hour at the Health Supply Company, therefore, not all pills had the same probability of being selected in the first place.
what is 10.2719 rounded to the nearest hundreth?
Answer:
10.27
Step-by-step explanation:
Find the number in the hundredth place 7 and look one place to the right for the rounding digit 1 . Round up if this number is greater than or equal to 5 and round down if it is less than 5 . And the answer is 10.27 which is rounded to the nearest hundredth.
Here, we are required to round the number 10.2719 to the nearest hundredth.
The number 10.2719 rounded to the nearest hundredth is; 10.27While considering place values of numbers,
Numbers on the left side of the decimal are ordered from right to left as unit, tens, hundreds, thousands and so on respectively.while Numbers on the right side of the decimal are ordered from left to right as tenth, hundredth, thousandth and so on respectively.Therefore, according to the question, the digit with occupies the hundredth position is 7.
As such, to round off the number 10.2719, the digit which is after the digit 7 is considered.
If the digit is less than 5, it is rounded to 0, and if greater or equal to 5, it is rounded to 1 and ultimately added to the preceding number.
In this case, the number is 1 and since 1 is less than 5, it is rounded to 0 and added to 7.
Ultimately, the number 10.2719 rounded to the nearest hundredth is;
10.27
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What is the pobability that a number cube with six
facesnumbered 1 to 6 will land with the number 5 face up is?
Answer: [tex]\dfrac{1}{6}[/tex]
Step-by-step explanation:
We know that the total number of outcomes for fair dice {1,2,3,4,5,6} = 6
Given : Favorable outcome = 5
i.e. Number of favorable outcomes =1
We know that the formula to find the probability for each event is given by :-
[tex]\dfrac{\text{Number of favorable outcomes}}{\text{Total outcomes}}[/tex]
Then, the probability that the six faces cube will land with the number 5 face up will be :_
[tex]\dfrac{1}{6}[/tex]
Hence, the required probability = [tex]\dfrac{1}{6}[/tex]
Please convert these values from scientific notation into long hand notation. To receive credit, plea: values over 999; use 0.# for values less than one; keep the two significant figuares. 3.3x1010 5.4x10-6 7.2x10-1 4.5x100=
Answer:
The required answers are:
33,000,000,000
0.0000054
0.72
4.5
Step-by-step explanation:
Consider the provided information.
We can convert the scientific notation into long hand notation as shown:
If the exponent of 10 is a positive number then move the decimal point right as much as the exponent value.
If the exponent of 10 is a negative number then move the decimal point left as much as the exponent value.
Part (A) [tex]3.3\times 10^{10}[/tex]
Here the exponent of 10 is 10 which is a positive number, so move the decimal point right as shown:
[tex]3.3\times 10^{10}=33,000,000,000[/tex]
Part (B) [tex]5.4\times 10^{-6}[/tex]
Here the exponent of 10 is -6 which is a negative number, so move the decimal point left as shown:
[tex]5.4\times 10^{-6}=0.0000054[/tex]
Part (C) [tex]7.2\times 10^{-1}[/tex]
Here the exponent of 10 is -1 which is a negative number, so move the decimal point left as shown:
[tex]7.2\times 10^{-1}=0.72[/tex]
Part (D) [tex]4.5\times 10^0[/tex]
Here the exponent of 10 is 0. Use the property of exponent [tex]a^0=1[/tex]
[tex]4.5\times 10^0=4.5[/tex]
Determine if the following system is linear and/or time-invariant y(t) = cos(3t) x(t)
Answer:
The following system is not linear.
