Use Definition 7.1.1, DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = [infinity] e−stf(t) dt 0 is said to be the Laplace transform of f, provided that the integral converges. to find ℒ{f(t)}. (Write your answer as a function of s.) f(t) = te9t ℒ{f(t)} = (s > 9)
Answers
Answer:
e^-s/s + e^-s/s^2
Step-by-step explanation:
See the attachment please
The question asks for the Laplace transform of f(t) = te^9t using the given definition of the Laplace Transform. This can be calculated using the integral ℒ{f(t)} = ∫ (from 0 to ∞) e^{-st} te^9t dt and likely requires the technique of integration by parts for evaluation.
Explanation:The question is asking for the Laplace transform of the function f(t) = te9t, using the definition of the Laplace transform. The Laplace Transform is a method that can be used to solve differential equations. In general, the Laplace Transform of a function f(t) is defined as ℒ{f(t)} = ∫ (from 0 to ∞) e-st f(t) dt, provided that the integral converges.
In this case, f(t) is equal to te9t so the integral becomes ℒ{f(t)} = ∫ (from 0 to ∞) e-st te9t dt. To find the integral, you would generally need to use integration by parts, which is a method of integration that is typically taught in a calculus course. Note that the given condition (s > 9) will affect the convergence of the integral.
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Related Questions
Two thirds plus one sixth equals
Answers
Answer:
5/6
Step-by-step explanation:
2/3 is equivalent to 4/6
4/6 plus 1/6= 5/6
5/6
Hope this helps :)
Answer:
The answer is 5/6
Step-by-step explanation:
First Step you need to convert 2/3 into 6th'sYou need to multiply the denominator by 2 to get 6th'sRemember whatever you multiply the bottom by, you do the same to the topSo 2/3 = 4/6Finally you add 4/6+1/6= 5/6What is the interquartile range of the data ? 0,2,4,0,2,3,8,6
Answers
(arrange the data set from least to greatest)
0, 0, 2, 2, 3, 4, 6, 8
(find the median: *the middle number*)
Median: 2.5
Lower quartile: 1
Upper quartile: 5
Interquartile range : upper quartile - lower quartile = answer
Interquartile range: 5 - 1 = 4
So the IQR or interquartile for the following data set is 4.
A certain small town has a population of 5000 residents. You want to calculate a confidence interval for the average number of gallons of gas bought per month by the residents of this town. You want to be 95% confident that the true value of the population mean is within your interval, and you need to have a margin of error no higher than 10 gallons per month. Based on your previous research in simlar towns, you believe that the population standard deviation is 50 galons per month Give the appropriate statistical symbol or formula for each of the foliowing numbers in this 1, 5000= 2. 50 4 What is the value of a? 5, What is the value of α/2 ? 三6. What is the value of Zan? 1. What wil be the width of your confidence interval from the smaller number to the larger number? (Hint Think about how the MOE is related to the width of the confidence interval)
Answers
Answer:
1. 5000 = N
2. 50 = σ
3. 95% = confidence level (1-α)
4. α = 0.05
5. α/2=0.025=2.5%
6. z_(α/2)=-1.96
1. Width of confidence interval UL-LL=20
Step-by-step explanation:
We have to calculate a 95% confidence interval for the mean (average number of gallons of gas bought per month by the residents of this town).
The margin of error has to be below 10 gallons/month.
The population standard deviation is considered 50 gallons/month.
1. 5000 = N.
This is the population size N for this study, as this is the total population of the town.
2. 50 = σ
This is the value of the population standard deviation σ, as it is estimated from other studies.
3. 95% = (1-α)
This is the confidence level of the interval, and is equal to 1 less the significance level α.
4. α = 0.05
This is calculated from the confidence level. As the confidence level is 95%, the level of significance is 5%.
[tex]1-\alpha=0.95\\\\\alpha=1-0.95\\\\\alpha=0.05[/tex]
5. α/2=0.05/2=0.025=2.5%
6. The value os z_α/2 is obtained from a standard normal distribution table, where:
[tex]P(z<z_{\alpha/2})=0.025\\\\z_{\alpha/2}=-1.96[/tex]
z_α/2=-1.96
1. As the margin of error is 10 (maximum value), the difference between upper and lower bound is:
[tex]UL-LL=2*MOE=2*10=20[/tex]
The wait times in line at a grocery store are roughly distributed normally with an average wait time of 7.6 minutes and a standard deviation of 1 minute 45 seconds. What is the probability that the wait time is less than 7.9 minutes
Answers
Answer:
[tex]0.56749[/tex].
Step-by-step explanation:
We have been given that the wait times in line at a grocery store are roughly distributed normally with an average wait time of 7.6 minutes and a standard deviation of 1 minute 45 seconds. We are asked to find the probability that the wait time is less than 7.9 minutes.
