Answer:
(a) 99%
(b) 50%
Step-by-step explanation:
(a) All factorials after 1! have 2 as a factor, so are even. Thus 99 of the 100 factorials are even, for a probability of 99%.
__
(b) The first two sums are odd; the next two sums are even. The pattern repeats every four sums. There are 25 repeats of that pattern in 100 sums, so 2/4 = 50% of sums are even.
a theater has two screens and shows its movies continuously. a 30 minute documentary is shown on one. a 120 minute film is shown on the other. If both showings start at noon, how many minutes will pass before both movies begin again at the same time?
Write the standard equation of a circle with center (-4 0) and radius 3 brainly
F(x)=x^2-14x+33 enter the quadratic function in factored form
Answer:
[tex]F(x)=(x-11)(x-3)[/tex]
Step-by-step explanation:
we have
[tex]F(x)=x^{2} -14x+33[/tex]
Find the zeros of the function
F(x)=0
[tex]0=x^{2} -14x+33[/tex]
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]-33=x^{2} -14x[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]-33+49=x^{2} -14x+49[/tex]
[tex]16=x^{2} -14x+49[/tex]
Rewrite as perfect squares
[tex]16=(x-7)^{2}[/tex]
square root both sides
[tex](x-7)=(+/-)4[/tex]
[tex]x=(+/-)4+7[/tex]
[tex]x=(+)4+7=11[/tex]
[tex]x=(-)4+7=3[/tex]
so
The factors are
(x-11) and (x-3)
therefore
[tex]F(x)=(x-11)(x-3)[/tex]
The probability that an adult possesses a credit card is .70. A researcher selects two adults at random. By assuming the independence, the probability that the first adult possesses a credit card and the second adult does not possess a credit card is:
Answer: 0.21
Step-by-step explanation:
We know that if two events A and B are independent , then the probability of A and B is given by :-
[tex]\text{P and B}=P(A)\times P(B)[/tex]
Given: The probability that an adult possesses a credit card P(A)= 0 .70
The probability that an adult does not possess a credit card[tex]P(B)= 1-P(A)=0 .30[/tex]
By assuming the independence, the probability that the first adult possesses a credit card and the second adult does not possess a credit card is given by :-
[tex]0.70\times0.30=0.21[/tex]
Hence, the probability that the first adult possesses a credit card and the second adult does not possess a credit card is 0.21.
Final answer:
To find the probability that the first adult selected at random has a credit card and the second does not, multiply the probability of the first event (0.70) by the probability of the second event (0.30), which yields 0.21 or 21%.
Explanation:
The subject of this question is Mathematics, specifically dealing with probability. The question is at a High School level, focusing on the concept of independent events in probability. To calculate the probability that the first adult possesses a credit card and the second adult does not possess a credit card, we use the rule of independent events:
The probability of the first adult having a credit card is 0.70 (given).
The probability of the second adult not having a credit card is 1 - 0.70 = 0.30.
Since these two events are independent, we multiply the probabilities of each event occurring:
P(First has a credit card AND Second does not have a credit card) = P(First has a credit card) * P(Second does not have a credit card) = 0.70 * 0.30
The answer is therefore 0.21 or 21%
Determine which statements are true in reals3. (Selectall that apply.)
(a)Two lines parallel to a third line are parallel.
(b) Twolines perpendicular to a third line are parallel.
(c) Twoplanes parallel to a third plane are parallel.
(d) Twoplanes perpendicular to a third plane are parallel.
(e) Twolines parallel to a plane are parallel.
(f) Twolines perpendicular to a plane are parallel.
(g) Twoplanes parallel to a line are parallel.
(h) Twoplanes perpendicular to a line are parallel.
(i) Twoplanes either intersect or are parallel.
(j) Twolines either intersect or are parallel.
(k) A plane and a line either intersector are parallel.
Answer:
(a)Two lines parallel to a third line are parallel.
(c) Two planes parallel to a third plane are parallel.
(f) Two lines perpendicular to a plane are parallel.
(h) Two planes perpendicular to a line are parallel.
(i) Two planes either intersect or are parallel.
(k) A plane and a line either intersect or are parallel
Step-by-step explanation:
(b) Two lines perpendicular to a third line are parallel. -- No. The y-, and z-axes are perpendicular to the x-axis, but are not parallel.
(d) Two planes perpendicular to a third plane are parallel. -- No. The x-y and y-z coordinate planes are both perpendicular to the x-z coordinate plane, but are at right angles to each other.
(e) Two lines parallel to a plane are parallel. -- No. Two intersecting lines in the plane z=0 are both parallel to the plane z=1, but are not parallel to each other.
(g) Two planes parallel to a line are parallel. -- No. Both the x-z plane and the y-z plane are parallel to the line (x, y, z) = (1, 1, z), but those coordinate planes are perpendicular to each other.