The following system is time-invariant
Step-by-step explanation:
To determine whether a system is linear, the following condition must be satisfied:
[tex]f(a) + f(b) = f(a+b)[/tex]
For [tex]y(t) = cos(3t)[/tex], we have
[tex]y(a) = cos(3at)[/tex]
[tex]y(b) = cos(3bt)[/tex]
[tex]y(a+b) = cos(3(a+b)t) = cos(3at + 3bt)[/tex]
In trigonometry, we have that:
[tex]cos(a+b) = cos(a)cos(b) - sin(a)sin(b)[/tex]
So
[tex]cos(3at + 3bt) = cos(3at)cos(3bt) - sin(3at)sin(3bt)[/tex]
[tex]y(a) + y(b) = cos(3at) + cos(3bt)[/tex]
[tex]y(a+b) = cos(3(a+b)t) = cos(3at + 3bt) = cos(3at)cos(3bt) - sin(3at)sin(3bt)[/tex]
Since [tex]y(a) + y(b) \neq y(a+b)[/tex], the system [tex]y(t) = cos(3t)[/tex] is not linear.
If the signal is not multiplied by time, it is time-invariant. So [tex]y(t) = cos(3t)[/tex]. Now, for example, if we had [tex]y(t) = t*cos(3t)[/tex] it would not be time invariant.
Prove the identity (a + b)^2 = a ^2 + 2ab + b^2 for all natural numbers a, b.
Step-by-step explanation:
To prove the identity we just manually compute the left hand side of it, simplify it and check that we do get the right hand side of it:
[tex](a+b)^2=(a+b)\cdot(a+b)[/tex] (that's the definition of squaring a number)
[tex]=a\cdot a + a\cdot b + b \cdot a +b \cdot b[/tex] (we distribute the product)
[tex]=a^2+ab+ba+b^2[/tex] (we just use square notation instead for the first and last term)
[tex]=a^2+ab+ab+b^2[/tex] (since product is commutative, so that ab=ba)
[tex]=a^2+2ab+b^2[/tex] (we just grouped the two terms ab into a single term)
Senior management of a consulting services firm is concerned about a growing decline in the firm’s weekly number of billable hours. The firm expects each professional employee to spend at least 40 hours per week on work. In an effort to understand this problem better, management would like to estimate the standard deviation of the number of hours their employees spend on work-related activities in a typical week. Rather than reviewing the records of all the firm’s full-time employees, the management randomly selected a sample of size 51 from the available frame. The sample mean and sample standard deviations were 48.5 and 7.5 hours, respectively. Construct a 99% confidence interval for the standard deviation of the number of hours this firm’s employees spend on work-related activities in a typical week
Answer: (45.79, 51.21)
Step-by-step explanation:
Given : Significance level : [tex]\alpha: 1-0.99=0.01[/tex]
Sample size : n= 51 , which is a large sample (n>30), so we use z-test.
Critical value: [tex]z_{\alpha/2}=2.576[/tex]
Sample mean : [tex]\overline{x}= 48.5\text{ hours}[/tex]
Standard deviation : [tex]\sigma=7.5\text{ hours}[/tex]
The confidence interval for population means is given by :-
[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]
i.e. [tex]48.5\pm(2.576)\dfrac{7.5}{\sqrt{51}}[/tex]
i.e.[tex]48.5\pm2.70534112234\\\\\approx48.5\pm2.71\\\\=(48.5-2.71, 48.5+2.71)=(45.79, 51.21)[/tex]
Hence, 99% confidence interval for the standard deviation of the number of hours this firm’s employees spend on work-related activities in a typical week = (45.79, 51.21)
Final answer:
To construct a 99% confidence interval for the standard deviation of firm's employees' work hours, follow the steps to calculate the interval between 6.26 and 9.08 hours.
Explanation:
Confidence Interval Calculation:
Calculate the degrees of freedom (df) using the formula df = n - 1. For a sample size of 51, df = 51 - 1 = 50.
Determine the critical values from the chi-squared distribution for a 99% confidence level with 50 degrees of freedom. These critical values are 30.984 and 73.361.
Next, calculate the confidence interval for the standard deviation using the formula CI = sqrt((n-1) ×s² / chi-squared upper) to sqrt((n-1) ×s² / chi-squared lower), which results in 6.26 to 9.08 hours.
In how many ways can 3 girls divide 10 pennies if each must end up with at least one penny?
Answer:
36.
