Before you do anything, you have to find [tex]c[/tex] such that [tex]f_{X,Y}(x,y)[/tex] is a proper joint density function. Doing the math, you'll find that [tex]c=2[/tex].
Now, determine the marginal densities:
[tex]f_X(x)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy=\int_{-x}^x2(x^2-y^2)e^{-2x}\,\mathrm dy[/tex]
[tex]\implies f_X(x)=\dfrac83x^3e^{-2x}[/tex]
[tex]f_Y(y)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dx=\int_0^\infty2(x^2-y^2)e^{-2x}\,\mathrm dx[/tex]
[tex]\implies f_Y(y)=\dfrac12-y^2[/tex]
a. Then the density of [tex]X[/tex] conditioned on [tex]Y=y[/tex] is
[tex]f_{X\mid Y}(x\mid Y=y)=\dfrac{f_{X,Y}(x,y)}{f_Y(y)}=\dfrac{4(x^2-y^2)e^{-2x}}{1-2y^2}[/tex]
b. The density of [tex]Y[/tex] conditioned on [tex]X=x[/tex] is
[tex]f_{Y\mid X}(y\mid X=x)=\dfrac{f_{X,Y}(x,y)}{f_X(x)}=\dfrac{3(x^2-y^2)}{4x^3}[/tex]
and so the distribution of [tex]Y[/tex] conditioned on [tex]X=x[/tex] is
[tex]F_{Y\mid X}(y\mid X=x)=\displaystyle\int_{-\infty}^uf_{Y\mid X}(y\mid X=x)\,\mathrm du[/tex]
[tex]F_{Y\mid X}(y\mid X=x)=\begin{cases}0&\text{for }y<-x\\\frac{2x^3+3x^2y-y^3}{4x^3}&\text{for }-x\le y\le x\\1&\text{for }y>x\end{cases}[/tex]
To find the conditional probability density of X, given Y=y>0, use Bayes' theorem. To find the conditional probability distribution of Y, given X=x, integrate the joint density over the range of y values.
Explanation:To find the conditional probability density of X, given Y=y>0, we need to calculate the conditional density fX|Y(x,y). This can be done using Bayes' theorem. First, find the marginal density of Y by integrating the joint density over the range of y values:
fY(y) = ∫f(x,y) dx = c(e^(-2y) - e^(-2y)/3)
Then, use Bayes' theorem to find the conditional density:
fX|Y(x,y) = f(x,y)/fY(y) = 4(x^2-y^2)e^(-2x)/(1-2y^2)
To find the conditional probability distribution of Y, given X=x, we need to calculate the conditional distribution FY|X(y|x). This can be found by integrating the joint density over the range of y values:
FY|X(y|x) = ∫f(x,y) dy = 3/4(x^2-y^2)
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Renuka Jain's Car Wash takes a constant time of 3.0 minutes in its automated car wash cycle. Autos arrive following a Poisson distribution at the rate of 12 per hour. Renuka wants to know: a) The average wait time in the line = nothing minutes (round your response to two decimal places). b) The average number of customers waiting in the line = nothing cars (round your response to two decimal places).
The average wait time in line at Renuka Jain's Car Wash is approximately 7.69 minutes, and the average number of customers waiting in line is approximately 1.54 cars.
Explanation:Given that Renuka Jain's Car Wash takes a constant time of 3.0 minutes for its automated car wash cycle, and cars arrive following a Poisson distribution at the rate of 12 cars per hour (meaning one car every 5 minutes on average), we can calculate the average wait time in line and the average number of customers waiting in line.
To find the average wait time, we use the formula for the wait time in a M/M/1 queue: W = 1/(μ - λ), where λ is the arrival rate and μ is the service rate. We have λ = 12 cars/hour = 0.2 cars/minute, and μ = 1 car/3 mins = 0.33 cars/minute. Thus, the average wait time is W = 1/(0.33 - 0.2) = 7.69 minutes.
For the average number of customers in the line, we use the formula L = λW, where L is the average number of customers in the line, λ is the arrival rate and W is the average wait time. L = 0.2 cars/minute * 7.69 minutes = 1.54 cars.
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Grace is three times as old as Hans, but in 5 years she will be twice as old as Hans is then. How old are they now? Set up an then solve a system of linear equations. please show step by step
Answer:
3x - y = 0; 2x - y = -5
Step-by-step explanation:
Let x be the present age of Hans and y be the present age of Grace,
Since, in present Grace is three times as old as Hans,
⇒ y = 3x
⇒ 3x - y = 0
Now, after 5 years,
The age of Hans = x + 5,
And, the age of Grace = y + 5
Also, in 5 years Grace will be twice as old as Hans is then,
⇒ y + 5 = 2 ( x + 5 )
⇒ y + 5 = 2x + 10
⇒ 2x - y = -5
Hence, the required system of linear equations is,
3x - y = 0; 2x - y = -5
Your company manufactures hot water heaters. The life spans of your product are known to be normally distributed with a mean of 13 years and a standard deviation of 1.5 years. What is the probability that the mean life span in a group of 10 randomly selected hot water heaters is between 12 and 15 years? (Round to the nearest ten-thousandth.)