First of all, we will convert 45 seconds into minutes by dividing by 60 as:
[tex]45\text{ Seconds}=\frac{45}{60}\text{ minutes}=0.75\text{ minutes}[/tex]
So 1 minute 45 seconds will be equal to 1.75 minutes.
Now, we will find z-score corresponding to 7.9 minutes using z-score formula.
[tex]z=\frac{x-\mu}{\sigma}[/tex]
z = z-score,
x = Random sample score,
[tex]\mu[/tex] = Mean,
[tex]\sigma[/tex] = Standard deviation.
[tex]z=\frac{7.9-7.6}{1.75}[/tex]
[tex]z=\frac{0.3}{1.75}[/tex]
[tex]z=0.17142\approx 0.17[/tex]
Now we will use normal distribution to find area under normal curve that corresponds to a z-score of 0.17 that is [tex]P(z<0.17)[/tex].
[tex]P(z<0.17)=0.56749[/tex]
Therefore, the probability that wait time is less than 7.9 minutes would be 0.56749.
Lily built a rectangular prism with cubes. The area of the base is 16 connecting cubes. It is 6 layers high what is the volume of his rectangular prism?
Answers
Answer:
Volume of prism is equal to [tex]96[/tex] units
Step-by-step explanation:
let the side of the cube be of one unit.
Then, the area of the base is equal to [tex]16[/tex] connecting cubes
Assuming a square base of the rectangular prism.
Length of one side of the base of rectangular prism [tex]= 4[/tex] units
Similarly, width of the base of rectangular prism [tex]= 4[/tex] units
The height of the rectangular prism is [tex]6[/tex] units
Volume of the prim [tex]=[/tex] Length [tex]*[/tex] Width [tex]*[/tex] Height
Substituting the given values in above equation, we get -
[tex]V = 4* 4* 6\\V = 96[/tex]
Volume of prism is equal to [tex]96[/tex] units
Final answer:
To find the volume of a rectangular prism, multiply the base area by the height. In this case, the volume of Lily's rectangular prism would be 96 cubic units.
Explanation:
The volume of a rectangular prism is calculated by multiplying the length, width, and height of the prism.
For this specific rectangular prism with a base area of 16 connecting cubes and 6 layers high, we can calculate the volume as follows:
Volume = Base Area x Height
Volume = 16 x 6
Volume = 96 cubic units
Brandon has an extension ladder that can only be used at a length of 10 feet, 15 feet, or 20 feet. He places the base of the latter 9 feet from the wall I need the top of the ladder to reach 12. Which ladder laying with Benton need to use to reach the height of the wall?
Answers
Answer:
15
Step-by-step explanation:
Use Pythagorean theorem.
c² = a² + b²
c² = 9² + 12²
c² = 225
c = 15
If f(x)= -x+5. Compute if f(1) + (5). Please show steps
Answers
Answer:
4
Step-by-step explanation:
[tex]f(x) = - x + 5 \\ \: f(1) = - 1+ 5 = 4 \\ \: f(5) = - 5+ 5 = 0\\ f(1) +f(5) =4 + 0 \\ \huge \red{ \boxed{f(1) +f(5) = 4}}[/tex]
The account executive of a brokerage firm has recommended that one of her customers consider stock from five aerospace companies, three energy development companies, and four electronics companies. After some research, the customer has decided to purchase shares in the stocks of two aerospace companies, two energy development companies, and two electronics companies. In how many ways can the investor select the group of six companies for the investment from the recommended list of five aerospace companies, three energy development companies, and four electronics companies
Answers
Answer:
180 ways
Step-by-step explanation:
Aerospace companies (A) = 5
Energy development companies (D) = 3
Electronics companies (E) = 4
Number of aerospace stocks bought (a) = 2
Number of Energy development stocks bought (d) = 2
Number of Electronics stocks bought (e) = 2
The number of ways that the investor can select his six investments from the recommended list is given by the combination of picking two out five aerospace companies, multiplied by picking two out three energy development companies, multiplied by picking two out four electronics companies:
[tex]n=\frac{5!}{(5-2)!2!}*\frac{3!}{(3-2)!2!}*\frac{4!}{(4-2)!2!}\\ n=10*3*6\\n=180\ ways[/tex]
There are 180 ways for the investor to select the six companies.
There are 180 ways for the investor to select the group of six companies from the recommended list of aerospace, energy development, and electronics companies.
To determine the number of ways the investor can select the group of six companies, we need to use combinations. The combination formula is given by C(n, k) = n! / [k!(n-k)!], where n is the total number of items, and k is the number of items to choose.