Step-by-step explanation:
We are asked to find the number of ways in which 3 girls can divide 10 pennies such that each must end up with at least one penny.
The selection can be done by selecting two dividing likes between the 10 pennies such that the set is divided into three parts.
Since each girl must have one penny, so no girl can have 0 penny. So the dividing like cannot be placed at end points, beginning and at the end. Therefore, we are left with 9 positions.
Now we need to find number of ways to select two positions out of the 9 positions that is C(9,2).
[tex]_{2}^{9}\textrm{C}=\frac{9!}{7!*2!}=\frac{9*8*7!}{7!*2*1}=\frac{9*8}{2}=9*4=36[/tex]
Therefore, there are 36 ways to divide 10 pennies between 3 girls.
The slope distance between two points was measured with an EDM and found to be 1223.88 ft. If the zenith angle (the angle from the vertical to the line) is 95°25'14", compute the horizontal distance. (Ans.: 1218.41 ft) 385.11 ft) 2138.77 ft) 10
Answer:
Horizontal distance = 1218.41 ft
Step-by-step explanation:
Given data:
Slope distance = 1223.88 ft
Zenith angle is = 95°25'14"
converting zenith angle into degree
Zenith angle is [tex]= 95 +\frac{25}{60} + \frac{14}{3600} =[/tex] 94.421°
Horizontal distance[tex]= S\times sin(Z)[/tex]
putting all value to get horizontal distance value
Horizontal distance [tex]= 1223.88\times sin(95.421)[/tex]
Horizontal distance = 1218.41 ft
A patient is ordered 25 mg/kg/day of a drug that comes in 250 mg tablets. The patient weighs 90 kg. The dose is to be divided into 3 daily doses and the patient is to take the medications for 15 days. How many tablets should be dispensed? DO NOT include units
Answer:
n = 135 tablets
Step-by-step explanation:
See in the picture.
A basketball team has 2 point guards, 3 shooting guards, 3 small forwards, 3 power forwards, and 2 centers. How many line-ups are possible (assum- ing that each line up consists of a point guard, a shooting guard, a small forward, a power forward, and a center).
The line ups for the basketball team is 108.
What is the combination?The arrangement of the different things or numbers in a number of ways is called the combination.
It is given that a basketball team has 2 point guards, 3 shooting guards, 3 small forwards, 3 power forwards, and 2 centres.
The Line up will be:-
Combination = 2x3x3x3x2
Combination = 108.
Hence, the lineup will have 108 arrangements.
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Final answer:
The subject of the question is Mathematics, combinatorics specifically, and we're calculating the number of basketball line-ups possible. By multiplying the choices for each position (2 point guards, 3 shooting guards, 3 small forwards, 3 power forwards, 2 centers), we find that there are 108 possible line-ups.
Explanation:
The subject of the question is Mathematics, specifically combinatorics, which is concerned with counting the different ways in which objects can be arranged or combined. In this particular problem, we are looking to find the number of possible basketball line-ups given a certain number of players available for each position.
To solve this, we need to calculate the product of the different choices for each position: 2 choices for point guard, 3 choices for shooting guard, 3 choices for small forward, 3 choices for power forward, and 2 choices for center. Since we are choosing one player for each position to form a single line-up, we multiply the number of choices for each position together.
The calculation is as follows:
For Small Forward: 3 possible choices
For Center: 2 possible choices
Thus, the total number of possible line-ups is:
2 (Point Guards) * 3 (Shooting Guards) * 3 (Small Forwards) * 3 (Power Forwards) * 2 (Centers) = 108 possible line-ups.
Suppose you are planning to sample cat owners to determine the average number of cans of cat food they purchase monthly. The following standards have been set: a confidence level of 99 percent and an error of less than 5 units. Past research has indicated that the standard deviation should be 6 units. What is the final sample required? If only 30 percent of households have a cat, what is the initial number of households that need to be contacted?
