Answer:
5y = -× + c
Step-by-step explanation:
If two lines are perpendicular, the product of their gradient equals -1.
We should note that the general equation of a line is y = mx + c, where m is the gradient.
For the line y = 5x - 2 , the slope is 5. The line that will be perpendicular to this line will have a slope of -1/5.
A line like 5y = -x + c fits the description
Suppose Adam wants to have $750,000 in his IRA at the end of 30 years. He decides to invest in an annuity paying 6% interest, compounded annually. What does he have to contribute each year to reach this goal?
Answer:
$9486.68
Step-by-step explanation:
The future value of a annuity formula can be used:
FV = P((1+r)^n -1)/r
750000 = P(1.06^30 -1)/0.06
P = 750000(.06)/(1.06^30 -1) = 9486.68
Adam has to contribute $9,486.68 each year to reach his goal.
Adam can calculate how much he needs to contribute to his IRA each year using the future value of an annuity formula. By substituting the known values into the rearranged formula, he can find the required annual payment.
Explanation:To find out how much Adam needs to contribute each year, we can use the formula for the future value of an annuity. The future value (FV) of an annuity formula is:
FV=P*[((1+r)^n -1)/r]
Where:
P is the annual payment, r is the annual interest rate (expressed as a decimal), n is the number of periods, which in this case will be years.
In this question, we want to find P, so we rearrange the formula as follows:
P = FV / [((1+r)^n -1)/r]
Substituting the given values:
P= $750,000 / [((1+0.06)^30 -1)/0.06]
By calculating the above expression, we will get the amount Adam needs to contribute annually to reach his goal.
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Jimmy walks 9 miles West and 12 miles North. How much shorter is the diagonal distance from point A to point B than walking the distance of both sides?
Answer:
6 miles shorter
Step-by-step explanation:
Right now, Jimmy walked 21 miles. If he had gone diagonally, he would've walked only 15 miles. This is 6 miles shorter than before.
The half life of a certain tranquilizer in the bloodstream is 37 hours. How long will it take for the drug to decay to 86% of the original decay model,A=A
Answer:
8.1 hours
Step-by-step explanation:
A model of the fraction remaining can be ...
f = (1/2)^(t/37) . . . . t in hours
So, for the fraction remaining being 86%, we can solve for t using ...
0.86 = 0.5^(t/37)
log(0.86) = (t/37)log(0.5)
t = 37·log(0.86)/log(0.5) ≈ 8.0509 ≈ 8.1 . . . hours
It takes about 8.1 hours to decay to 86% of the original concentration.
An adult can lose or gain two pounds of water in the course of a day. Assume that the changes in water weight are uniformly distributed between minus two and plus two pounds in a day. What is the standard deviation of your weight over a day?
Answer: 1.15 pounds
Step-by-step explanation:
For uniform distribution.
The standard deviation is :
[tex]\sigma=\sqrt{\dfrac{(b-a)^2}{12}}[/tex]
, where a = Lower limit of interval [a,b].
b = Upper limit of interval [a,b].
Given : The changes in water weight are uniformly distributed between minus two and plus two pounds in a day.
i.e. Interval = [-2 , +2]
Here , a= -2 and b= 2
Then, the standard deviation is :
[tex]\sigma=\sqrt{\dfrac{(2-(-2))^2}{12}}[/tex]
[tex]\sigma=\sqrt{\dfrac{(2+2)^2}{12}}[/tex]
[tex]\sigma=\sqrt{\dfrac{16}{12}}=\sqrt{1.3333}=1.15468610453\approx1.15[/tex]
Hence, the standard deviation of your weight over a day = 1.15 pounds
The standard deviation of the uniform distribution representing an adult's daily change in weight due to water is around 1.155 pounds.
Explanation:The question is about the standard deviation of the adult weight changes due to gain or loss in water content which is uniformly distributed between minus two and plus two pounds in a day.
To calculate the standard deviation for this uniform distribution, you need to follow these steps:
The range of the distribution is the difference between the highest and lowest values. In this case, the range is 4 pounds (2 pounds of gain - (-2 pounds of loss)). The formula of standard deviation for a uniform distribution is: sqrt((range^2) / 12). Substituting the values, the answer would be sqrt((4^2) / 12), which equals to 1.155 pounds.
So, the standard deviation of your weight changes over a day due to the water flux is approximately 1.155 pounds.