(j) Two lines either intersect or are parallel. -- No. The lines may be skew, running different directions in parallel panes.
Final answer:
The student's inquiry into the truth of various geometric statements has been addressed, confirming the true relationships and correcting the false ones, based on the principles of Euclidean geometry which govern lines and planes.
Explanation:
When it comes to geometry in the context of Euclidean space, which is the setting for high school mathematics, the rules governing the behavior of lines and planes can be understood through the principles of parallel and perpendicular relationships. Now, let's assess each of the statements given by the student:
(a) True: Two lines parallel to a third line are parallel to each other based on the Transitive Property of parallel lines.(b) True: Two lines perpendicular to a third line are parallel to each other as they both create right angles with the third line, leading to them being parallel.(c) True: Two planes parallel to a third plane are parallel to each other, by the definition of parallel planes.(d) False: Two planes perpendicular to a third plane need not be parallel as they can intersect along a line.(e) True: Two lines parallel to a plane are parallel to each other as they never intersect with the plane or each other.(f) False: Two lines perpendicular to a plane are not necessarily parallel; they can intersect each other at a point.(g) False: Two planes parallel to a line are not necessarily parallel to each other; they could intersect along lines that are both parallel to the given line.(h) True: Two planes perpendicular to a line are parallel to each other as the line is a line of intersection for the planes, and they do not intersect each other anywhere else.(i) True: Two planes either intersect or are parallel, this is a foundational concept in Euclidean geometry.(j) True: Two lines either intersect or are parallel, as there is no other possibility for their relationship in Euclidean space.(k) True: A plane and a line either intersect or are parallel.These principles form the basis for understanding the complex relations of geometric shapes and objects which are important for most geometrical problems and real-world applications.
If $1000 is invested in an account earning 3% compounded monthly, how long will it take the account to grow in value to $1500? Round to the nearest month.
To calculate the time required for an investment of $1000 at 3% interest compounded monthly to grow to $1500, use the compound interest formula. Solve for 't' using natural logarithms and rounding to the nearest month.
Explanation:To determine how long it takes for $1000 invested at 3% interest compounded monthly to grow to $1500, we use the formula for compound interest:
Final Amount = Principal (1 + (Interest Rate / Number of Compounding Periods in a Year))^(Total Number of Compounding Periods)
Plugging in the values we have:
$1500 = $1000 (1 + 0.03/12)^(12t)
Where 't' is in years. To find 't', we need to isolate it in the equation:
1.5 = (1 + 0.03/12)^(12t)
Take the natural logarithm of both sides:
ln(1.5) = 12t * ln(1 + 0.03/12)
Then, solve for 't' by dividing both sides by 12 * ln(1 + 0.03/12), and round to the nearest month:
t = ln(1.5) / (12 * ln(1 + 0.03/12))
The number of geese is modeled by the function G(t) that satisfies the differential equation dG dt equals the product of G divided by 5 and the quantity 350 minus G where t is the time in years and G(0) = 100 . What is the goose population when the population is increasing most rapidly?
Answer:
175
Step-by-step explanation:
The rate of change of the goose population is a function of the population:
G'(x) = (x/5)(350 -x)
This function describes a downward-opening parabola with zeros at x=0 and x=350. The value of x halfway between these zeros, at x = 175, is where the maximum value of G'(x), hence the maximum rate of change, is located.
The goose population is increasing most rapidly when it is 175.
Six customers enter a three-floor restaurant. Each customer decides on which floor to have dinner. Assume that the decisions of different customers are independent, and that for each customer, each floor is equally likely. Find the probability that exactly one customer dines on the first floor.
The probability that exactly one customer dines on the first floor is:
0.26337
Step-by-step explanation:We need to use the binomial theorem to find the probability.
The probability of k success in n experiments is given by:
[tex]P(X=k)=n_C_k\cdot p^k\cdot (1-p)^{n-k}[/tex]
where p is the probability of success.
Here p=1/3
( It represents the probability of choosing first floor)
k=1 ( since only one customer has to chose first floor)
n=6 since there are a total of 6 customers.
This means that:
[tex]P(X=1)=6_C_1\times (\dfrac{1}{3})^1\times (1-\dfrac{1}{3})^{6-1}\\\\\\P(X=1)=6\times (\dfrac{1}{3})\times (\dfrac{2}{3})^5\\\\\\P(X=1)=0.26337[/tex]
Using the binomial distribution, it is found that there is a 0.2634 = 26.34% probability that exactly one customer dines on the first floor.
----------------
For each customer, there are only two possible outcomes, either they dine on the first floor, or they do not. The probability of a customer dining on the first floor is independent of any other customer, which means that the binomial probability distribution is used to solve this question.
----------------
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of a success on a single trial.