Given : Significance level : [tex]\alpha: 1-0.99=0.01[/tex]
Critical value : [tex]z_{\alpha/2}=2.576[/tex]
Margin of error : [tex]E=5[/tex]
Standard deviation : [tex]\sigma=6[/tex]
The formula to find the sample size :-
[tex]n=(\dfrac{z_{\alpha/2}\times\sigma}{E})^2[/tex]
Then, the sample size will be :-
[tex]n=(\dfrac{(2.576)\times6}{5})^2\\\\=(3.0912)^2=9.55551744\approx10[/tex]
The minimum final size sample required is 10.
If only 30 percent of households have a cat, then the proportion of households have a cat = 0.3
The formula to find the sample size :-
[tex]n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2[/tex]
Then, the sample size will be :-
[tex]n=0.3(1-0.3)(\dfrac{(2.576)}{5})^2\\\\=(0.21)(0.5152)^2=0.0557405184\approx1[/tex]
Hence, the initial number of households that need to be contacted =1
Final answer:
The final sample size required is 94 cat owners and the initial number of households that need to be contacted is approximately 314, accounting for the 30% ownership rate of cats among households.
Explanation:
To determine the final sample size required for a survey that aims to establish the average number of cans of cat food purchased by cat owners monthly, we use the formula for sample size in estimating a mean:
[tex]n = \left(\frac{Z \cdot \sigma}{E}\right)^2[/tex]
Where n is the sample size, Z is the Z-score associated with the confidence level, σ is the standard deviation, and E is the margin of error.
For a 99% confidence level, the Z-score is approximately 2.576 (from the Z-table). Given that the standard deviation (σ) is 6 units and the margin of error (E) is 5 units, the calculation is as follows:
[tex]n = (2.576 * 6 / 5)^2 \approx 9.68^2 \approx 93.7[/tex]
So, we would need at least 94 cat owners (rounding up since we cannot have a fraction of a person).
To calculate the initial number of households to be contacted, considering that only 30% have cats, you would divide the needed sample size by the proportion of households with cats:
Initial Households = Final Sample Size / Proportion of Cat Owners ≈ 94 / 0.30 ≈ 313.3
Therefore, approximately 314 households should be initially contacted to ensure that the final sample of cat owners is achieved.
What is the quotient of the fractions below? 4/7 + 7/5
Answer:
(4/7) ÷ (7/5) = 20/49
Step-by-step explanation:
You show a sum, not a quotient.
Perhaps you want the quotient ...
(4/7) ÷ (7/5) = (4/7)×(5/7) = (4·5)/(7·7) = 20/49
A bicycle shop sells only bicycles and tricycles. Altogther there are 23 seats and 50 wheels. How many bicycles and tricycles are there?
Answer:
The number of bicycles is 19.
and the number of tricycles is 4.
Step-by-step explanation:
As we know that the number of seats in bicycles as well as in tricycles is 1.
The number of wheels in bicycles is 2
and the number of wheels in tricycles is 3.
Let the number of bicycles be x.
and the number of tricycles be y.
Thus using the given information we can make equation as,
x + y = 23
and, 2x + 3y = 50
Solving these two equations:
We get, x = 19 and y = 4
Thus the number of bicycles is 19.
and the number of tricycles is 4.
If you roll one die and flip one coin, what is the probability of rolling a 2 and flipping a head? Why?
Answer:
Probability of rolling a 2 and flipping a head will be [tex]\frac{1}{12}[/tex]
Step-by-step explanation:
If we roll one die then probability to get any one side is [tex]\frac{1}{6}[/tex]
Therefore, probability to get 2 by rolling the die will be P(A) = [tex]\frac{1}{6}[/tex]
Now we flip a coin then getting head or tale probability is [tex]\frac{1}{2}[/tex]
Or probability to get head by flipping the coin P(B) = [tex]\frac{1}{2}[/tex]
Probability of happening both the events (rolling a 2 and flipping a head) will be denoted by
P(A∩B) = P(A)×P(B)
= [tex]\frac{1}{6}\times \frac{1}{2}[/tex]
= [tex]\frac{1}{12}[/tex]
Therefore, probability of rolling a 2 and flipping a head will be [tex]\frac{1}{12}[/tex]
Answer:
The probability of rolling a 2 and flipping a head is [tex]\frac{1}{12}[/tex]
Step-by-step explanation:
Notice that rolling a die and flipping a coin are two independent events this means that the probability that one event occurs in no way affects the probability of the other event occurring. When we determine the probability of two independent events we multiply the probability of the first event by the probability of the second event.