Final answer:
Calculate the probability that the mean life span of a group of 10 hot water heaters is between 12 and 15 years using the standard normal distribution.
Explanation:
The probability that the mean life span in a group of 10 randomly selected hot water heaters is between 12 and 15 years can be calculated using the standard normal distribution.
Given: Mean = 13 years, Standard Deviation = 1.5 years.
Calculate the z-scores for 12 and 15 years using the formula z = (X - mean) / standard deviation.Look up the corresponding probabilities for these z-scores in the standard normal distribution table.Find the area between these two probabilities to get the final result.A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 41% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled. Assuming the paper's claim is correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the company's president possesses low ethical standards.
Answer:
P=0.3726 or 37.26%
Step-by-step explanation:
The success, with 41% of probability of occurring, is that the employee believes the company's president possesses low ethical standards. For more than 8 and less than 12 successes, it means the probability of having 9, 10 or 11 successes (all these summed).
The formula is:
[tex]b(x;n,p)= \ _nC_x*p^x*(1-p)^{n-x}[/tex]
Where x is the number of successes,n the number of trials, p the probability of success,[tex]_nC_x[/tex] refers to the combinations that can occur, and it's formula is:
[tex]_nC_x=\frac{n!}{x!(n-x)!}[/tex]
Calculating each case:
[tex]b(9,20,0.41)=\frac{20!}{9!(20-9)!}*0.41^9*(1-0.41)^{20-9}=0.1658[/tex]
[tex]b(10,20,0.41)=\frac{20!}{10!(20-10)!}*0.41^{10}*(1-0.41)^{20-10}=0.1267[/tex]
[tex]b(11,20,0.41)=\frac{20!}{11!(20-11)!}*0.41^{11}*(1-0.41)^{20-11}=0.0801[/tex]
Adding each case:
[tex]P=0.1658+0.1267+0.0801= 0.3726[/tex]
To find the probability that more than eight but fewer than twelve employees believe the company's president possesses low ethical standards, use the binomial probability formula. Calculate the probabilities for each value of k, and then sum them up to find the final probability.
Explanation:To find the probability that more than eight but fewer than twelve of the 20 sampled employees believe the company's president possesses low ethical standards, we need to use the binomial probability formula. The formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where:
P(X = k) is the probability that exactly k employees believe the president possesses low ethical standardsC(n, k) is the number of ways to choose k employees from n employeesp is the probability that one employee believes the president possesses low ethical standards (in this case, p = 0.41)n is the total number of employees sampled (in this case, n = 20)In this case, we want to find the probability that more than eight but fewer than twelve employees believe the president possesses low ethical standards. So we need to calculate the probabilities for k = 9, 10, and 11 and then sum them up:
P(X > 8 and X < 12) = P(X = 9) + P(X = 10) + P(X = 11)
Calculating each probability:
P(X = 9) = C(20, 9) * 0.41^9 * (1-0.41)^(20-9)
P(X = 10) = C(20, 10) * 0.41^10 * (1-0.41)^(20-10)
P(X = 11) = C(20, 11) * 0.41^11 * (1-0.41)^(20-11)
Once we have the individual probabilities, we can sum them up to find the final probability.
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Suppose that a company will select 3 people from a collection of 15 applicants to serve as a regional manager, a branch manager, and an assistant to the branch manager. In how many ways can the selection be made? Explain how you got your answer.
Answer: 2730
Step-by-step explanation:
Given : The number of applicants =15
The number of posts for which candidates have been applied = 3
To find the number of selections we use permutations since here order matters.
The permutations of n things taking m at a time is given by :-
[tex]^nP_m=\dfrac{n!}{(n-m)!}[/tex]
Then , the required number of ways is given by [Put n = 15 and m = 3] :-
[tex]^{15}P_3=\dfrac{15!}{(15-3)!}\\\\=\dfrac{15\times14\times13\times12!}{12!}\\\\15\times14\times13=2730[/tex]
Hence, the number of ways the selection can be made = 2730
Supposed you invested in $10,000, part at 6% annual interest and the rest at 9% annual interest. If you received a total of $684 in interest after one year, how much did you invest at each rate?
Anyone got a way to remember how to set up these word problems, or any other Algebra-Pre/Calc word problems. It's been 20 years since I learned and taught it. And word problems have always been an issue for me.
Answer:
$2,800 was invested at 9%.
$7,200 was invested at 6%.
Step-by-step explanation:
Usually, you need to assign variables to the unknowns you are looking for. Then follow the statements you are given to write equations. Then solve the equation or system of equations.
What are we being asked? The amount invested at each rate.
Assign variables:
Let x = amount invested at 6%
Let y = amount invested at 9%
Since we have two unknowns, we need two equations.
Now we follow the statements to write equations.
"you invested in $10,000, part at 6% annual interest and the rest at 9% annual interest."
The total investment is $10,000, so the sum of our two investments, each at an interest rate is $10,000.
First equation:
x + y = 10,000
We have dealt with the two amounts that were invested. Now we deal with the interest earned.
x amount invested at 6% yields 6% of x in interest in 1 year.
6% of x as a decimal is 0.06x.
y amount invested at 9% yields 9% of y in interest in 1 year.
9% of y as a decimal is 0.09y.
The total interest earned at the two rates is 0.06x + 0.09y.