----------------
Six customers, thus [tex]n = 6[/tex].They are equally as likely to dine on any of the three floors, thus [tex]p = \frac{1}{3} = 0.3333[/tex].----------------
The probability that exactly one customer dines on the first floor is P(X = 1), thus:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 1) = C_{6,1}.(0.3333)^{1}.(0.6667)^{5} = 0.2634[/tex]
0.2634 = 26.34% probability that exactly one customer dines on the first floor.
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You work as a cashier for a bookstore and earn $6 per hour. You also baby sit and earn $6 per hour. You want to earn at least $60 per week, but would like to work no more than 12 hours per week.
Which system of inequalities, along with y ≥ 0 and x ≥ 0, would you use to solve the real-world problem?
Final answer:
To solve the problem, use the system of inequalities: x + y ≥ 0, x ≥ 0, y ≥ 0, 6x + 6y ≥ 60, and x + y ≤ 12.
Explanation:
To solve the given real-world problem, the system of inequalities you would use is:
x + y ≥ 0x ≥ 0y ≥ 06x + 6y ≥ 60x + y ≤ 12These inequalities represent the conditions that need to be met: x and y (representing the number of hours worked as a cashier and as a babysitter, respectively) must be greater than or equal to 0, and the total income from both jobs (6x + 6y) must be greater than or equal to $60, and the total number of hours worked (x + y) must be less than or equal to 12.
A family has four children. If the genders of these children are listed in the order they are born, there are sixteen possible outcomes: BBBB, BBBG, BBGB, BGBB, GBBB, BGBG, GBGB, BGGB, GBBG, BBGG, GGBB, BGGG, GBGG, GGBG, GGGB, and GGGG. Assume these outcomes are equally likely. Let represent the number of children that are girls. Find the probability distribution of .
The probability distribution of the number of female children in a family with 4 children, assuming male and female children are equally likely, is calculated by enumerating combinations for each possible number of girls and dividing by the total number of outcomes.
Explanation:This problem involves understanding the concept of probability distribution. Let's denote 'G' for girl and 'B' for boy. In a family with 4 children, every child can be either a boy or a girl which gives us 2*2*2*2 = 16 possible combinations which we see listed in the problem.
Let's represent 'X' as the number of girls in the family. X could be 0, 1, 2, 3 or 4. For each of these values of X, we need to calculate the probability, i.e., the number of combinations which satisfy each X, divided by 16 (the total possibilities).
For X=0(genders: BBBB), there is only 1 combination. Therefore, P(X=0) = 1/16.For X=1 (genders: BBBG, BBGB, BGBB, GBBB), there are 4 combinations. Therefore, P(X=1) = 4/16 = 1/4.For X=2 (genders: BGBG, BBGG, GBGB, GBBG, BGGB, GGBB), there are 6 combinations. Therefore, P(X=2) = 6/16 = 3/8.For X=3 (genders: BGGG, GBGG, GGBG, GGGB), there are 4 combinations. Therefore, P(X=3) = 4/16 = 1/4.For X=4 (gender: GGGG), there is 1 combination. Therefore, P(X=4) = 1/16.So the probability distribution of X is: P(X=0) = 1/16, P(X=1) = 1/4, P(X=2) = 3/8, P(X=3) = 1/4, P(X=4) = 1/16.
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The probability distribution of the number of girls in a family with four children is as follows: P(X = 0) = 1/16, P(X = 1) = 4/16, P(X = 2) = 6/16, P(X = 3) = 4/16, P(X = 4) = 1/16.
Explanation:The probability distribution of the number of girls in a family with four children can be determined by analyzing the possible outcomes. There are 16 possible outcomes, ranging from all boys (BBBB) to all girls (GGGG) and various combinations in between. To find the probability distribution, we need to calculate the probability of each outcome. Since all outcomes are equally likely, the probability of each outcome is 1/16. Therefore, the probability distribution is as follows:
P(X = 0) = 1/16P(X = 1) = 4/16P(X = 2) = 6/16P(X = 3) = 4/16P(X = 4) = 1/16Learn more about Probability distribution here:https://brainly.com/question/14210034
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Suppose that the distribution of touchdown passes (in football) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 touchdowns.
What is the probability that the 49 touchdowns traveled an average of less than 245 feet? Please explain how you derived your answer.
Answer: 0.2420
Step-by-step explanation:
Given: Mean : [tex]\mu = 250 \text{ feet}[/tex]
Standard deviation : [tex]\sigma =50\text{ inch}[/tex]
Sample size : [tex]n=49[/tex]
The formula to calculate z is given by :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x= 245
[tex]z=\dfrac{245-250}{\dfrac{50}{\sqrt{49}}}=-0.7[/tex]
The P Value =[tex]P(Z<245)=P(z<-0.7)=0.2419637\approx0.2420[/tex]
Hence, the probability that the 49 touchdowns traveled an average of less than 245 feet= 0.2420
The probability that the 49 touchdowns traveled an average of less than 245 feet is approximately 0.2438, or 24.38%.