[tex]P(X and \:Y) =P(X) \cdot P(Y)[/tex]
When you roll a die there six outcomes from 1 to 6 and when you flip a coin are two possible outcomes (heads or tails).
We know that the probability of an event is
[tex]P=\frac{the \:number \:of \:wanted\:outcomes}{the \:number \:of \:possible \:outcomes}[/tex]
So the probability of rolling a 2 and flipping a head is
[tex]P(2 \:and \:H) = P(2) \cdot P(H)\\P(2 \:and \:H) = \frac{1}{2} \cdot \frac{1}{6}\\P(2 \:and \:H) = \frac{1}{12}[/tex]
Divide and simplify to the form a+bi. 5+6i 5+6i 6+ i (Simplify your answer. Type an integer or a fraction. Type your answer in the form a+bi.)
Answer:
[tex]-\frac{6}{37} + \frac{371}{37}i[/tex]
Step-by-step explanation:
We need to evaluate [tex]\frac{(5+6i)(5+6i)}{6+i}[/tex]
(5+6i)(5+6i) = (25 + 36i² + 60i) = (25 - 36 + 60i) = -11 + 60i
= [tex]\frac{-11+60i}{6+i}[/tex]
Now we rationalize the denominator.
Now, multiplying both the numerator and denominator by (6-i)
[tex]\frac{-66 + 11i + 360i - 60i^2}{36 - i^2} = \frac{-66 + 60 + 371i}{37} = \frac{-6 + 371i}{37}[/tex]
= [tex]-\frac{6}{37} + \frac{371}{37}i[/tex]
Formula used:
(a+b)² = a² + b² + 2ab
i² = -1
a shop has 8 different types of postcards and at least 13 of each type. how many selections of 12 postcards are there if
1. you can choose any number of each type?
2. You can choose at most one of type 1 and any number of the other types?
Answer:
Answered
Step-by-step explanation:
It is a combbinatorics problem. let's think as we need to do 8 partitions
of these 13 to separate the postcards of different types. So The number of
partitions of n=13 into r=8 terms counting 0's as terms as C(n+r-1,r-1).
(a)
Here n=13 and r=8, put it in the above formula so we get C(13+8-1,8-1)= C(20,7)= 77520 selections.
b).
Here, either (i) we can choose none of type I or (ii) we choose one of type I
Case(i): r=7, n=12 (Here we have only 7 types to choose from)
Case(ii): r=7, n=11 (Here we have only 11 cards to choose and only 7 types to choose them from)
Case (i) + Case(ii) = ,C(12+7-1,7-1) + C(11+7-1,7-1) = C(18,6) + (17,6) = 18564+12376 = 30940 selections.
Karen is planning to drive 848 miles on a road trip. If she drives 212 miles a day, how many days will it take to complete the trip? # days License Points possible: 2
Answer: It will take 4 days to complete the whole trip.
Step-by-step explanation:
Given : Karen is planning to drive 848 miles on a road trip.
The number of miles she travels in a day = 212 miles
Then, the number of days taken to complete the whole trip will be [Divide 848 by 212]:_
[tex]\dfrac{848}{212}=4[/tex]
Hence, it will take 4 days to complete the whole trip.
Final answer:
To find the number of days Karen needs to complete her 848-mile road trip at a rate of 212 miles per day, divide the total miles by the daily miles, which results in 4 days.
Explanation:
The student's question asks: Karen is planning to drive 848 miles on a road trip. If she drives 212 miles a day, how many days will it take to complete the trip? This is a simple division problem in mathematics. To find out how many days it will take for Karen to complete the trip, you divide the total miles of the trip (848 miles) by the number of miles she can drive in a day (212 miles/day).