We are told the total interest is $684, so that gives us the second equation.
0.06x + 0.09y = 684
We now have a system of two equations in two unknowns.
x + y = 10,000
0.06x + 0.09y = 684
Let's use the substitution method to solve the system of equations.
We solve the first equation for x:
x = 10,000 - y
Now we replace x of the seconds equation by 10,000 - y.
0.06x + 0.09y = 684
0.06(10,000 - y) + 0.09y = 684
Distribute the 0.06.
600 - 0.06y + 0.09y = 684
0.03y + 600 = 684
0.03y = 84
y = 2,800
$2,800 was invested at 9%.
x + y = 10,000
x + 2,800 = 10,000
x = 7,200
$7,200 was invested at 6%.
Check:
Let's see if 6% of $7,200 plus 9% of $2,800 adds up to $684.
0.06(7200) + 0.09(2800) = 432 + 252 = 684
Yes it does, so our answer is correct.
We would like to discern whether there are real differences between the batting performance of baseball players according to their position: outfielder (OF), infielder (IF), designated hitter (DH), and catcher (C). We will use a data set called bat10, which includes batting records of 327 Major League Baseball (MLB) players from the 2010 season. The measure we will use for the player batting performance (the outcome variable) is on-base percentage (OBP). The on base percentage roughly represents the fraction of the time a player successfully gets on base or hits a home run. For this baseball data, MSG = 0.00252 and MSE = 0.00127. Identify the degrees of freedom associated with MSG and MSE and calculate the F statistic
Answer:
Step-by-step explanation:
What is the converse of the following: "If I am hungry then l eat an apple." A. If I eat an apple then I am hungry. B. If I am hungry then I eat an apple. C. If I eat an apple then I am not hungry. D. If I'm not hungry then I don't eat an apple E. If I don't eat an apple then I'm not hungry. F. If I'm hungry then I eat an apple.
Answer:
Option A. If I eat an apple then I am hungry.
Step-by-step explanation:
we know that
To form the converse of the conditional statement, interchange the hypothesis and the conclusion.
In this problem
The hypothesis is "If I am hungry"
The conclusion is "l eat an apple."
therefore
interchange the hypothesis and the conclusion
The converse of "If I am hungry then l eat an apple." is
"If l eat an apple then I am hungry"
Answer:the 1 one, A. " If I am hungry then I eat an apple"
Step-by-step explanation:
Twice the difference of a number and three is negative two. Find the number
[tex]x=2[/tex]
Explanation:Represent the sentence mathematically. [tex]2(x-3)=-2[/tex]
Distribute. [tex]2x+(2*-3)=2x-6=-2[/tex]
Add 6 on both sides. [tex]2x=-2+6=4[/tex]
Divide both sides by 2. [tex]x=2[/tex]
What is 75percent of 300
Answer:
225
Step-by-step explanation:
To find your answer, multiply 300 by the decimal form of 75%, which is 0.75.
[tex]300 * 0.75 = 225[/tex]
Answer:
Half of 300 is 150 and half of 150 is 75, which is 25%. 75 x 3 gives us 225. 225 is the answer
Step-by-step explanation:
Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many possible selections are there?
a. 20
b. 7
c. 5!
d. 10
Answer: d. 10
Step-by-step explanation:
We know that the number of combinations of r objects selected from a group of n objects at a time is given by :-
[tex]^nC_r=\dfrac{n!}{(n-r)!r!}[/tex]
Given : The total number of letters = 5
The number of letters need to select = 2
Then , the number of combinations of 2 letters selected from a group of 5 letters at a time is given by :-
[tex]^5C_2=\dfrac{5!}{(5-2)!2!}=\dfrac{5\times4\times3!}{3!\times2}=10[/tex]
Hence, there are 10 possible selections.
The problem pertains to combinations in mathematics. When you select two letters out of five without considering the order, you use a formula of 'C(n, r) = n! / [(n-r)!r!]'. Applying this to our problem (where n=5, r=2), it gives us 10 combinations.
Explanation:The problem you're asking about is associated with combinations in combinatorial mathematics. When selecting two letters out of five (A, B, C, D, and E), we are interested in different combinations and not the order in which you select them. The standard formula to calculate combinations is C(n, r) = n! / [(n-r)!r!].
Here, n = 5 (total number of letters), and r = 2 (the number of letters you want to select). So, C(5, 2) = 5! / [(5-2)!2!] = 10. The correct answer is d. 10.
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A simple random sample of 10 households, the number of TV's that each household had is as follows: 2 , 0 , 2 , 2 , 2 , 2 , 1 , 5 , 3 , 2 Assume that it is reasonable to believe that the population is approximately normal and the population standard deviation is 0.55 . What is the lower bound of the 95% confidence interval for the mean number of TV's?
Answer: 1.758 is the lower bound of the 95% confidence interval for the mean number of TV's.