Step 1:
To find the probability that the 49 touchdowns traveled an average of less than 245 feet, we can use the Central Limit Theorem (CLT) since we have a large enough sample size (49) to assume that the sample mean follows a normal distribution.
The CLT states that the distribution of sample means of a sufficiently large sample size will be approximately normal, regardless of the distribution of the original population, as long as the sample size is large enough.
Given:
- Population mean mu = 250 feet
- Population standard deviation [tex](\( \sigma \))[/tex] = 50 feet
- Sample size n = 49
Step 2:
The standard error of the sample mean SE is given by:
[tex]\[SE = \frac{\sigma}{\sqrt{n}}\][/tex]
Substituting the given values:
[tex]\[SE = \frac{50}{\sqrt{49}} = \frac{50}{7} \approx 7.14\][/tex]
Step 3:
Now, we can calculate the z-score for the sample mean of 245 feet using the formula:
[tex]\[z = \frac{\bar{x} - \mu}{SE}\][/tex]
Where:
- [tex]\( \bar{x} \)[/tex] is the sample mean
- [tex]\( \mu \)[/tex] is the population mean
- [tex]\( SE \)[/tex] is the standard error of the sample mean
Step 4:
Substituting the given values:
[tex]\[z = \frac{245 - 250}{7.14} \approx -0.6993\][/tex]
Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to this z-score.
The probability that the sample mean is less than 245 feet can be found by finding the area to the left of the z-score on the standard normal distribution curve.
From the standard normal distribution table, we find that the probability corresponding to a z-score of -0.6993 is approximately 0.2438.
Therefore, the probability that the 49 touchdowns travelled an average of less than 245 feet is approximately 0.2438, or 24.38%.
If random samples of size 525 were taken from a very large population whose population proportion is 0.3. The standard deviation of the sample proportions (i.e., the standard error of the proportion) is
Answer: 0.02
Step-by-step explanation:
Given: Sample size : [tex]n= 525[/tex]
The population proportion [tex]P=0.3[/tex]
Then, [tex]Q=1-P=1-0.3=0.7[/tex]
The formula to calculate the standard error is given by :-
[tex]S.E.\sqrt{\dfrac{PQ}{n}}[/tex]
[tex]\Rightarrow\ S.E.=\sqrt{\dfrac{0.3\times0.7}{525}}=0.02[/tex]
Hence, the standard deviation of the sample proportions (i.e., the standard error of the proportion) is 0.02.
Which of the following statements is CORRECT? a. A graph of the SML as applied to individual stocks would show required rates of return on the vertical axis and standard deviations of returns on the horizontal axis. b. An increase in expected inflation, combined with a constant real risk-free rate and a constant market risk premium, would lead to identical increases in the required returns on a riskless asset and on an average stock, other things held constant. c. If two "normal" or "typical" stocks were combined to form a 2-stock portfolio, the portfolio's expected return would be a weighted average of the stocks' expected returns, but the portfolio's standard deviation would probably be greater than the average of the stocks' standard deviations. d. If investors become more risk averse, then (1) the slope of the SML would increase and (2) the required rate of return on low-beta stocks would increase by more than the required return on high-beta stocks. e. The CAPM has been thoroughly tested, and the theory has been confirmed beyond any reasonable doubt.
b. An increase in expected inflation, combined with a constant real risk-free rate and a constant market risk premium, would lead to identical increases in the required returns on a riskless asset and on an average stock, other things held constant.
Hope this helps :)
The mean number of words per minute (WPM) read by sixth graders is 93 with a standard deviation of 22.If 30 sixth graders are randomly selected, what is the probability that the sample mean would be greater than 97.95 WPM? (Round your answer to 4 decimal places)
Answer: 0.1093
Step-by-step explanation:
Given: Mean : [tex]\mu=93[/tex]
Standard deviation : [tex]\sigma = 22[/tex]
Sample size : [tex]n=30[/tex]
The formula to calculate z-score is given by :_
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x= 97.95, we have
[tex]z=\dfrac{97.95-93}{\dfrac{22}{\sqrt{30}}}\approx1.23[/tex]
The P-value = [tex]P(z>1.23)=1-P(z<1.23)=1-0.8906514=0.1093486\approx0.1093[/tex]
Hence, the probability that the sample mean would be greater than 97.95 WPM =0.1093
Solve the equation for x. If a solution is extraneous, be sure to identify it in your final answer.
Square root of x-2+8=x
ANSWER
Extraneous solution: x=6
Real solution: x=11
EXPLANATION
The given expression is
[tex] \sqrt{x - 2} + 8 = x[/tex]
Add -8 to both sides:
[tex]\sqrt{x - 2} + 8 + - 8= x + - 8[/tex]
[tex] \implies\sqrt{x - 2} = x - 8[/tex]
Square both sides.