So the calculation would be: 848 miles ÷ 212 miles/day = 4 day
Therefore, it will take Karen 4 days to complete her road trip if she drives 212 miles each day.
Show that 2^2n-1 +1 is divisible by 3 for all n > 1.
Answer:
The proof makes use of congruences as follows:
Step-by-step explanation:
We can prove this result using congruences module 3. First of all we shall show that
[tex]2^{2n-1}\equiv 2 \pmod{3}[/tex] for all [tex]n\in \mathbb{N}[/tex]. By induction we have
[tex]n=2[/tex]. For [tex]n=2[/tex] we have [tex]2^{4-1}=8\equiv 2 \pmod{3}[/tex]Suppose that the statement is true for [tex]n=k[/tex] and let's prove that it is also true for [tex]n=k+1[/tex]. In fact, [tex]2^{2(k+1)-1}=2^{2k-1+2}=2^{2k-1}2^{2}\equiv 2\cdot 2^{2}\equiv 8 \equiv 2 \pmod{3}[/tex]Then induction we proved that [tex]2^{2n-1}\equiv 2 \pmod{3}[/tex] for all [tex]n>1[/tex]. Then
[tex]2^{2n-1}+1\equiv 2+1\equiv 3\equiv 0 \pmod{3}[/tex]
From here we conclude that the expression [tex]2^{2n-1}+1[/tex] is divisible by 3.
Solve the differential equation t dy/dt + dy/dt = te ^y
The answer is
t dy / dt + dy / dt = te ^ y
I apply common factor 1/dt*(tdy+dy)=te^y
I pass "dt" tdy+dy=(te^y)*dt
I apply common factor (t+1)*dy=(te^y)*dt
I pass "e^y" (1/e^y)dy=((t+1)*t)*dt
I apply integrals ∫ (1/e^y)dy= ∫ (t^2+t)*dt
by property of integrals ∫ (1/e^y)dy= ∫ (e^-y)dy
∫ (e^-y)dy= ∫ (t^2+t)*dt
I apply integrals
-e^-y=(t^3/3)+(t^2/2)+C
I apply natural logarithm to eliminate "e"
-ln (e^-y)=-ln(t^3/3)+(t^2/2)+C
y=ln(t^3/3)+(t^2/2)+C
The general solution of the differential equation is
[tex]y = - ln( - ln |t + 1| + c)[/tex]
How to find the particular solution.
Given differential equation
[tex]t \frac{dy}{dt} + \frac{dy}{dt} = t {e}^{y} [/tex]
we can rearrange it as:
[tex](t + 1) \frac{dy}{dt} = t {e}^{y} [/tex]
Now, we can separate variables:
[tex] \frac{dy}{t {e}^{y} } = \frac{dt}{t + 1} [/tex]
Integrating both sides
[tex]∫ \frac{1}{t {e}^{y} } dy = ∫ \frac{1}{t + 1} dt[/tex]
[tex] - {e}^{ - y} = ln |t + 1| + c[/tex]
[tex] {e}^{ - y} = ln |t + 1| + c[/tex]
[tex]y = - ln( - ln |t + 1| + c[/tex]
Where C1 is the constant of integration.
The atmospheric pressures at the top and the bottom of a building are read by a barometer to be 96.0 and 98.0 kPa. If the density of air is 1.0 kg/m^3, the height of the building is (a) 17 m (b) 20 m (c) 170 m (d) 204 m (e) 252 m e
Answer:
Option d - 204 m
Step-by-step explanation:
Given : The atmospheric pressures at the top and the bottom of a building are read by a barometer to be 96.0 and 98.0 kPa. If the density of air is 1.0 kg/m³.
To find : The height of the building ?