First, we calculate the ways to choose the aerospace companies:
C(5, 2) = 5! / [2!(5-2)!] = 10
Next, calculate the ways to choose the energy development companies:
C(3, 2) = 3! / [2!(3-2)!] = 3
Finally, calculate the ways to choose the electronics companies:
C(4, 2) = 4! / [2!(4-2)!] = 6
To find the total number of ways to select the group of six companies, multiply the three results:
Total = C(5, 2) * C(3, 2) * C(4, 2) = 10 * 3 * 6 = 180 ways
An educational psychologist wishes to know the mean number of words a third grader can read per minute. She wants to make an estimate at the 99% level of confidence. For a sample of 1584 third graders, the mean words per minute read was 35.7. Assume a population standard deviation of 3.3. Construct the confidence interval for the mean number of words a third grader can read per minute. Round your answers to one decimal place.
Answers
Answer:
99% confidence interval for the true mean number of words a third grader can read per minute is [35.5 , 35.9].
Step-by-step explanation:
We are given that a sample of 1584 third graders, the mean words per minute read was 35.7. Assume a population standard deviation of 3.3.
Firstly, the pivotal quantity for 99% confidence interval for the population mean is given by;
P.Q. = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\bar X[/tex] = sample mean words per minute read = 35.7
[tex]\sigma[/tex] = population standard deviation = 3.3
n = sample of third graders = 1584
[tex]\mu[/tex] = population mean number of words
Here for constructing 99% confidence interval we have used One-sample z test statistics as we know about the population standard deviation.
So, 99% confidence interval for the population mean, [tex]\mu[/tex] is ;
P(-2.58 < N(0,1) < 2.58) = 0.99 {As the critical value of z at 0.5% level
of significance are -2.58 & 2.58}
P(-2.58 < [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 2.58) = 0.99
P( [tex]-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.99
P( [tex]\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.99
99% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] , [tex]\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ]
= [ [tex]35.7-2.58 \times {\frac{3.3}{\sqrt{1584} } }[/tex] , [tex]35.7+2.58 \times {\frac{3.3}{\sqrt{1584} } }[/tex] ]
= [35.5 , 35.9]
Therefore, 99% confidence interval for the true mean number of words a third grader can read per minute is [35.5 , 35.9].
The following data show the brand, price ($), and the overall score for 6 stereo headphones that were tested by Consumer Reports. The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 (lowest) to 100 (highest). The estimated regression equation for these data is = 24.9 + 0.301x, where x = price ($) and y = overall score.Brand Price ScoreBose 18 76Scullcandy 150 71Koss 95 62Phillips/O'Neill 70 57Denon 70 30JVC 35 34Round your answers to three decimal places.a. Compute SST, SSR, and SSE.
Answers
Complete Question
The complete question is shown on the first uploaded image
Answer:
a
SST = 1800
SSR = 1512.376
SSE = 287.624
b
coefficient of determination is [tex]r^2 \approx 0.8402[/tex]
What this is telling us is that 84.02% variation in dependent variable y can be fully explained by variation in the independent variable x
c
The correlation coefficient is [tex]r = 0.917[/tex]
Step-by-step explanation:
The table shown the calculated mean is shown on the second uploaded image
Let first define some term
SST (sum of squares total) : This is the difference between the noted dependent variable and the mean of this noted dependent variable
SSR(sum of squared residuals) : this can defined as a predicted shift from the actual observed values of the data
SSE (sum of squared estimate of errors): this can be defined as the sum of the square difference between the observed value and its mean
From the table
[tex]SST = SS_{yy} = 1800[/tex]
[tex]SSR = \frac{SS^2_{xy}}{SS_{xx}} = \frac{4755^2}{14950} = 1512.376[/tex]
[tex]SSE =SST-SSR[/tex]
[tex]=1800 - 1512.376[/tex]
[tex]= 287.62[/tex]
The coefficient of determination is mathematically represented as
[tex]r^2 = \frac{SSR}{SST}[/tex]
[tex]= 1-\frac{SSE}{SST}[/tex]
[tex]r^2= 1-\frac{287.6237}{1800}[/tex]
[tex]r^2 \approx 0.8402[/tex]
The correlation coefficient is mathematically represented as
[tex]r = \pm\sqrt{r^2}[/tex]
Substituting values
[tex]r = \sqrt{0.84020}[/tex]
[tex]r = 0.917[/tex]
this value is + because the value of the coefficient of x in estimated regression equation([tex]24.9 + 0.301x,[/tex]) is positive
To compute SST, SSR, and SSE, we calculate the variability of the dependent variable around the mean, the variability explained by the regression model, and the variability not explained by the regression model.
Explanation:To compute SST, SSR, and SSE, we need to understand what each of them represents. SST (the total sum of squares) measures the total variability of the dependent variable (overall score) around the mean. SSR (the regression sum of squares) measures the amount of variability in the dependent variable that is explained by the regression model. Finally, SSE (the error sum of squares) measures the amount of variability in the dependent variable that is not explained by the regression model.