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Find an equation of the line that passes through the point (-1, 7) and is parallel to the line passing through the points (-3, -4) and (1, 4). (Let x be the independent variable and y be the dependent variable.)
To find a line parallel to another, determine the slope of the original line, which is 2 in this case, and then use the point-slope form with a given point and the same slope to find the equation, resulting in y = 2x + 9.
Explanation:To find an equation of the line that is parallel to another, we must first determine the slope of the given line. The line passing through the points (-3, -4) and (1, 4) has a slope calculated by the formula ∆y/∆x = (4 - (-4))/(1 - (-3)) = 8/4 = 2.
Since parallel lines have the same slope, our new line will also have a slope of 2. We can use the point-slope form of a line's equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Substituting our known point (-1, 7) and the slope 2, we get y - 7 = 2(x - (-1)). Simplified, the equation of our parallel line is y = 2x + 9.
At the center of espionage in Kznatropsk one is thinking of a new method for sending Morse telegrams. Instead of using the traditional method, that is, to send letters in groups of 5 according to a Poisson process with intensity 1, one might send them one by one according to a Poisson process with intensity 5. Before deciding which method to use one would like to know the following: What is the probability that it takes less time to send one group of 5 letters the traditional way than to send 5 letters the new way (the actual transmission time can be neglected).
Answer:
It takes less time sending 5 letters the traditional way with a probability of 36.7%.
Step-by-step explanation:
First we must take into account that:
- The traditional method is distributed X ~ Poisson(L = 1)
- The new method is distributed X ~ Poisson(L = 5)
[tex]P(X=x)=\frac{L^{x}e^{-L}}{x!}[/tex]
Where L is the intensity in which the events happen in a time unit and x is the number of events.
To solve the problem we must calculate the probability of events (to send 5 letters) in a unit of time for both methods, so:
- For the traditional method:
[tex]P(X=5)=\frac{1^{5}e^{-1}}{1!}\\\\P(X=5) = 0.367[/tex]
- For the new method:
[tex]P(X=5)=\frac{5^{5}e^{-5}}{5!}\\\\P(X=5) = 0.175[/tex]
According to this calculations we have a higher probability of sending 5 letters with the traditional method in a unit of time, that is 36.7%. Whereas sending 5 letters with the new method is less probable in a unit of time. In other words, we have more events per unit of time with the traditional method.
A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses, estimate the probability of getting at least 20% correct. 0.1492 0.3508 0.0901 0.8508 Normal approximation is not suitable.
Answer:
Option 4 - 0.8508
Step-by-step explanation:
Given : A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses.
To find : Estimate the probability of getting at least 20% correct ?
Solution :
20% correct out of 60,
i.e. [tex]20\%\times 60=\frac{20}{100}\times 60=12[/tex]
Minimum of 12 correct out of 60 i.e. x=12
Each question has 4 possible answers of which one is correct.
i.e. probability of answering question correctly is [tex]p=\frac{1}{4}=0.25[/tex]
Total question n=60.
Using a binomial distribution,
[tex]P(X\geq 12)=1-P(X\leq 11)[/tex]
[tex]P(X\geq 12)=1-[P(X=0)+P(X=1)+P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)][/tex]
[tex]P(X\geq 12)=1- [^{60}C_0(0.25)^0(1-0.25)+^{60-0}+^{60}C_1(0.25)^1(1-0.25)^{60-1}+^{60-1}+^{60}C_2(0.25)^2(1-0.25)^{60-2}+^{60}C_3(0.25)^3(1-0.25)^{60-3}+^{60}C_4(0.25)^4(1-0.25)^{60-4}+^{60}C_5(0.25)^5(1-0.25)^{60-5}+^{60}C_6(0.25)^6(1-0.25)^{60-6}+^{60}C_7(0.25)^7(1-0.25)^{60-7}+^{60}C_8(0.25)^8(1-0.25)^{60-8}+^{60}C_9(0.25)^9(1-0.25)^{60-9}+^{60}C_{10}(0.25)^{10}(1-0.25)^{60-10}+^{60}C_{11}(0.25)^{11}(1-0.25)^{60-11}][/tex]
[tex]P(X\geq 12)\approx 0.8508 [/tex]
Therefore, option 4 is correct.
Derrick adds equations A and B to solve this system of equations. What makes this approach a valid method in general for solving a system of equations?
Answer:
The substitution and addition properties of equality.