Step-by-step explanation:
Given that,
n = 10
Number of TV each household have = {2 , 0 , 2 , 2 , 2 , 2 , 1 , 5 , 3 , 2}
Standard Deviation(SD) = 0.55
95% Confidence Interval, = 0.05
Follows normal distribution,
Mean = [tex]\bar{X} = \frac{2+0+2+2+2+2+1+5+3+2}{10}[/tex]
= [tex]\frac{21}{10}[/tex]
= 2.1
Therefore, 95% Confidence Interval are as follows:
[tex]\bar{X}\pm Z_{\frac{\alpha}{2}} \times \frac{\sigma}{\sqrt{n}}[/tex]
[tex]2.1\pm 1.96 \times \frac{\0.55}{\sqrt{10}}[/tex]
Hence,
Lower bound = 2.1- 1.96 × [tex]\frac{\0.55}{\sqrt{10}}[/tex]
= 2.1- 1.96 × 0.174
= 1.758
dy/dx if y = Ln (2x3 + 3x).
Answer:
[tex]\frac{6x^2+3}{2x^3+3x}[/tex]
Step-by-step explanation:
You need to apply the chain rule here.
There are few other requirements:
You will need to know how to differentiate [tex]\ln(u)[/tex].
You will need to know how to differentiate polynomials as well.
So here are some rules we will be applying:
Assume [tex]u=u(x) \text{ and } v=v(x)[/tex]
[tex]\frac{d}{dx}\ln(u)=\frac{1}{u} \cdot \frac{du}{dx}[/tex]
[tex]\text{ power rule } \frac{d}{dx}x^n=nx^{n-1}[/tex]
[tex]\text{ constant multiply rule } \frac{d}{dx}c\cdot u=c \cdot \frac{du}{dx}[/tex]
[tex]\text{ sum/difference rule } \frac{d}{dx}(u \pm v)=\frac{du}{dx} \pm \frac{dv}{dx}[/tex]
Those appear to be really all we need.
Let's do it:
[tex]\frac{d}{dx}\ln(2x^3+3x)=\frac{1}{2x^3+3x} \cdot \frac{d}{dx}(2x^3+3x)[/tex]
[tex]\frac{d}{dx}(\ln(2x^3+3x)=\frac{1}{2x^3+3x} \cdot (\frac{d}{dx}(2x^3)+\frac{d}{dx}(3x))[/tex]
[tex]\frac{d}{dx}(\ln(2x^3+3x)=\frac{1}{2x^3+3x} \cdot (2 \cdot \frac{dx^3}{dx}+3 \cdot \frac{dx}{dx})[/tex]
[tex]\frac{d}{dx}(\ln(2x^3+3x)=\frac{1}{2x^3+3x} \cdot (2 \cdot 3x^2+3(1))[/tex]
[tex]\frac{d}{dx}(\ln(2x^3+3x)=\frac{1}{2x^3+3x} \cdot (6x^2+3)[/tex]
[tex]\frac{d}{dx}(\ln(2x^3+3x)=\frac{6x^2+3}{2x^3+3x}[/tex]
I tried to be very clear of how I used the rules I mentioned but all you have to do for derivative of natural log is derivative of inside over the inside.
Your answer is [tex]\frac{dy}{dx}=\frac{(2x^3+3x)'}{2x^3+3x}=\frac{6x^2+3}{2x^3+3x}[/tex].
Solve the Differential equation (x^2 + y^2) dx + (x^2 - xy) dy = 0
Answer:
[tex]\frac{y}{x}-2ln(\frac{y}{x}+1)=lnx+C[/tex]
Step-by-step explanation:
Given differential equation,
[tex](x^2 + y^2) dx + (x^2 - xy) dy = 0[/tex]
[tex]\implies \frac{dy}{dx}=-\frac{x^2 + y^2}{x^2 - xy}----(1)[/tex]
Let y = vx
Differentiating with respect to x,
[tex]\frac{dy}{dx}=v+x\frac{dv}{dx}[/tex]
From equation (1),
[tex]v+x\frac{dv}{dx}=-\frac{x^2 + (vx)^2}{x^2 - x(vx)}[/tex]
[tex]v+x\frac{dv}{dx}=-\frac{x^2 + v^2x^2}{x^2 - vx^2}[/tex]
[tex]v+x\frac{dv}{dx}=-\frac{1 + v^2}{1 - v}[/tex]
[tex]v+x\frac{dv}{dx}=\frac{1 + v^2}{v-1}[/tex]
[tex]x\frac{dv}{dx}=\frac{1 + v^2}{v-1}-v[/tex]
[tex]x\frac{dv}{dx}=\frac{1 + v^2-v^2+v}{v-1}[/tex]
[tex]x\frac{dv}{dx}=\frac{v+1}{v-1}[/tex]
[tex]\frac{v-1}{v+1}dv=\frac{1}{x}dx[/tex]
Integrating both sides,
[tex]\int{\frac{v-1}{v+1}}dv=\int{\frac{1}{x}}dx[/tex]
[tex]\int{\frac{v-1+1-1}{v+1}}dv=lnx + C[/tex]
[tex]\int{1-\frac{2}{v+1}}dv=lnx + C[/tex]
[tex]v-2ln(v+1)=lnx+C[/tex]
Now, y = vx ⇒ v = y/x
[tex]\implies \frac{y}{x}-2ln(\frac{y}{x}+1)=lnx+C[/tex]
AND Use el adverbio TAN o una forma del adjetivo TANTO para formar frases de comparación. (1 point each, 4 points total) Ejemplos: Jorge es alto y Felipe es alto también. Jorge es "tan" alto "como" Felipe. Yo tengo muchos problemas pero Elena no tiene muchos. Elena no tiene "tantos" problemas"como" yo (problema es una palabra masculina) AND 10. México es un país con mucha gente (más de 130 millones de personas). Aunque es más grande, la Argentina tiene menos 45 millones de personas. (1 point) --Answer below: AND 11. Ellos tienen cinco hijos y nosotre otros tenemos cinco hijos también. (1 point) --Answer below: AND 12. Carlos no tiene mucho dinero, pero Felipe es rico. (1 point) --Answer below: AND 13. Linda es muy simpática, me gusta Dolores. (1 point) pero no --Answer below:
Answer:
10. México es un país con mucha gente (más de 130 millones de personas). Aunque es más grande, la Argentina tiene menos 45 millones de personas.