[tex]\implies(\sqrt{x - 2} )^{2} =( x - 8)^{2} [/tex]
[tex]x - 2=( x - 8)^{2} [/tex]
We expand the to get
[tex]x - 2 = {x}^{2} - 16x + 64[/tex]
Write in standard quadratic form.
[tex] {x}^{2} - 16x - x + 64 + 2 = 0[/tex]
[tex] {x}^{2} - 17x + 66 = 0[/tex]
Factor to get:
[tex](x - 6)(x - 11) = 0[/tex]
[tex]x = 6 \: or \: \: x = 11[/tex]
We check for extraneous solutions by substituting each value of x into the original equation.
When x=6
[tex]\sqrt{6 - 2} + 8 = 6[/tex]
[tex]\sqrt{4} + 8 =6[/tex]
[tex]2 + 8 = 10 \ne8[/tex]
Hence x=6 is an extraneous solution.
When x=11
[tex]\sqrt{11- 2} + 8 = 11[/tex]
[tex]\sqrt{9} + 8 = 11[/tex]
[tex]3 + 8 = 11[/tex]
This statement is true.
Hence x=11 is the only solution.
find the solution of the following system of equations -5+2y=9 3x+5y=7
The solution to the system of equations is [tex]\(x = -1\) and \(y = 2\)[/tex].
To solve the system of equations:
[tex]\[ \begin{cases} -5x + 2y = 9 \\ 3x + 5y = 7 \end{cases} \][/tex]
We can use either the substitution method or the elimination method. Let's use the elimination method here.
First, let's rewrite the equations in standard form:
Equation 1: [tex]\( -5x + 2y = 9 \)[/tex]
Equation 2: [tex]\( 3x + 5y = 7 \)[/tex]
To eliminate one of the variables, let's multiply Equation 1 by 3 and Equation 2 by 5 to make the coefficients of x the same:
[tex]\[ \begin{cases} -15x + 6y = 27 \\ 15x + 25y = 35 \end{cases} \][/tex]
Now, let's add the two equations:
[tex]\[ (-15x + 6y) + (15x + 25y) = 27 + 35 \]\[ -15x + 6y + 15x + 25y = 62 \]\[ 31y = 62 \][/tex]
Now, let's solve for y:
[tex]\[ y = \frac{62}{31} \][/tex]
y=2
Now that we have found the value of y, let's substitute it back into one of the original equations to find x. Let's use Equation 1:
[tex]\[ -5x + 2(2) = 9 \]\[ -5x + 4 = 9 \]\[ -5x = 9 - 4 \]\[ -5x = 5 \]\[ x = \frac{5}{-5} \][/tex]
[tex]\[ x = -1 \][/tex]
Complete question: Find the solution of the following system of equations
-5x+2y=9
3x+5y=7
\[ x = -\frac{28}{3} \] and \( y = 7 \) are the solutions to the system of equations.
To solve the system of equations:
1. -5 + 2y = 9
2. 3x + 5y = 7
We can start by solving equation 1 for [tex]\( y \):[/tex]
[tex]\[ -5 + 2y = 9 \][/tex]
Add 5 to both sides:
[tex]\[ 2y = 9 + 5 \]\[ 2y = 14 \][/tex]
Divide both sides by 2:
[tex]\[ y = \frac{14}{2} \]\[ y = 7 \][/tex]
Now that we have the value of [tex]\( y \)[/tex], we can substitute it into equation 2 and solve for [tex]\( x \):[/tex]
[tex]\[ 3x + 5(7) = 7 \]\[ 3x + 35 = 7 \][/tex]
Subtract 35 from both sides:
[tex]\[ 3x = 7 - 35 \]\[ 3x = -28 \][/tex]
Divide both sides by 3:
[tex]\[ x = \frac{-28}{3} \][/tex]
So, the solution to the system of equations is [tex]\( x = -\frac{28}{3} \) and \( y = 7 \).[/tex]
The Beardstown Bearcats baseball team plays 60 percent of its games at night and 40 percent in the daytime. It wins 55 percent of its night games but only 35 percent of its day games. You read in the paper that the Bearcats won their last game against the Manteno Maulers. What is the probability that it was played at night?
Answer: 0.7021
Step-by-step explanation:
Let D be the event that team plays in day , N be the event that the team plays in night and W be the event when team wins.
Then , [tex]P(D)=0.40\ \ \ P(N)=0.60[/tex]
[tex]P(W|D})=0.35\ \ \ \ P(W|N)=0.55[/tex]
Using the law of total probability , we have
[tex]P(W)=P(D)\timesP(W|D)+P(N)\timesP(W|N)\\\\\Rightarrow\ P(W)=0.40\times0.35+0.60\times0.55=0.47[/tex]
Using Bayes theorem ,
The required probability :[tex]P(N|W)=\dfrac{P(N)P(W|N)}{P(W)}[/tex]
[tex]=\dfrac{0.60\times0.55}{0.47}=0.702127659574\approx0.7021[/tex]
Using composition of functions, determine if the two functions are inverses
of each other. Will Mark Brainliest!