Step-by-step explanation:
The substitution property tells you that equals may be substituted for each other at any time.
The addition property of equality tells you that the same quantity can be added to both sides of an equation without violating the equal sign.
So, if you start with the equation ...
a = b
and you add c to both sides (addition property), you get
a + c = b + c
and if c = d, this can become (substitution property) ...
a + c = b + d . . . . . d substituted for c
In other words, we have added the equations
a = b
c = d
to get ...
a + c = b + d
The addition and substitution properties of equality make this valid.
it is b cuz im smart and i know t
QUESTION 1 A researcher compares differences in positivity between participants in a low-, middle-, or upper-middle-class family. If she observes 15 participants in each group, then what are the degrees of freedom for the one-way between-subjects ANOVA?
Answer:
The degrees of freedom for the denominator on this case is given by [tex]df_{den}=df_{between}=N-K=3*15-3=42[/tex].
Step-by-step explanation:
Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".
The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"
If we assume that we have [tex]3[/tex] groups and on each group from [tex]j=1,\dots,15[/tex] we have [tex]15[/tex] individuals on each group we can define the following formulas of variation:
[tex]SS_{total}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x)^2 [/tex]
[tex]SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2 [/tex]
[tex]SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2 [/tex]
And we have this property
[tex]SST=SS_{between}+SS_{within}[/tex]
The degrees of freedom for the numerator on this case is given by [tex]df_{num}=df_{within}=k-1=3-1=2[/tex] where k =3 represent the number of groups.
The degrees of freedom for the denominator on this case is given by [tex]df_{den}=df_{between}=N-K=3*15-3=42[/tex].
And the total degrees of freedom would be [tex]df=N-1=3*15 -1 =44[/tex]
And the F statistic to compare the means would have 2 degrees of freedom on the numerator and 42 for the denominator.
A farmer packed 3 pints of strawberries every 4 minutes. In the afternoon she packed 2 pints of strawberries every 3 minutes. What was the difference between her morning and afternoon packing rates in pints per hour?
Answer:
5 (strawberries / hours)
Step-by-step explanation:
calculation fro morning
strawberries / minutes x minutes / hours = strawberries / hours
so after adding the value in above equation
3/4* 60/1 = 45 strawberries / hours
calculation in the afternoon
strawberries / minutes x minutes / hours = strawberries / hours
2/3 x 60/1 = 40 strawberries / hours
so now by calculating difference between morning and afternoon packing rates, you can easily calculate
45-40 = 5 (strawberries / hours)
Two cars leave towns 400 kilometers apart at the same time and travel toward each other. One car's rate is 14 kilometers per hour less than the other's. If they
meet in 2 hours, what is the rate of the slower car?
Do not do any rounding.
Answer:
Step-by-step explanation:
v*2+(v-14)*2=400
2v+2v-28=400
4v=400+28
4v=428
v=107 km/h
speed of slowest car=107-14=93 km/h
A phone store employee earns a salary of $450 per week plus 10% comission on her weekly sales. A) What function rule models the employee's weekly earnings? B)If the employee earned $570 in a week, what was the amount of her sales for that week?
Answer:
A) [tex]E(w)=450+0.10w[/tex]
B) $1200
Step-by-step explanation:
Let w represent employee's weekly sales and [tex]E(w)[/tex] be total weekly earnings.
We have been given that a phone store employee earns a salary of $450 per week plus 10% commission on her weekly sales.
A) The total weekly earnings of employee would be weekly salary plus 10% of weekly sales.
10% of weekly sales, w, would be [tex]\frac{10}{100}*w=0.10w[/tex]
[tex]E(w)=450+0.10w[/tex]
Therefore, the function [tex]E(w)=450+0.10w[/tex] models the employee's weekly earnings.
B) To find the weekly sales in the week, when employee earned $570, we will substitute [tex]E(w)=570[/tex] in our formula and solve for w as:
[tex]570=450+0.10w[/tex]
[tex]570-450=450-450+0.10w[/tex]
[tex]120=0.10w[/tex]
[tex]0.10w=120[/tex]
[tex]\frac{0.10w}{0.10}=\frac{120}{0.10}[/tex]
[tex]x=1200[/tex]
Therefore, the amount of employee's weekly sales was $1200.
Identify the graph for the complex number 3 − 5i. HELP ASAP!