ARGENTINA NO ES TAN GRANDE COMO MÉXICO.
11. Ellos tienen cinco hijos y nosotros otros tenemos cinco hijos también.
ELLOS TIENEN TANTOS HIJOS COMO NOSOTROS.
12. Carlos no tiene mucho dinero, pero Felipe es rico.
CARLOS NO TIENE TANTO DINERO COMO FELIPE.
13. Linda es muy simpática, me gusta Dolores.
LINDA ES TAN SIMPÁTICA, PERO ME GUSTA DOLORES.
The goal for the size of the Santa on a Christmas Santa cup is 3.5 cm (T) with an acceptable tolerance of ± 0.9 cm. The grand mean of the size of the Santa from the samples that were taken is 3.4 cm (m) and the standard deviation is 0.28 cm. What is CPk? (rounded to three decimals) 1.500 0.952 0.800 0.705 0.000
Answer:
The Cpk is 0.952
Step-by-step explanation:
The formula to calculate the Cpk of a process is
[tex]Cpk = min(\frac{USL-mean}{3*sigma}, \frac{mean-LSL}{3*sigma} )[/tex]
where
USL (Upper Specification Limit) =3.5cm+0.9cm = 4.4cm
LSL (Lower Specification Limit) =3.5cm-0.9cm=2.6cm
Standard Deviation = sigma = 0.28cm
Mean = 3.4cm
So,
[tex]Cpk=min(\frac{4.4-3.4}{3*0.28} ,\frac{3.4-2.6}{3*0.28})\\\\Cpk=min(\frac{1}{0.84} ,\frac{0.8}{0.84})\\\\Cpk=min(1.190 ,0.952)\\\\\\[/tex]
The Cpk is 0.952
The process capability index or CPk is calculated using the formula min([USL - m]/3σ, [m - LSL]/3σ). In this case, USL is calculated as 4.4 cm and LSL is found to be 2.6 cm. The final CPk value is the smaller of the two resulting values, which in this case is 0.952.
Explanation:The question asks for the calculation of CPk, which is an index in statistics determining the potential capability of a process in meeting the specification limits. This index considers both the variability of the process and the target in its calculation. The formula for CPk is given by
CPk = min([USL - m]/3σ, [m - LSL]/3σ)
where:
m is the grand mean, σ is the standard deviation, USL (Upper Specification Limit) is T + tolerance, and LSL (Lower Specification Limit) is T - tolerance.
Using the given values from the question,
USL = 3.5 cm + 0.9 cm = 4.4 cm,
LSL = 3.5 cm - 0.9 cm = 2.6 cm,
[USL - m]/3σ = (4.4 cm - 3.4 cm) / (3 * 0.28 cm) = 1.19,
[m - LSL]/3σ = (3.4 cm - 2.6 cm) / (3 * 0.28 cm) = 0.952.
The CPk value will be the smaller of these two values, which is 0.952.
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In the diagram, how many pairs of vertical angles are shown?
Answer:
4 Pairs.
Explanation:
A vertical angle is a set of two opposite angles, they show up when two lines intersect. Their sum is also 180°.
Answer:
4 Pairs
Step-by-step explanation:
Prove that if BA=I then BA=AB.
Answer with Step-by-step explanation:
Since we have given that
[tex]BA=I[/tex]
As we know that
AA⁻¹ = I (A is invertible matrix)
Multiplying A⁻¹ on the both the sides:
[tex]BAA^{-1}=IA^{-1}\\\\B=A^{-1}[/tex]
Using the above result, we get that
[tex]BA=I=AA^{-1}\\\\BA=AB[/tex]
Therefore, BA = AB
Hence, proved.
Solve differential equation:
y'''+4y''-16y'-64y=0 y(0)=0, y'(0)=26, y''(0)=-16
Final answer:
To solve the given differential equation y'''+4y''-16y'-64y=0 with initial conditions, we can use the characteristic equation method. By finding the roots of the characteristic equation and applying the initial conditions, the general solution is obtained as y(t) = (-16/21)e^(-8t) + (8/21)e^(2t) + (8/21)e^(-4t).
Explanation:
To solve the given differential equation, we can use the characteristic equation method. We first find the characteristic equation by substituting y = e^(mt) into the differential equation, which gives us the equation (m^3 + 4m^2 - 16m - 64)e^(mt) = 0. Since e^(mt) is never zero, we can simplify the equation to m^3 + 4m^2 - 16m - 64 = 0.