The functions F(x) and G(x) are not inverses of each other.
The correct answer is B. No, because the functions contain different operations.
Given are composition of functions, F(x) = √(x) -6G(x) = (x+6)²
We need to determine if the two functions are inverses of each other.
To determine if the functions F(x) = √(x) - 6 and G(x) = (x + 6)² are inverses of each other using composition of functions, we need to check if their compositions result in the identity function.
Let's calculate the composition:
F(G(x)) = F((x + 6)²) = √((x + 6)²) - 6 = |x + 6| - 6
Now, let's calculate the composition in the reverse order:
G(F(x)) = G(√(x) - 6) = (√(x) - 6 + 6)² = (√(x))² = x
Since F(G(x)) = |x + 6| - 6 and G(F(x)) = x, we can see that they are not equal for all values of x.
Therefore, the functions F(x) and G(x) are not inverses of each other.
The correct answer is B. No, because the functions contain different operations.
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Two functions are inverses if both (f o g)(x) and (g o f)(x) are equal to x. If they are, their composition will yield x, indicating that the two functions are indeed inverses.
Explanation:To determine if two functions are inverses of each other using composition of functions, you should perform the operation (f o g)(x) and (g o f)(x). If f and g are inverse functions, both of these compositions will yield x.
Let's take the example of functions f(x) = 2x + 3 and g(x) = (x - 3) / 2. To check if they are inverses:
Compute (f o g)(x) = f(g(x)) = f((x - 3) / 2) = 2((x - 3) / 2) + 3 = xCompute (g o f)(x) = g(f(x)) = g(2x + 3) = (2x + 3 - 3) / 2 = xSince both (f o g)(x) and (g o f)(x) equals x, so f(x) and g(x) are inverses of each other.
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Which of the following is the major negative aspect of crossover designs for research studies? A. Prohibitive cost B. Residual effects C-Subject drepout D. Incomplete randomization E. Large sample size required
Answer:
D. Incomplete randomization
Step-by-step explanation:
Find the volume of the solid whose base is the circle x2+y2=25 and the cross sections perpendicular to the x-axis are triangles whose height and base are equal. Find the area of the vertical cross section A at the level x=4.
Triangles with height [tex]h[/tex] and base [tex]b[/tex], with [tex]b=h[/tex] have area [tex]\dfrac{b^2}2[/tex].
Such cross sections with the base of the triangle in the disk [tex]x^2+y^2\le25[/tex] (a disk with radius 5) have base with length
[tex]b(x)=\sqrt{25-x^2}-\left(-\sqrt{25-x^2}\right)=2\sqrt{25-x^2}[/tex]
i.e. the vertical (in the [tex]x,y[/tex] plane) distance between the top and bottom curves describing the circle [tex]x^2+y^2=25[/tex].
So when [tex]x=4[/tex], the cross section at that point has base
[tex]2\sqrt{25-16}=6[/tex]
so that the area of the cross section would be 6^2/2 = 18.
In case it's relevant, the entire solid would have volume given by the integral
[tex]\displaystyle\int_{-5}^5\frac{b(x)^2}2\,\mathrm dx=4\int_0^5(25-x^2)\,\mathrm dx=\frac{1000}3[/tex]
The question is about finding the volume of a solid with a circular base and equilateral triangular cross-sections, and the area of a cross section at x = 4. The base is defined by the circle equation x2 + y2 = 25 and the height and base of triangles are equal.
Explanation:The question relates to the calculation of the volume of a solid object and the area of its cross section. The base of the solid is a circle defined by x2 + y2 = 25, which is a circle of radius 5. As the cross sections perpendicular to the x-axis are equal in height and base, they form equilateral triangles.
So the area A of the triangle at x = 4 is given by A = 1/2 * Base * Height. But in an equilateral triangle, the base and height are equal, so A = 1/2 * b2. From the equation of circle, the value of 'b' at x = 4 can be calculated as √(25 - 42) = 3. To get the volume we integrate the area A over the x domain of [-5,5].
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math problem The number of incarcerated adults N (measured in thousands) in a certain country can be approximated by the equation N = -2.7 x^2 + 72.4x + 1911, where x is the number of years since 2000. In 2013, the number of incarcerated adults peaked. How many adults were incarcerated in that year?
Answer:
Step-by-step explanation:
-2.7(13)^2 + 72.4(13) + 1911
1,232.01 + 941.2 + 1911 = 4084.21
Explain why f(x) = x^2+4x+3/x^2-x-2 is not continuous at x = -1.