Answer:
Plot the point (3, -5)
Step-by-step explanation:
Recall that the general number a+bi can be plotted on a complex plane with the x axis as the real part and the y axis as the imaginary part.
In short,
a = real part = x
b = imaginary part = y
So (a,b) = (x,y)
In this case, a = 3 and b = -5 which is how I got (3, -5).
It might help to rewrite 3 - 5i into 3 + (-5)i
A road that is 11 miles long is represented on a map that shows a scale of 1 centimeter being equivalent to 10 kilometers. How many centimeters long does the road appear on the map? Round your result to the nearest tenth of a centimeter.
The 11 kilometer road will be 1.1 cm long on the map
Step-by-step explanation:
Scale is used on maps to show large locations or roads as small representatives of the larger objects.
The scale factor is usually in proportion to the original length or dimensions.
Given that
1 cm = 10 km
Then, we will divide the number of kilometers by 10 to find the length of road on map
11 km on map = [tex]\frac{11}{10}[/tex]
Hence,
The 11 kilometer road will be 1.1 cm long on the map
Keywords: Maps, Scales
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An astronaut is returning to earth in a spacecraft. If the spcecraft is descending at a rate of 13.81 kilometers per minute, what will its change be in height after 5 1/2 minutes ?
Answer:
-75.955 kilometers
Step-by-step explanation:
multiply the speed by the time to get distance
the spacecraft is descending, so change in height will most likely be answered as a negative number
13.81 × 5.5 = 75.955
The required change in the height of the spacecraft will be 76 kilometers down.
Given that, an astronaut is returning to earth in a spacecraft. If the spacecraft is descending at a rate of 13.81 kilometers per minute, what will its change be in height after 5 1/2 minutes is to be determined.
What is Distance?Distance is defined as the object traveling at a particular speed in time from one point to another.
Here,
Speed = 13.81 km/s
Time = 5 1 /2 minute = 5+0.5 minute = 5.5 minutes
Distance traveled = speed * time
Distance traveled = 13.81 * 5.5 ≈ 76 km (down)
Thus, the required change in the height of the spacecraft will be 76 kilometers down.
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Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter, she prepared an envelope with its correct address. If the 4 letters are to be put into the 4 envelopes at random, what is the probability that only 1 letter will be put into the envelope with its correct address?
A. 1/24
B. 1/8
C. 1/4
D. 1/3
E. 3/8
Answer:
The probability that only 1 letter will be put into the envelope with its correct address is [tex]\frac{1}{3}[/tex]
Step-by-step explanation:
Given:
Number of Letters=4
Number of addresses= 4
To Find:
The probability that only 1 letter will be put into the envelope with its correct address=?
Solution:
Let us assume first letter goes in correct envelope and others go in wrong envelopes, then
=> Probability putting the first letter in correct envelope =[tex]\frac{1}{4}[/tex]
=> Probability putting the second letter in correct envelope =[tex]\frac{2}{3}[/tex]
=> Probability putting the third letter in correct envelope= [tex]\frac{1}{2}[/tex]
=> Probability putting the fourth letter in correct envelope = 1;
( only 1 wrong addressed envelope is left);
This event can occur with other 3 envelopes too.
Hence total prob. = [tex]4\times(\frac{1}{4}\times\frac{2}{3}\times\frac{1}{2}\times1)[/tex]
=> [tex]\frac{1}{3}[/tex]
A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 30000 feet. The pilot would like to estimate her distance from Gateway Arch. She finds that the angle of depression to a point on the ground at the arch is 23°. Find the distance between the plane and the arch. Round your answer to the nearest foot. (Do not include ft in your answer)
Answer: 70676
Step-by-step explanation:
First we draw a diagram representing the problem, which can be found in the picture uploaded,
Point a is the point of the plane, you can see where the angle of depression is imputed in the diagram, point C is the point where the gateway arch is, and drawing a vertical line to the ground from the point of the plane, point Blank is where that vertical line touches the ground
So we can tell the the angle of depression from the plane to the arch is the same as the angel of elevation from the arch to the plane
And we are to look for the distance between B and C which is labeled x in the diagram
So looking at the right angle triangle made from this question, we can see we have the opposite length which the angle of elevation from the arch is looking at, and we are looking for the adjacent length, so we use SOH, CAH, TOA, to solve
Choosing TOA which means
Tan(angle) = (opposite length)/(adjacent length)
Tan 23 = 30000/x
Multiplying both sides by x
xtan23 = 30000
Dividing both sides by tan23
x = 30000/tan23
x = 70675.57
Approximately 70676
Which number completes the inequality?