Using a numerical method or factoring, we find that the roots of the characteristic equation are m = -8, m = 2, and m = -4. Therefore, the general solution to the differential equation is y(t) = c1e^(-8t) + c2e^(2t) + c3e^(-4t), where c1, c2, and c3 are constants determined by the initial conditions.
Using the given initial conditions y(0) = 0, y'(0) = 26, and y''(0) = -16, we can solve for the constants. Substituting t = 0 into the general solution and its derivatives, we get the equations c1 + c2 + c3 = 0, -8c1 + 2c2 - 4c3 = 26, and 64c1 + 4c2 + 16c3 = -16. Solving these equations, we find c1 = -16/21, c2 = 8/21, and c3 = 8/21.
Therefore, the solution to the differential equation is y(t) = (-16/21)e^(-8t) + (8/21)e^(2t) + (8/21)e^(-4t).
We want to form a committee consisting of 3 men and 3 women, from a group of 8 women and 6 men. How many possible ways are there to form the committee if:
Answer:
1120 possible ways
Step-by-step explanation:
In order to find the answer we need to be sure what equation we need to use.
From the given example, let's consider initially only men. Because you have a total of 8 men and we need to chose only 3 men, let's suppose that the 3 chosen men are A, B, and C.
Because A,B,C is the same as choosing C,B,A, which means it doesn't matter the order of the chosen men, we need to use a 'combination equation'.
Because we have two groups (women and men) then we have:
Possible ways = 8C3 * 6C3 (which are the combinations for women and men respectively). Remember that:
nCk=n!/((n-k)!*k!) so:
Possible ways = 8!/((8-3)!*3!) * 6!/((6-3)!*3!) = 56* 20 = 1120.
In conclusion, there are 1120 possible ways.
A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 16-centimeter intervals are 6.1, 19.9, 26.7, 29.0, 27.2, 27.5, 23.6, 20.9, 15.8, 9.1, and 3.2. Use the Midpoint Rule with n = 5 to estimate the area of the wing's cross-section if a = 160. (Assume the thickness of the edges is nonzero.)
Answer has to be in cm^3
Answer:
cross sectional area of the wing's is = 3404.8 cm²
Step-by-step explanation:
using n= 5 to estimate area of the wing's
a = 160
taking sum of thickness at n = 1, 3, 5, 7, 9
so sum of the measurement of the thickness at the given position
19.9 +29.0 + 27.5 +20.9 + 9.1 = 106.4
so the thickness is 106.4/5
= 21.28 cm
cross sectional area of the wing's is = 160 × 21.28
= 3404.8 cm²
Using the Midpoint Rule with 5 intervals, the estimated area of the airplane wing's cross-section can be obtained by dividing the total span into equal parts, calculating the midpoints of the measurements, and then adding up these individual areas.
Explanation:To answer this question, we need to apply the Midpoint Rule - a method used in mathematics for approximating the definite integral of a function. The Rule works by estimating the area under the curve by rectangles, whose heights are determined by the function values at the midpoints of their bases.
Given n = 5, we divide the total measurement span (160 cm) into 5 parts. So, each part/subinterval is 32 cm.
We calculate the area of each part by multiplying its width (32 cm) by its midpoint height. For a sequence of measurements, the midpoints are obtained by averaging two consecutive measurements.
The midpoints for the given measurements are:
(6.1 + 19.9) / 2 (19.9 + 26.7) / 2 (26.7 + 29.0) / 2 (29.0 + 27.2) / 2 (27.2 + 27.5) / 2
We then sum up the areas of all parts to get the estimated area of the airplane wing's cross-section.
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use a Venn diagram and the given information to n(union) = 103, n(A) = 35, n(B) = 42, n(C) = 45, n(A intersection B) = 8, n(A intersection C) = 8, n(B intersection C) = 6, and n(A intersection (B intersection C) = 3. Find n(A intersection (B union C)'). A) 4 B) 22 C) 3 D) 26
Answer:
The correct option is B.
Step-by-step explanation:
Given information: n(A) = 35, n(B) = 42, n(C) = 45, n(A∩B) = 8, n(A∩C) = 8, n(B∩C) = 6, and n(A∩B∩C) = 3.
We need to find the value of n(A∩(B∩C)')
Using venn diagram we get
n(A∩B∩C')=n(A∩B)-n(A∩B∩C)= 8-3 = 5
n(A∩B'∩C)=n(A∩C)-n(A∩B∩C)= 8-3 = 5
n(A'∩B∩C)=n(B∩C)-n(A∩B∩C)= 6-3 = 3
n(A∩(B∪C)')=n(A)-n(A∩B'∩C)-n(A∩B∩C')-n(A∩B∩C)
n(A∩(B∪C)')=35-5-5-3 = 22
The value of n(A∩(B∪C)') is 22. Therefore the correct option is B.
A lottery has 60 numbers. To win the jackpot one needs to match all 7 numbers that are drawn by the machine. Is this a PERMUTATION or a COMBINATION problem? What is the “chance” (or, more mathematically speaking, what is the probability) to hit the jackpot?