Answer:
The value of x = -1 makes the denominator of the function equal to zero. That is why this value is not included in the domain of f(x)
Step-by-step explanation:
We have the following expression
[tex]f(x) = \frac{x^2+4x+3}{x^2-x-2}[/tex]
Since the division between zero is not defined then the function f(x) can not include the values of x that make the denominator of the function zero.
Now we search that values of x make 0 the denominator factoring the polynomial [tex]x^2-x-2[/tex]
We need two numbers that when adding them get as a result -1 and when multiplying those numbers, obtain -2 as a result.
These numbers are -2 and 1
Then the factors are:
[tex](x-2) (x + 1)[/tex]
We do the same with the numerator
[tex]x^2+4x+3[/tex]
We need two numbers that when adding them get as a result 4 and when multiplying those numbers, obtain 3 as a result.
These numbers are 3 and 1
Then the factors are:
[tex](x+3)(x + 1)[/tex]
Therefore
[tex]f(x) = \frac{(x+3)(x+1)}{(x-2)(x+1)}[/tex]
Note that [tex]\frac{(x+1)}{(x+1)}=1[/tex] only if [tex]x \neq -1[/tex]
So since [tex]x = -1[/tex] is not included in the domain the function has a discontinuity in [tex]x = -1[/tex]
Final answer:
The function f(x) = (x²+4x+3)/(x²-x-2) is not continuous at x = -1 because the denominator becomes zero at that point, rendering the function undefined.
Explanation:
The function f(x) = (x²+4x+3)/(x²-x-2) is not continuous at x = -1 primarily because the denominator of the function becomes zero at x = -1.
Specifically, the denominator factors as (x-2)(x+1), and when x equals -1, the denominator equals zero, which makes the function undefined at that point.
Therefore, the function has a discontinuity at x = -1, and by definition, a function is not continuous at points where it is not defined.
What is the area of a square that measures 3.1 m on each side?
The area of a square that measures 3.1 m on each side will be 9.61 m².
How to find the area of the square?The area of the square is found as the square of the length of its side. If the length of a side is a;
Area of a square = side²
Given data;
S is the length of the side= 3.1 m
Area of a square = a²
A=a²
A= (3.1 m)²
A = 9.61 m²
Hence, the area of a square will be 9.61 m².
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The area of a square With each side measuring 3.1 m is 9.61 m², and this answer is provided with three significant figures.
Explanation:The area of a square is calculated as the product of its side lengths. Since all sides of a square are equal, if a square measures 3.1 m on each side, the area will be:
Area = side × side = 3.1 m × 3.1 m
To find this product, you multiply 3.1 by itself:
3.1 m × 3.1 m = 9.61 m²
To report this area, we express it in square meters (m²) and use the correct number of significant figures, which in this case is three, based on the given measurements of the sides of the square.
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Use a proof by contradiction to prove that underroot 3 is irrational.
Let assume that [tex]\sqrt3[/tex] is rational. Therefore we can express it as [tex]\dfrac{a}{b}[/tex] where [tex]a,b\in \mathbb{Z}[/tex] and [tex]\text{gcd}(a,b)=1[/tex].
[tex]\dfrac{a}{b}=\sqrt3\\\dfrac{a^2}{b^2}=3\\a^2=3b^2[/tex]
It means that [tex]3|a^2[/tex] and so also [tex]3|a[/tex].
Therefore [tex]a=3k[/tex] where [tex]k\in\mathbb{Z}[/tex].
[tex](3k)^2=3b^2\\9k^2=3b^2\\b^2=3k^2[/tex]
It means that [tex]3|b^2[/tex] and so also [tex]3|b[/tex].
If both [tex]a[/tex] and [tex]b[/tex] are divisible by 3, then it contradicts our initial assumption that [tex]\text{gcd}(a,b)=1[/tex]. Therefore [tex]\sqrt3[/tex] must be an irrational number.
Fewer young people are driving. In 1983, 87% of 19-year-olds had a driver’s license. Twenty-five years later that percentage had dropped to 75% (University of Michigan Transportation Research Institute website, April 7, 2012). Suppose these results are based on a random sample of 1200 19-year-olds in 1983 and again in 2008.
a. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 1983?
b. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 2008?
c. Is the margin of error the same in parts (a) and (b)? Why or why not?
Answer: the answer to this question is B
Step-by-step explanation: Hope This Helps
Formula to find the margin of error :
[tex]E=z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex] , where n= sample size , [tex]\hat{p}[/tex] is the sample proportion and z*= critical z-value.
Let p be the proportion of 19-year-olds had a driver’s license.