2/3 < <7/9
3/5
6/9
3/4
6/7
Answer:
3/4
Step-by-step explanation:
It can be helpful to use a common denominator for comparison. That denominator can be 100, meaning we can make them all decimal fractions. Approximate (2 digit) values are good enough for the purpose.
2/3 ≈ 0.67 . . . . left end of the range
7/9 ≈ 0.78 . . . . right end of the range
3/5 = 0.60
6/9 ≈ 0.67 . . . . = 2/3, so is not greater than 2/3
3/4 = 0.75
6/7 ≈ 0.86
The only decimal value between 0.67 and 0.78 is 0.75, corresponding to the fraction 3/4.
2/3 < 3/4 < 7/9
A spotlight is made by placing a strong light bulb inside a reflective paraboloid formed by rotating the parabola x^2=4y around its axis of symmetry (assume that x and y are in units of inches). In order to have the brightest, most concentrated light beam, how far from the vertex should the bulb be placed? Express your answer as a fraction or a decimal rounded to two decimal places.
Answer:
1.00 inches
Step-by-step explanation:
The distance from vertex to focus is "p" in the quadratic equation ...
x^2 = 4py
In the given equation, p=1. Since units are inches, ...
the bulb should be placed 1.00 inches from the vertex.
What is the solution to the following system of equations?
x − 4y = 6
2x + 2y = 12
answer choices
(0,10)
(10,0)
(6,0)
(0,6)
Answer:
The answer to your question is (6, 0)
Step-by-step explanation:
Solve the system of equations by elimination
x - 4y = 6 (I)
2x + 2y = 12 (II)
Multiply (II) by 2
x - 4y = 6
4x + 4y = 24
Simplify
5x + 0 = 30
Find x
5x = 30
x = 30/ 5
x = 6
Find "y"
6 - 4y = 6
-4y = 6 - 6
-4y = 0
y = 0/-4
y = 0
Answer:
(6,0)
Step-by-step explanation:
Given equations are:
\[x - 4y = 6\] -------------------- (1)
\[2x + 2y = 12\] -------------------- (2)
Multiplying (1) by 2 :
\[2x - 8y = 12\] -------------------- (3)
Calculating (2) - (3) :
\[2x + 2y -2x + 8y = 12 - 12\]
=> \[10y =0\]
=> \[y = 0\]
Substituting the value of y in (1):
\[ x = 6 \]
So the required solution of the system of equations is x=6,y=0. This can be alternatively expressed in coordinate notation as (6,0).
An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population has only this risk factor (and no others). For any two of the three factors, the probability is 0.12 that she has exactly these two risk factors (but not the other). The probability that a woman has all three risk factors, given that she has A and B, is 1/3. Calculate the probability that a woman has none of the three risk factors, given that she does not have risk factor A?
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The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is calculated to be 0.54.
Explanation:To find the probability that a woman has none of the three risk factors, given that she does not have risk factor A, we will use the given probabilities and apply the principles of probability. Let's denote the probabilities of having only the risk factors as P(A), P(B), and P(C), the probabilities of having exactly two risk factors as P(A and B), P(A and C), and P(B and C), and the probability of having all three risk factors as P(A and B and C).
Given:
P(A) = P(B) = P(C) = 0.1
P(A and B) = P(A and C) = P(B and C) = 0.12
P(A and B and C | A and B) = 1/3
We can calculate P(A and B and C) using the conditional probability:
P(A and B and C) = P(A and B) × P(A and B and C | A and B) = 0.12 × 1/3 = 0.04
To find the probability of not having A, denoted as P(A'), we can use the complement rule:
P(A') = 1 - P(A) - P(A and B) - P(A and C) - P(A and B and C) = 1 - 0.1 - 0.12 - 0.12 - 0.04 = 0.62
Since P(A') includes probabilities of women with neither of the risk factors or only with B or C, we need to subtract the probabilities of having only risk factors B and C:
P(None | A') = P(A') - P(B) - P(C) + P(B and C) = 0.62 - 0.1 - 0.1 + 0.12 = 0.54
The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is thus 0.54.
Find the area. The figure is not drawn to scale.
Answer:
15
Step-by-step explanation:
to find area you need to do height times base and divide by 2 or multiply it by 1/2.