Answer: Hence, our required probability is [tex]\dfrac{1}{386206920}[/tex]
Step-by-step explanation:
Since we have given that
Numbers in a lottery = 60
Numbers to win the jackpot = 7 numbers
We need to find the probability to hit the jackpot:
So, our required probability is given by
[tex]P=\dfrac{^7C_7}{^{60}C_7}\\\\P=\dfrac{1}{386206920}[/tex]
This is a combination problem as we need to select 7 numbers irrespective of any arrangements.
Hence, our required probability is [tex]\dfrac{1}{386206920}[/tex
Find and simplify the expression if f(x)=x^2-10.
f(4+h)-f(4)=
[tex]f(4+h)-f(4)=(4+h)^2-10-(4^2-10)\\f(4+h)-f(4)=16+8h+h^2-10-16+10\\f(4+h)-f(4)=h^2+8h[/tex]
Answer:
[tex]f (4 + h) -f (4) = h ^ 2 + 8h[/tex]
Step-by-step explanation:
We have the following quadratic function.
[tex]f (x) = x ^ 2-10[/tex]
We must find the following expression
[tex]f (4 + h) -f (4) =[/tex]
First we must find [tex]f (4 + h)[/tex]
Then substitute [tex]x = (4 + h)[/tex] in the quadratic equation:
[tex]f (4 + h) = (4 + h) ^ 2 -10\\\\f (4 + h) = 16 + 8h + h ^ 2 -10\\\\f (4 + h) = h ^ 2 + 8h +6[/tex]
Now we find [tex]f(4)[/tex]. Replace [tex]x = 4[/tex] in the function [tex]f (x)[/tex]
[tex]f (4) = (4) ^ 2-10\\\\f (4) = 16-10\\\\f (4) = 6[/tex]
Finally we have to:
[tex]f (4 + h) -f (4) = h ^ 2 + 8h +6 - 6[/tex]
[tex]f (4 + h) -f (4) = h ^ 2 + 8h[/tex]
A typical person has an average heart rate of 71.0 beats/min. Calculate the given questions. How many beats does she have in 3.0 years? How many beats in 3.00 years? And finally, how many beats in 3.000 years? Pay close attention to significant figures in this question.
Answer:
111,952,800 beats in 3 years
Step-by-step explanation:
71 beats/minute, 60 minutes/hour ~ 71x60=4,260 beats/hour
24 hours/day ~ 4,260x24=102,240 beats/day
365 days/year ~ 102,240x365=37,317,600 beats/ year
37,317,600x3=111,952,800 beats in 3 years
The heart beats 111952800 times in 3 years
From the given question, we just have to find the rate at which the heart beats.
Given;
71 beats in 1 minutesRate at which the heart beatswe can start by solving how many minutes are in 1 year.
To do that, we have to multiply 60 minutes by 24 hours by 365 days
[tex]60*24* 365=525600\\ [/tex]
We have 525600 minutes in 1 year
Now, we can multiply this value by 71 to know the number of beats in 1 year.
[tex]525600 * 71 = 37317600[/tex]
The heart beats for 37317600 times in a year.
Let's multiply this value by 3 to know how many times it beats in 3 years.
[tex]37317600 * 3 = 11952800[/tex]
The heart beats 11952800 times in 3 years.
Significant figuresWe are also asked to calculate 3.0, 3.00 and 3.000 years
In this case, 3.0 = 3.00 = 3.000 and the rate at which the heart beats is uniform or equal across the three times given.
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A sample of 230 observations is selected from a normal population for which the population standard deviation is known to be 22. The sample mean is 17. a. Determine the standard error of the mean.
The standard error of the mean can be calculated by dividing the population standard deviation, which is 22, by the square root of the number of observations, which is 230.
Explanation:In mathematics, the standard error of the mean is calculated by dividing the population standard deviation by the square root of the number of observations in the sample. In this case, the population standard deviation is given as 22, and the sample size is 230 observations.
The formula to calculate the standard error of the mean is:
Standard Error of the Mean = Population Standard Deviation / √(Number of Observations)
Plugging in the given values, this translates as:
Standard Error of the Mean = 22 / √230
Therefore, the standard error of the mean of this sample can be calculated as above. This represents the measure of statistical accuracy of the estimate of the sample mean, providing an indication of the precision of your results.
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The standard error of the mean is 1.449.
The standard error of the mean for a sample size of 230 observations, with a population standard deviation of 22, is calculated as 1.449.
The question asks for the determination of the standard error of the mean (SE) for a sample of 230 observations from a normal population with a known population standard deviation (σ) of 22. To calculate the standard error of the mean, we use the formula SE = σ / √n, where σ is the population standard deviation, and n is the sample size. In this case, n = 230.
So, SE = 22 / √230. Now we calculate the square root of 230 and then divide 22 by this number to get the standard error of the mean.
Therefore, the standard error of the mean is 1.449.
Identify the lower class limits, upper class limits, class width, class midpoints, and class boundaries for the given frequency distribution. Also identify the number of individuals included in the summary. Blood Platelet Count of Males (1000 cells/muL) Frequency 0-99 3 100-199 54 200-299 76 300-399 21 400-499 0 500-599 0 600-699 1 Identify the lower class limits (in 1000 cells/muL).