A) As per given , In 1983
[tex]\hat{p}=0.87[/tex]
n= 1200
Critical value for 95% confidence level is 1.96 (By z-table)
So ,Margin of error : [tex]E=(1.96)\sqrt{\dfrac{0.87(1-0.87)}{1200}}\approx0.019[/tex]
Interval : [tex](\hat{p}-E , \ \hat{p}+E)=(0.87-0.019 ,\ 0.87+0.019)[/tex]
[tex]=(0.851,\ 0.889)[/tex]
B) In 2008 ,
[tex]\hat{p}=0.75[/tex]
Margin of error : [tex]E=(1.96)\sqrt{\dfrac{0.75(1-0.75)}{1200}}\approx0.0245[/tex]
Interval : [tex](\hat{p}-E, \hat{p}+E)=(0.75-0.0245,\ 0.75+0.0245)[/tex]
[tex]=(0.7255,\ 0.7745)[/tex]
c. The margin of error is not the same in parts (a) and (b) because the sample proportion of 19-year-olds had a driver’s license are not same in both parts.
If f (x) =1/9x-2 what is f1(x)?
Answer:
[tex]\large\boxed{f^{-1}(x)=9x+18}[/tex]
Step-by-step explanation:
[tex]f(x)=\dfrac{1}{9}x-2\to y=\dfrac{1}{9}x-2\\\\\text{Exchange x to y and vice versa}\\\\x=\dfrac{1}{9}y-2\\\\\text{solve for}\ y:\\\\\dfrac{1}{9}y-2=x\qquad\text{add 2 to both sides}\\\\\dfrac{1}{9}y=x+2\qquad\text{multiply both sides by 9}\\\\9\!\!\!\!\diagup^1\cdot\dfrac{1}{9\!\!\!\!\diagup_1}y=9x+(9)(2)\\\\y=9x+18[/tex]
.....Help Please......
Answer:
i cant see the picture
Step-by-step explanation:
Solve the following system of equations.
9x + 4y = 4
-5x + 7y = 7
Answer:
this is the answer with steps
hope it helps!
Answer:
The solution is:
[tex](0, 1)[/tex]
Step-by-step explanation:
We have the following equations
[tex]9x + 4y = 4[/tex]
[tex]-5x + 7y = 7[/tex]
To solve the system multiply by [tex]\frac{9}{5}[/tex] the second equation and add it to the first equation
[tex]-5*\frac{9}{5}x + 7\frac{9}{5}y = 7\frac{9}{5}[/tex]
[tex]-9x + \frac{63}{5}y = \frac{63}{5}[/tex]
[tex]9x + 4y = 4[/tex]
---------------------------------------
[tex]\frac{83}{5}y=\frac{83}{5}[/tex]
[tex]y=1[/tex]
Now substitute the value of y in any of the two equations and solve for x
[tex]9x + 4(1) = 4[/tex]
[tex]9x +4 = 4[/tex]
[tex]9x = 4-4[/tex]
[tex]9x = 0[/tex]
[tex]x=0[/tex]
The solution is:
[tex](0, 1)[/tex]
At The Car rental Company , You must play a rate of $ 130 and then a daily fee of $ 17 Per day . Wrote a Linear Equation to describe the total Cost , y, of renting the car for x days . What is the Cost of renting a Car for 9 days With this Company...
Answer:
y = 17x + 130
For 9 days, you would pay $283.
Step-by-step explanation:
y = 17x + 130
Total cost = 17$ a day, plus the 130$ fee.
x = 9
y = (17)(9) + 130
y = 153 + 130
y = 283
The required cost of renting a car for 9d days with the company is $283.
What are equation models?The equation model is defined as the model of the given situation in the form of an equation using variables and constants.
here,
At The Car rental Company, You must pay a rate of $ 130 and then a daily fee of $ 17 Per day.
Let the number of days be x for renting a car,
According to the question,
Total cost(y) = 130 + 17x
Put x = 9
Total cost = 130 + 17×9
= $283
Thus, the required cost of renting a car for 9d days with the company is $283.
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your bike lock has 4 digits numbered 0-9. Find the total number of possible combinations for the lock.
Answer:
10,000
Step-by-step explanation:
With a bike lock, you can usually use the same number more than one... you can have them the same (7777) if you want.
So, you have 10 possibilities for the first digit, 10 again for the second digit, 10 for the third digit... and also 10 for the last digit. So...
10 * 10 * 10 * 10 = 10,000
These combinations range from 0000 to 9999
Final answer:
The total number of possible combinations for a 4-digit bike lock with each digit ranging from 0-9 is 10,000, as determined by the fundamental counting principle and calculated by multiplying 10 choices for each digit.
Explanation:
To find the total number of possible combinations for a 4-digit bike lock where each digit can range from 0-9, you apply the fundamental counting principle. This principle states that if there are n ways to perform one task and m ways to perform another task, then there are n × m ways to perform both tasks in sequence.
For the bike lock, each of the 4 digits can be chosen in 10 ways (0 through 9). Since the choice of each digit is independent of the others, you multiply the number of choices for each digit together:
Choice 1: 10 waysChoice 2: 10 waysChoice 3: 10 waysChoice 4: 10 waysTherefore, the total number of combinations is 10 × 10 × 10 × 10 = 10,000.