10 x 3 = 30
30/2 = 15
A study found out that 1% of social security recipients are too young to vote. If 800 social security recipients are randomly selected, find the mean, variance and standard deviation of the number of recipients who are too young to vote
Answer:
Mean : [tex]\mu=8[/tex]
Variance : [tex]\sigma^2=7.92[/tex]
Standard deviation = [tex]\sigma=2.81[/tex]
Step-by-step explanation:
We know that , in Binary Distribution having parameters p (probability of getting success in each trial) and n (Total number trials) , the mean and variance is given by:-
Mean : [tex]\mu=np[/tex]
Variance : [tex]\sigma^2=np(1-p)[/tex]
We are given that ,
Total social security recipients : n=800
The probability of social security recipients are too young to vote : p=1%= 0.01
Here success is getting social security recipients are too young to vote .
Then, the mean, variance and standard deviation of the number of recipients who are too young to vote will be :-
Mean : [tex]\mu=800\times0.01=8[/tex]
Variance : [tex]\sigma^2=800\times 0.01(1-0.01)=8\times0.99=7.92[/tex]
Standard deviation = [tex]\sigma=\sqrt{\sigma^2}=\sqrt{7.92}=2.81424945589\approx2.81[/tex]
Hence, the mean, variance and standard deviation of the number of recipients who are too young to vote :
Mean : [tex]\mu=8[/tex]
Variance : [tex]\sigma^2=7.92[/tex]
Standard deviation = [tex]\sigma=2.81[/tex]
In a sample of 800 social security recipients, with 1% being too young to vote, the mean is 8, the variance is 7.92, and the standard deviation is approximately 2.81.
The study indicates that 1% of social security recipients are too young to vote. When sampling 800 social security recipients, we treat the number of recipients too young to vote as a binomial random variable (since each recipient is either too young or not, with a fixed probability of being too young).
To find the mean of the binomial distribution, we use the formula:
Mean = n * p
Where n is the sample size (800) and p is the probability of success (0.01).
Mean = 800 * 0.01 = 8
The variance of the binomial distribution is given by the formula:
Variance = n * p * (1 - p)
Variance = 800 * 0.01 * (1 - 0.01) = 7.92
To calculate the standard deviation, we take the square root of the variance.
Standard Deviation = √(Variance) = √(7.92) = 2.81
Let H be a subgroup of a group G. We call H characteristic in G if for any automorphism σ∈Aut(G) of G, we have σ(H)=H.
(a) Prove that if σ(H)⊂H for all σ∈Aut(G), then H is characteristic in G.
(b) Prove that the center Z(G) of G is characteristic in G.
Answer:Problem 1. Let G be a group and let H, K be two subgroups of G. Dene the set HK = {hk : h ∈ H,k ∈ K}.
a) Prove that if both H and K are normal then H ∩ K is also a normal subgroup of G.
b) Prove that if H is normal then H ∩ K is a normal subgroup of K.
c) Prove that if H is normal then HK = KH and HK is a subgroup of G.
d) Prove that if both H and K are normal then HK is a normal subgroup of G.
e) What is HK when G = D16, H = {I,S}, K = {I,T2,T4,T6}? Can you give geometric description of HK?
Solution: a) We know that H ∩ K is a subgroup (Problem 3a) of homework 33). In order to prove that it is a normal subgroup let g ∈ G and h ∈ H ∩ K. Thus h ∈ H and h ∈ K. Since both H and K are normal, we have ghg−1 ∈ H and ghg−1 ∈ K. Consequently, ghg−1 ∈ H ∩ K, which proves that H ∩ K is a normal subgroup.
b) Suppose that H G. Let K ∈ k and h ∈ H ∩ K. Then khk−1 ∈ H (since H is normal in G) and khk−1 ∈ K (since both h and k are in K), so khk−1 ∈ H ∩ K. This proves that H ∩ K K.