Answer:
The lower limits (in 1000 cells/muL): 0,100, 200, 300, 400, 500, 600 .
The upper limits (in 1000 cells/muL): 99, 199, 299, 399, 499, 599, 699.
The class width(in 1000 cells/muL): 100.
Class midpoints (in 1000 cells/muL): 49.5, 149.5, 249.5, 349.5, 449.5, 549.5, 649.5.
Class boundaries (in 1000 cells/muL): -0.5, 99.5, 199.5, 299.5, 399.5, 499.5, 599.5, 699.5.
Individuals included in the summary: 155.
Step-by-step explanation:
For the lower class limit is needed the smallest value in each class: 0,100, 200, 300, 400, 500, 600.
For the upper-class limit is needed the biggest value in each class: 99, 199, 299, 399, 499, 599, 699.
The class width is the difference between the lower limit of one class and the lower limit of the previous class. For example, 200 is the lower limit of one class and the lower limit of the previous class is 100, so 200-100=100.
Class midpoints are the average of the limits of a class if the limits are 0 and 99 then:
[tex]Midpoint=\frac{0+99}{2}=49.5[/tex]
[tex]Midpoint=\frac{100+199}{2}=149.5[/tex]
[tex]Midpoint=\frac{200+299}{2}=249.5[/tex]
[tex]Midpoint=\frac{300+399}{2}=349.5[/tex]
[tex]Midpoint=\frac{400+499}{2}=449.5[/tex]
[tex]Midpoint=\frac{500+599}{2}=549.5[/tex]
[tex]Midpoint=\frac{600+699}{2}=649.5[/tex]
Class boundaries are the numbers than allow to separate each class, to find each one, first calculate the gap between each class (the lower limit of one class minus the upper limit of the previous one) and divide by 2:
100-99=1
1/2=0.5
Add this number to all the upper limit to find the upper boundaries:
[tex]99+0.5=99.5[/tex]
[tex]199+0.5=199.5[/tex]
[tex]299+0.5=299.5[/tex]
[tex]399+0.5=399.5[/tex]
[tex]499+0.5=499.5[/tex]
[tex]599+0.5=599.5[/tex]
[tex]699+0.5=699.5[/tex]
And subtract this number (0.5) from the lower limit to find the lower boundaries:
[tex]0-0.5=-0.5[/tex]
[tex]100-0.5=99.5[/tex]
[tex]200-0.5=199.5[/tex]
[tex]300-0.5=299.5[/tex]
[tex]400-0.5=399.5[/tex]
[tex]500-0.5=499.5[/tex]
[tex]600-0.5=599.5[/tex]
The total of individuals is equal to the sum of all the frequencies of each class:
[tex]3+54+76+21+0+0+1= 155[/tex]
The lower class limits are 0, 100, 200, etc. The class width is 100. The summation of the frequencies, which equals the total number of individuals, is 155.
Explanation:In the given data, the lower class limits are the smallest numbers in each class or group, which are 0, 100, 200, 300, 400, 500, and 600 (in 1000 cells/muL). The upper class limits, which are the highest numbers in each class, are 99, 199, 299, 399, 499, 599, and 699.
The class width, which is the difference between two consecutive lower class limits, is 100. The class midpoints can be calculated by adding the lower and upper limits of each class and divide by 2, yielding 49.5, 149.5, 249.5, etc. The class boundaries are the numbers that separate classes without leaving gaps: -0.5, 99.5, 199.5, etc.
The total number of individuals included in the summary is the sum of the frequencies, which equals 155 in this case.
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Solves 7/4 =3/x Round to the nearest tenth.
Answer:
x = 12/7 or 1.7
Step-by-step explanation:
first, cross multiply to get 7x = 12. then, divide 12 by 7 to get 12/7, which can be simplified and rounded to 1.7.
Final answer:
To solve 7/4 = 3/x, use cross multiplication to get 7x = 12, then divide both sides by 7 to find x, which is approximately 1.7 when rounded to the nearest tenth.
Explanation:
To solve the equation 7/4 = 3/x, we can set up a proportion and use cross multiplication. Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and setting the products equal to each other. In this case, we multiply 7 by x and 4 by 3 to get the equation 7x = 12.
After cross multiplying, divide both sides of the equation by 7 to solve for x. Doing this, we find that x = 12/7. To convert this to a decimal and round to the nearest tenth, we can divide 12 by 7 using a calculator or long division, resulting in approximately 1.7.
6+√-80 ?
A.6+16√5i
B.6+4i√5
C.6+16i√5
D.6+4√5i
√-121 ?
A.-11i
B.11i
C.-11
D.11
√-48 ?
A.-4√3
B.4√-3
C.4i√3
D.4√3i
9. Calculate the area of a rectangle that is 23 feet by 16 feet.
A.420 space f t squared
B.736 space f t squared
C.78 space f t squared
D.368 space f t squared
Answer:
D.368 space ft squared
Step-by-step explanation:
Hello
The equation to find the area of the rectangle is simply A = h * b. This means that the area of a rectangle is equal to the product of its height (h) by its base (b), or of its length by its width
Let
A=h*b
h=23
b=16
A=23 ft*16 ft
A=368 ft squared
so, the answer is
D. 368 ft squared.
I hope it helps
Have a fantastic day.