Solution :
We have given atmospheric pressures,
[tex]P_{\text{top}}=96\ kPa[/tex]
[tex]P_{\text{bottom}}=98\ kPa[/tex]
The density of air is 1.0 kg/m³ i.e. [tex]\rho_a=1\ kg/m^3[/tex]
Atmospheric pressure reduces with altitude,
The height of the building is given by formula,
[tex]H=\frac{\triangle P}{\rho_a\times g}[/tex]
[tex]H=\frac{P_{\text{bottom}}-P_{\text{top}}}{\rho_a\times g}[/tex]
[tex]H=\frac{(98-96)\times 10^3}{1\times 9.8}[/tex]
[tex]H=\frac{2000}{9.8}[/tex]
[tex]H=204\ m[/tex]
Therefore, Option d is correct.
The height of the building is 204 meter.
Using the barometric pressure formula and the given atmospheric pressures at the top and bottom of the building, the height is calculated to be approximately 204 meters, which matches option (d).
To calculate the height of the building using the difference in atmospheric pressure at the top and bottom, we can use the barometric pressure formula P = h ρ g, where P is the pressure, h is the height, ρ is the density of the fluid (or air in this case), and g is the acceleration due to gravity (approximately 9.8 m/s²).
The difference in pressure between the two points can be used to solve for h.
Given:
Difference in atmospheric pressure ΔP = 98.0 kPa - 96.0 kPa = 2.0 kPa
Density of air ρ = 1.0 kg/m3
Acceleration due to gravity g = 9.8 m/s²
We rearrange the formula to solve for h: h = ΔP / ( ρg)
h = (2.0 kPa) / (1.0 kg/m³ · 9.8 m/s²)
h = (2000 Pa) / (9.8 N/kg)
h = 204.08 m
The height of the building is therefore approximately 204 meters, making option (d) the correct answer.
POPICS! 1-22C Is the weight of a system an extensive or intensive property?
Answer:
Extensive property
Step-by-step explanation:
The intensive properties does not depend on the amount of mass of the system or the size of the system, for example the density [tex]\rho[/tex] is a quantity that is already defined for the system, because the density of water is equal for a drop or for a pool of water.
In the case of extensive properties the value of them is proportional to the size or mass of the system. For example mass is an extensive property because depend on the amount of substance. Other example could be the enegy.
In the case of weigth is a quantity that depends on mass value, then weigth is an extensive property.
Water leaks from a vertical cylindrical tank through a small hole in its base at a volumetric rate proportional to the square root of the volume of water remaining. The tank initially contains 300 liters and 22 liters leak out during the first day A. When will the tank be half empty?t16 day (include units.) B. How much water will remain in the tank after 5 days? volume 198 L (include units)
Answer:
After 7.84 the tank be half empty. The water remain in the tank after 5 days is 198.401 L.
Step-by-step explanation:
Consider the provided information.
It is given that a small hole in its base at a rate proportional to the square root of the volume of water remaining. The tank initially contains 300 liters and 22 liters leak out during the first day.
The rate of water leak can be written as:
[tex]\frac{dV}{dt}\propto \sqrt{V}[/tex]
Let k be the constant of proportionality.
[tex]\frac{dV}{dt}=k \sqrt{V}[/tex]
Integrate both the sides as shown:
[tex]\frac{dV}{\sqrt{V}}=k dt\\\int\frac{dV}{\sqrt{V}}=\int k dt\\2\sqrt{V} =kt+c[/tex]
Since for t=0 the volume was 300.
[tex]2\sqrt{300} =k(0)+c\\20\sqrt{3} =c\\c=34.641[/tex]
Now substitute the value of c in above equation.
[tex]2\sqrt{V} =kt+34.641[/tex]
22 liters leak out during the first day, thus now the remaining volume is 300-22=278 liters.
[tex]2\sqrt{278} =k(1)+34.641\\33.347 =k+34.641\\k=33.347 -34.641\\k=-1.294[/tex]
Thus, the required equation is:[tex]2\sqrt{V} =-1.294t+34.641[/tex]
Part (A) When will the tank be half empty.
Substitute v=150 liters for half empty in above equation.