To compute these values:
Calculate the mean of the overall scores. In this case, the mean is (76 + 71 + 62 + 57 + 30 + 34) / 6 = 50.Calculate the total sum of squares (SST) by subtracting the overall score for each observation from the mean, squaring the differences, and summing them. In this case, SST = (76 - 50)^2 + (71 - 50)^2 + (62 - 50)^2 + (57 - 50)^2 + (30 - 50)^2 + (34 - 50)^2 = 5424.Calculate the regression sum of squares (SSR) by subtracting the predicted overall score for each observation from the mean, squaring the differences, and summing them. In this case, SSR = (24.9 + 0.301*18 - 50)^2 + (24.9 + 0.301*150 - 50)^2 + (24.9 + 0.301*95 - 50)^2 + (24.9 + 0.301*70 - 50)^2 + (24.9 + 0.301*70 - 50)^2 + (24.9 + 0.301*35 - 50)^2 = 10435.558.Calculate the error sum of squares (SSE) by subtracting the predicted overall score for each observation from the actual overall score, squaring the differences, and summing them. In this case, SSE = (76 - (24.9 + 0.301*18))^2 + (71 - (24.9 + 0.301*150))^2 + (62 - (24.9 + 0.301*95))^2 + (57 - (24.9 + 0.301*70))^2 + (30 - (24.9 + 0.301*70))^2 + (34 - (24.9 + 0.301*35))^2 = 171.442.Hence, SST = 5424, SSR = 10435.558, and SSE = 171.442.
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Suppose Frances is a researcher at Beaded Gemsz, a company that makes beaded jewelry. She wants to evaluate whether using better equipment during the current year has increased jewelry-making productivity. The company's reporting team estimated an average daily production yield of 105 units per store from previous years. Frances conducts a one-sample z - test with a significance level of 0.08 , acquiring daily unit yield data from each of the stores' databases for 45 randomly selected days of the year. She obtains a P -value of 0.06 . The power of the test to detect a production increase of 12 units or more is 0.85 . What is the probability that Frances concludes that the new equipment increases the average daily jewelry production when in fact the new equipment has no effect
Answers
Answer:
There is 8% (P=0.08) that Frances concludes that the new equipment increases the average daily jewelry production when in fact the new equipment has no effect.
Step-by-step explanation:
We have one-sample z-test with a significance level of 0.08 and a power ot the test of 0.85.
In this test, the null hypothesis will state that the new equipment has the same productivity of the older equipment. The alternative hypothesis is that there is a significative improvement from the use of new equipment.
The probability that Frances concludes that the new equipment increases the average daily jewelry production when in fact the new equipment has no effect is equal to the probability of making a Type I error (rejecting a true null hypothesis).
The probability of making a Type I error is defined by the level of significance, and in this test this value is α=0.08.
Then, there is 8% that Frances concludes that the new equipment increases the average daily jewelry production when in fact the new equipment has no effect.
Birdseed costs $0.52 a pound and sunflower seeds cost $0.82 a pound. Angela Leinenbachs' pet store wishes to make a 40-pound mixture of birdseed and sunflower seeds that sells for $0.72 per a pound. How many pounds of each type of seed should she use?
Answers
Answer: she should use 13 pounds of birdseed and 27 pounds of sunflower seeds.
Step-by-step explanation:
Let x represent the number of pounds of birdseed that she should use.
Let y represent the number of pounds of sunflower seeds that she should use.
Angela Leinenbachs' pet store wishes to make a 40-pound mixture of birdseed and sunflower seeds. It means that
x + y = 40
Birdseed costs $0.52 a pound and sunflower seeds cost $0.82 a pound. If the mixture would sell for $0.72 per a pound, then the total cost of the mixture would be 0.72 × 40 = $28.8
The equation would be
0.52x + 0.82y = 28.8- - - - - - - - - -1
Substituting x = 40 - y into equation 1, it becomes
0.52(40 - y) + 0.82y = 28.8
20.8 - 0.52y + 0.82y = 28.8
- 0.52y + 0.82y = 28.8 - 20.8
0.3y = 8
y = 8/0.3 = 27
x + y = 40
x + 27 = 40
x = 40 - 27
x = 13
HELP ME PLEASE!!
I WILL MARK AS BRANLIEST!!
16 POINTS!!
Answers
Answer:
y = 1.8x + 32
or
y = 9/ 5x + 32
Step-by-step explanation:
Used a slope calculator
How many one-third cubes are needed to fill the gap in the prism shown below?
A. 4
B. 8
C. 16
D. 24
Answers
The one-third cubes which are needed to fill the gap in the prism shown below are [tex]976[/tex].