c) Let x ∈ HK. Then x = hk for some h ∈ H and k ∈ K. Note that x = hk = k(k−1hk). Since k ∈ K and k−1hk ∈ H (here we use the assumption that H G), we see that x ∈ KH. This shows that HK ⊆ KH. To see the opposite inclusion, consider y ∈ KH, so y = kh for some h ∈ H and k ∈ K. Thus y = (khk−1)k ∈ HK, which proves that KH ⊆ HK and therefoere HK = KH. To prove that HK is a subgroup note that e = e · e ∈ HK. If a,b ∈ HK then a = hk and b = h1k1 for some h,h1 ∈ H and k,k1 ∈ K. Thus ab = hkh1k1. Since HK = KH and kh1 ∈ KH, we have kh1 = h2k2 for some k2 ∈ K, h2 ∈ H. Consequently,
ab = h(kh1)k1 = h(h2k2)k1 = (hh2)(k2k1) ∈ HK
(since hh2 ∈ H and k2k1 ∈ K). Thus HK is closed under multiplication. Finally,
Step-by-step explanation:
Question 1(Multiple Choice Worth 1 points) (01.04 LC) Solve for x: −3x + 3 < 6 x > −1 x < −1 x < −3 x > −3
To solve -3x + 3 < 6 and find the value of x, subtract 3 from both sides and divide by -3. The solution is x > -1.
Explanation:To solve the inequality -3x + 3 < 6, we want to isolate the variable x. First, subtract 3 from both sides to get -3x < 3. Then, divide both sides by -3 (remember to flip the inequality sign when dividing by a negative) to obtain x > -1. Therefore, the solution is x > -1.
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What is the sum of a common geometric series if the first term is 8 and the common ratio is 1/2?
Answer: A
Step-by-step explanation:
The sum to infinity of a geometric series is
S (∞ ) = \frac{a}{1-r} ( - 1 < r < 1 )
where a is the first term 8 and r is the common ratio, hence
S(∞ ) = {8}{1-\{1}{2} } = {8}{1}{2} } = 16
Answer:
Step-by-step explanation:
32 i think
Joe has $200 in his savings account and is depositing $50 per month. Kathy has $50 in her account and is depositing $75 per month. In how many months will they have the same amount of money? Please show work
In 6 months, they will have same amount of money.
Step-by-step explanation:
Let,
x be the number of months
Given,
Joe's savings = $200
Per month deposit = $50
J(x)=200+50x Eqn 1
Kathy savings = $50
Per month deposit = $75
K(x)=50+75x Eqn 2
For same amount;
J(x)=K(x)
[tex]200+50x=50+75x\\50x-75x=50-200\\-25x=-150[/tex]
Dividing both sides by -25;
[tex]\frac{-25x}{-25}=\frac{-150}{-25}\\x=6[/tex]
In 6 months, they will have same amount of money.
Keywords: functions, division
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Add or subtract
x/x^2-4 -2/x^2-4
In a recent year, 32% of all college students were enrolled part-time. If 8.2 million college students were enrolled part-time that year, what was the total number of college students? Round your answer to the nearest million
Answer:
26 million
Step-by-step explanation:
8200000 / 0.32 = 25625000
25625000 rounded to nearest million = 26 million
The total number of college students is 26 million.
Given that, in a recent year, 32% of all college students were enrolled part-time and 8.2 million college students were enrolled part-time that year.
What is an equation?A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.
Let the total number of college students be x.
Now, 32% of x=8.2 million
⇒ 0.32 x=8200000
⇒ x = 8200000/3.2
⇒ x = 2562500
2562500 rounded to nearest million = 26 million
Therefore, the total number of college students is 26 million.
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A rectangular poster is to contain 81 square inches of print. The margins at the top and bottom and on each side are to be 5 inches. Find the dimensions of the page which will minimize the amount of paper used.
To minimize the paper used for a poster with a specific print area and margin size, we derive a formula for total paper area, take its derivative with respect to the print width, solve for the width, and then find the matching height.
To find the dimensions of the page that will minimize the amount of paper used for a rectangular poster that contains 81 square inches of print with 5-inch margins on all sides, we need to set up a function to minimize. Let the width of the print area be x inches, and the height be y inches. Therefore, the total dimensions of the poster will be (x + 10) inches wide and (y + 10) inches high due to the margins on each side.
The area of print is given, so x*y = 81. We will minimize the total area of the page, A = (x+10)(y+10). Substituting the value of y from the print area equation, y =[tex]\frac{81}{x}[/tex], we get A(x) = (x+10)([tex]\frac{81}{x}[/tex]+10).
Now, to find the dimensions that minimize the paper used, we will take the derivative of A(x) with respect to x, set it to zero, and solve for x. From there, we can find the corresponding value of y to get the dimensions that will use the least amount of paper while still fitting the print area and margins.