[tex]2\sqrt{150} =-1.294t+34.641[/tex]
[tex]24.495 =-1.294t+34.641[/tex]
[tex]-10.146 =-1.294t[/tex]
[tex]t=7.84[/tex]
Hence, after 7.84 the tank be half empty.
Part (B) How much water will remain in the tank after 5 days.
Substitute the value of t=5 in [tex]2\sqrt{V} =-1.294t+34.641[/tex]
[tex]2\sqrt{V} =-1.294(5)+34.641[/tex]
[tex]2\sqrt{V} =28.171[/tex]
[tex]\sqrt{V} =14.0855[/tex]
[tex]V =198.401[/tex]
Hence, the water remain in the tank after 5 days is 198.401 L.
The correct answer is A) The tank will be half empty in 16 days, B) The remaining volume after 5 days will be 198 L.
A) To find when the tank will be half empty, we need to solve the differential equation that models the rate of change of the volume of water in the tank.
Let V(t) be the volume of water remaining in the tank at time t.
The rate of change of the volume is proportional to the square root of the volume:
dV/dt = -k√V
where k is a constant that can be determined from the given information.
We know that V(0) = 300 L and V(1) = 300 - 22 = 278 L.
Substituting these values, we get:
k = 22 / √300 = 4
Solving the differential equation with the initial condition V(0) = 300, we get:
[tex]V(t) = 300^_{(1/2)}$-2t^_2[/tex]
Setting V(t) = 150 L (half of the initial volume), we get:
t = 16 days
B) To find the volume remaining after 5 days, we substitute t = 5 in the solution:
[tex]V(5) = (300^_(1/2)} - 2(5))^2 = (\sqrt{300} - 10)^2 = 198 L[/tex]
The rate of change of the volume is proportional to the square root of the volume, which leads to a separable differential equation. By using the given information to determine the constant of proportionality, we can solve the differential equation and find the time when the volume is halved. Substituting the desired time into the solution gives the remaining volume after that time.
A tank initially contains 60 gallons of brine, with 30 pounds of salt in solution. Pure water runs into the tank at 3 gallons per minute and the well-stirred solution runs out at the same rate. How long will it be until there are 23 pounds of salt in the tank? Answer: the amount of time until 23 pounds of salt remain in the tank is minutes.
Answer:
the amount of time until 23 pounds of salt remain in the tank is 0.088 minutes.
Step-by-step explanation:
The variation of the concentration of salt can be expressed as:
[tex]\frac{dC}{dt}=Ci*Qi-Co*Qo[/tex]
being
C1: the concentration of salt in the inflow
Qi: the flow entering the tank
C2: the concentration leaving the tank (the same concentration that is in every part of the tank at that moment)
Qo: the flow going out of the tank.
With no salt in the inflow (C1=0), the equation can be reduced to
[tex]\frac{dC}{dt}=-Co*Qo[/tex]
Rearranging the equation, it becomes
[tex]\frac{dC}{C}=-Qo*dt[/tex]
Integrating both sides
[tex]\int\frac{dC}{C}=\int-Qo*dt\\ln(\abs{C})+x1=-Qo*t+x2\\ln(\abs{C})=-Qo*t+x\\C=exp^{-Qo*t+x}[/tex]
It is known that the concentration at t=0 is 30 pounds in 60 gallons, so C(0) is 0.5 pounds/gallon.
[tex]C(0)=exp^{-Qo*0+x}=0.5\\exp^{x} =0.5\\x=ln(0.5)=-0.693\\[/tex]
The final equation for the concentration of salt at any given time is
[tex]C=exp^{-3*t-0.693}[/tex]
To answer how long it will be until there are 23 pounds of salt in the tank, we can use the last equation:
[tex]C=exp^{-3*t-0.693}\\(23/60)=exp^{-3*t-0.693}\\ln(23/60)=-3*t-0.693\\t=-\frac{ln(23/60)+0.693}{3}=-\frac{-0.959+0.693}{3}= -\frac{-0.266}{3}=0.088[/tex]