What is cube ?Cube is a [tex]3D[/tex] solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
Here we have,
Length [tex]=4\frac{2}{3}[/tex]
Breadth [tex]=4[/tex]
Height [tex]=2[/tex]
Here, we have the side of cube [tex]=\frac{1}{3}[/tex],
So, Total volume of the prism [tex]=l*b*h[/tex]
[tex]=2*4*4\frac{2}{3}[/tex]
[tex]=\frac{112}{3}[/tex]
Volume of a cube [tex]= a^{3}[/tex]
[tex]=(\frac{1}{3} )^{3}=\frac{1}{27}[/tex]
Total number of cubes [tex]= \frac{\frac{112}{3}}{\frac{1}{27} } =1008[/tex]
Volume of given cubes in prism [tex]= 32*\frac{1}{27} =\frac{32}{27}[/tex]
So, the gap left in the prism [tex]=\frac{112}{3} -\frac{32}{27} =\frac{976}{27}[/tex]
So, the number of cubes required [tex]=\frac{\frac{976}{27} }{\frac{1}{27} }=976[/tex]
Hence, we can say that the one-third cubes which are needed to fill the gap in the prism shown below are [tex]976[/tex].
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Lue is rolling a random number cube.the cube has six sides and each one is labeled with a different number 1 through 6.what is the probability that he will roll a sum of 12 in two rolls
Answers
since the cube is numbered 1-6, the only way to roll a 12 is by getting 2 six’s.
1/6 • 1/6 = 1/36
Hayden is 59 inches tall and is standing on top of a ladder that is 2 yards y’all in inches what is the distance from the top of Hayden’s head to the ground
Answers
Answer:
131 inches
Step-by-step explanation:
1 yard = 36 inches
36 x 2 = 72
72 + 59 = 131
c:
Which loan is likely to have the highest Annual Percentage Rate (APR)? All of the loans are $500 and include a finance charge of $20.
a. Term of 5 days
b. Term of 10 days
c. Term of 20 days
d. Term of 30 days
e. Term of 90 days
Answers
The loan with the shortest term, which is 5 days in this case, is likely to have the highest APR because the finance charge would have less time to be spread out.
Explanation:The loan that is likely to have the highest Annual Percentage Rate (APR) is the one with the shortest term when all other factors are equal. In this scenario, that would be the loan with a term of 5 days. This is due to the fact that APR is calculated by the annualizing the interest and fees associated with a loan, meaning the shorter the term of a loan, the higher the APR. For instance, a $500 loan with a $20 finance charge has a much higher APR over 5 days as compared to 30 days or 90 days as the finance charge would have lesser time to be spread out.
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8s²-t³;s=2,t=3 need help plzzz. All these numbers and symbols give me headaches.
Answers
Answer:
5
Step-by-step explanation:
8 s^2 - t^3
Let s =2 and t=3
8 (2)^2 - (3)^3
Exponents first
8 *4 - 27
Then multiply and divide
32 -27
5
Make x the subject of the formula
ax + 2c = bx + 3d
Answers
Answer:x=(3d-2c)/(a-b)
Step-by-step explanation:
ax+2c=bx+3d
Collect like terms
ax-bx=3d-2c
x(a-b)=3d-2c
Divide both sides by (a-b)
x(a-b)/(a-b) =(3d-2c)/(a-b)
x=(3d-2c)/(a-b)
The linear equation in two variables is the equation of a straight line. The given formula can be rewritten as x = (bx + 3d - 2c) / a.
What is a Linear equation?A linear equation is a equation that has degree as one.
To find the solution of n unknown quantities n number of equations with n number of variables are required.
The given equation is as follows,
ax + 2c = bx + 3d
It can be simplified and written in terms of x as,
ax + 2c = bx + 3d
=> ax = bx + 3d - 2c
=> x = (bx + 3d - 2c) / a
Hence, by making x as the subject the given formula can be written as,
x = (bx + 3d - 2c) / a.
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An article suggests that substrate concentration (mg/cm3) of influent to a reactor is normally distributed with μ = 0.50 and σ = 0.08. (Round your answers to four decimal places.) (a) What is the probability that the concentration exceeds 0.60?
Answers
Answer:
0.1056 = 10.56% probability that the concentration exceeds 0.60
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 0.5, \sigma = 0.08[/tex]
What is the probability that the concentration exceeds 0.60?
This is 1 subtracted by the pvalue of Z when X = 0.6. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{0.6 - 0.5}{0.08}[/tex]
[tex]Z = 1.25[/tex]
[tex]Z = 1.25[/tex] has a pvalue of 0.8944
1 - 0.8944 = 0.1056
0.1056 = 10.56% probability that the concentration exceeds 0.60
A woman puts $1 million dollars into a savings account that she is not going to touch. She isn’t going to work either but she will live off the interest. If the rate of interest on the account is 6 1/2% what is the amount she will make in interest? If her totally expenses last year is $47,000 will she be able to live off the interest? Justify your answer
Answers
Answer:
i. The amount made in interest is $65,000.
ii. Yes, because her expenses is lesser that her interest.
Step-by-step explanation:
i. The woman puts $1,000,000 in a savings account with an interest rate of 6[tex]\frac{1}{2}[/tex]%.
Interest rate = 6[tex]\frac{1}{2}[/tex]% = [tex]\frac{13}{200}[/tex]
The amount she would make in interest = [tex]\frac{13}{200}[/tex] × 1,000,000
= 65,000
The amount made in interest is $65,000.
ii. The total of her expenses last year is $47,0000 which is lesser than the amount made in inerest.
After her total expenses, should would have a remainder of;
= $65,000 - $47,0000
= $18,000
Therefore, she would be able to live off the interest.
Evaluate ∫SF⃗ ⋅dA⃗ , where F⃗ =(bx/a)i⃗ +(ay/b)j⃗ and S is the elliptic cylinder oriented away from the z-axis, and given by x2/a2+y2/b2=1, |z|≤c, where a, b, c are positive constants.
Answers
Answer:
Therefore surface integral is [tex]\pi(a^2+b^2)c-0-0=\pi(a^2+b^2)c[/tex].
Step-by-step explanation:
Given function is,
[tex]\vec{F}=\frac{bx}{a}\uvec{i}+\frac{ay}{b}\uvec{j}[/tex]
To find,
[tex]\int\int_{S}\vec{F}dS[/tex]
where S=A=surfece of elliptic cylinder we have to apply Divergence theorem so that,
[tex]\int\int_{S}\vec{F}dS[/tex]
[tex]=\int\int\int_V\nabla.\vec{F}dV[/tex]
[tex]=\int\int\int_V(\frac{b}{a}+\frac{a}{b})dV[/tex]
[tex]=\frac{a^2+b^2}{ab}\int\int\int_VdV[/tex]
[tex]=\frac{a^2+b^2}{ab}\times \textit{Volume of the elliptic cylinder}[/tex]
[tex]=\frac{a^2+b^2}{ab}\times \pi ab\times 2c=\pi (a^2+b^2)c[/tex]
If unit vector [tex]\cap{n}[/tex] directed in positive (outward) direction then z=c and,[tex]\int\int_{S_1}\vex{F}.dS_1=\int\int_{S_1}<\frac{bx}{a}, \frac{ay}{b}, 0> . <-z_x,z_y,1>dA[/tex]
[tex]=\int\int_{S_1}<\frac{bx}{a},\frac{ay}{b}, 0>.<0,0,1>dA=0[/tex]
If unit vector [tex]\cap{n}[/tex] directed in negative (inward) direction then z=-c and,[tex]\int\int_{S_2}\vex{F}.dS_2=\int\int_{S_2}<\frac{bx}{a}, \frac{ay}{b}, 0>. -<-z_x,z_y,1>dA[/tex]
[tex]=\int\int_{S_2}<\frac{bx}{a},\frac{ay}{b}, 0>. -<0,0,1>dA=0[/tex]
Therefore surface integral without unit vector of the surface is,
[tex]\pi(a^2+b^2)c-0-0=\pi(a^2+b^2)c[/tex]
The value of ∫SF ⋅dA where F =(bx/a)i +(ay/b)j and S is the elliptic cylinder oriented away from the z-axis is 2πc(a² + b²).
How to solve the elliptic cylinder?From the information, F = (b/ax) + (a/by)j and S is the elliptic cylinder.
To evaluate ∫F.dA goes thus:
divF = (I'd/dx + jd/dx + kd/dx) × (b/ax)i + (a/by)j
= b/a + a/b
= (a² + b²)/ab
Using Gauss divergence theorem, this will be further solved below:
∫∫∫v(a² + b²/ab)dV
= (a² + b²/ab)∫∫∫vdV
= (a² + b²/ab) × Volume of cylinder
= (a² + b²/ab) × πab(2c)
= 2πc(a² + b²)
Learn more about cylinder on:
https://brainly.com/question/76386
answer the question bellow
Answers
Answer:
c.
Step-by-step explanation:
Use the two-point form of the linear equation. Fill in the missing blanks using (1, 1) for (x1, y1). You will need both points to determine the slope, y2 − y1 x2 − x1 . y − 1 = − 2 3 − 1 6 − 1 x −
Answers
Answer: (-2, 3)
Step-by-step explanation:
Answer:
The point-slope form of a linear equation is a formula that allows a person to calculate the slope and point of intercept of a line, and then once you calculate the linear equation, you can calculate the x and y coordinate of any point on the line!
1. Choose two points on a line.
2. Indicate the y1 (the y coordinate of the first point) and y2 (the y coordinate of the second point).
3. Indicate the x1 (the x coordinate of the first point) and the x2 (the x coordinate of the second point)
4. Plug the previously identified variables into the slope formula where the slope is equal to (y2-y1)/(x2-x1)
5. Subtract y2 and y1.
6. Subtract x2 and x1.
7. Divide the quantity in #5 by the quantity in #6. This is your slope.
8. Observe the equation: y=mx+b where "m" is the slope calculated in #7.
9. Plug in the slope for "m"
10. Using one of the points identified earlier, plug in y1 into "y" and x1 into "x". Rearrange and solve for b.
11. Then plug in the value "b" into y=mx+b. Make sure to leave the unknown variables "x" and "y", but make sure to still plug in the "m" calculated earlier!
A right triangle whose hypotenuse is StartRoot 18 EndRoot18 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume that can be made this way.
Answers
Answer:
V = [tex]\frac{1}{3}[/tex]π(8 - [tex]\frac{8}{3}[/tex])[tex]\sqrt{\frac{8}{3} }[/tex] = 9.11 [tex]m^{3}[/tex]
r = 4[tex]\frac{\sqrt{3} }{3}[/tex]
h = [tex]\sqrt{\frac{8}{3} }[/tex]
Step-by-step explanation:
Given that the right triangle whose hypotenuse is [tex]\sqrt{8}[/tex]
Let r is the radius of the cone Let h is the height of the coneWe know that:
[tex]r^{2} + h^{2} = 8[/tex]
<=> [tex]r^{2} = 8 - h^{2}[/tex]
The volume of the cone is:
V = π[tex]r^{2} \frac{1}{3} h[/tex]
<=> V = π[tex]\frac{1}{3}(8 - h^{2} )h[/tex]
Differentiate w.r.t h
[tex]\frac{dV}{dh}[/tex] = π [tex]\frac{1}{3}[/tex] (8 - [tex]3h^{2}[/tex])
For maximum/minimum: [tex]\frac{dV}{dh}[/tex] = 0
<=> π [tex]\frac{1}{3}[/tex] (8 - [tex]3h^{2}[/tex]) = 0
<=> [tex]h^{2}[/tex] = [tex]\frac{8}{3}[/tex]
<=> h = [tex]\sqrt{\frac{8}{3} }[/tex]
=> [tex]r^{2}[/tex] = [tex]\frac{16}{3}[/tex]
<=> r = 4[tex]\frac{\sqrt{3} }{3}[/tex]
So the volume of the cone is:
V = [tex]\frac{1}{3}[/tex]π(8 - [tex]\frac{8}{3}[/tex])[tex]\sqrt{\frac{8}{3} }[/tex] = 9.11 [tex]m^{3}[/tex]
A grain silo is shaped like a cylinder with a cone-shaped top. The cylinder is 30 feet tall. The volume of the silo is 1152 cubic feet. Find the radius of the silo.
Answers
Answer:
The radius of the silo is 3.49 m.
Step-by-step explanation:
We have,
Height of the cylinder, h = 30 ft
Volume of cylindrical shaped grain silo, [tex]V=1152\ ft^3[/tex]
It is required to find the radius of silo. The formula of volume of cylinder is given by :
[tex]V=\pi r^2h[/tex]
r is radius
[tex]r=\sqrt{\dfrac{V}{\pi h}} \\\\r=\sqrt{\dfrac{1152}{\dfrac{22}{7}\times 30 }} \\\\r=3.49\ m[/tex]
So, the radius of the silo is 3.49 m.
Suppose that $20,000 is invested in an account for which interest is compounding continuously at 3.14%. What is the value after 5 years? After 10 years? (Round your answers to two decimal places.) After how many years will the original investment be doubled? (Round your answer to two decimal places.)
Answers
Answer:
value after 5 years = $23,399.91
after 10 years = $27,377.79
Time it takes for the amount to double = 22.07 years
Step-by-step explanation:
For amounts that are compounded continuously, it means that the interest rate is is added to the investment amount at an infinite number of time, and the formula is given as:
A = P [tex]e^{r.t}[/tex], where:
A = Future value
P = present value
e = constant ≈ 2.7183
r = interest rate in decimal form
t = years
Now for value after 5 years;
A = ???
P = $20,000
r = 3.14% = 0.0314
t = 5 years
∴ A = P [tex]e^{r.t}[/tex]
= 20,000 [tex]e^{0.0314*5}[/tex]
= 20,000 × [tex]e^{0.157}[/tex] = 20,000 × 1.169995 = $23,399.91 ( to 2 decimal places)
(Note that the function '[tex]e[/tex]' can be punched directly from the calculator)
value after 10 years;
A = ???
P = $20,000
r = 3.14% = 0.0314
t = 10 years
∴ A = P[tex]e^{r.t}[/tex]
= 20,000 × [tex]e^{0.0314 * 10}[/tex]
= 20,000 × [tex]e^{0.314}[/tex] = $27,377.79 ( to 2 decimal places)
Time it will take to double the original investment;
A = P [tex]e^{r.t}[/tex]
where;
A = 40,000
P = 20,000
r = 0.0314
t =???
40,000 = 20,000 × [tex]e^{0.0314 * t}[/tex]
[tex]\frac{40,000}{20,000} = \frac{20,000}{20,000} * e^{0.0314*t}[/tex] (divide both sides by 20,000)
2 = [tex]e^{0.0314 * t}[/tex]
Next take the natural logarithm of both sides
㏑(2) = ㏑[tex]e^{0.0314 *t}[/tex] (㏑[tex]e[/tex] = 1; and the exponent can be brought down )
= 0.6931 = 0.0314 × t × 1
∴ t = [tex]\frac{0.06931}{0.0314}[/tex] = 22.07 years ( to 2 decimal places)
Calculate the value of an investment after 5 and 10 years with continuous compounding at a certain interest rate. Determine the time needed for an investment to double using the continuous compounding formula.
When calculating the future value of an investment with compound interest, we can use the formula A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial sum of money), r is the annual interest rate (as a decimal), t is the time the money is invested for in years, and e is the base of the natural logarithm.
Applying this formula, the future value of a $20,000 investment at a 3.14% interest rate compounded continuously for 5 years would be
After 5 years: $20,000 × [tex]e^{(0.0314 \times 5)[/tex] ≈ $23,914.83After 10 years: $20,000 × [tex]e^{(0.0314 \times 10)[/tex] ≈ $28,717.83Time to double: $20,000 × [tex]e^{(0.0314 \times t)[/tex] = $40,000 ⇒ t ≈ 22.10 yearsA software developer wants to know how many new computer games people buy each year. A sample of 1233 people was taken to study their purchasing habits. Construct the 99% confidence interval for the mean number of computer games purchased each year if the sample mean was found to be 7.4. Assume that the population standard deviation is 1.4. Round your answers to one decimal place.
Answers
Answer:
The 99% confidence interval for the mean number of computer games purchased each year is between 7.3 and 7.5 games.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.99}{2} = 0.005[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.005 = 0.995[/tex], so [tex]z = 2.575[/tex]
Now, find the margin of error M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
[tex]M = 2.575\frac{1.4}{\sqrt{1233}} = 0.1[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 7.4 - 0.1 = 7.3.
The upper end of the interval is the sample mean added to M. So it is 7.4 + 0.1 = 7.5
The 99% confidence interval for the mean number of computer games purchased each year is between 7.3 and 7.5 games.
Answer: = ( 7.3, 7.5)
Therefore at 99% confidence interval (a,b) = ( 7.3, 7.5)
Step-by-step explanation:
Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.
The confidence interval of a statistical data can be written as.
x+/-zr/√n
Given that;
Mean gain x = 7.4
Standard deviation r = 1.4
Number of samples n = 1233
Confidence interval = 99%
z(at 99% confidence) = 2.58
Substituting the values we have;
7.4+/-2.58(1.4/√1233)
7.4+/-2.58(0.03987)
7.4+/-0.1028
7.4+/-0.1
= ( 7.3, 7.5)
Therefore at 99% confidence interval (a,b) = ( 7.3, 7.5)
Hector took out a loan of $900 for 18 months at a rate of 5.5% annually. How much will he pay in interest on the loan?
Answers
Answer:
Hector will pay $74.25 interest on the loan.
Step-by-step explanation:
Hector took out a loan of $900 for 18 months at a rate of 5.5% annually.
Principal amount = $900
Rate of interest = 5.5%
Time = 18 months = 1.5 years
Formula for interest :
[tex]I=\frac{P\times R\times T}{100}[/tex]
[tex]=\frac{900\times 5.5\times 1.5}{100}[/tex]
= 74.25
Hector will pay $74.25 interest on the loan.
The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 3.3 years and a standard deviation of 0.6 years. He then randomly selects records on 50 laptops sold in the past and finds that the mean replacement time is 3.1 years.
Assuming that the laptop replacement times have a mean of 3.3 years and a standard deviation of 0.6 years, find the probability that 50 randomly selected laptops will have a mean replacement time of 3.1 years or less.
P(M < 3.1 years) =
Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
Based on the result above, does it appear that the computer store has been given laptops of lower than average quality?
No. The probability of obtaining this data is high enough to have been a chance occurrence.
Yes. The probability of this data is unlikely to have occurred by chance alone
Answers
Answer:
Probability that the 50 randomly selected laptops will have a mean replacement time of 3.1 years or less is 0.0092.
Yes. The probability of this data is unlikely to have occurred by chance alone.
Step-by-step explanation:
We are given that the replacement times for the model laptop of concern are normally distributed with a mean of 3.3 years and a standard deviation of 0.6 years.
He then randomly selects records on 50 laptops sold in the past and finds that the mean replacement time is 3.1 years.
Let M = sample mean replacement time
The z-score probability distribution for sample mean is given by;
Z = [tex]\frac{ M-\mu}{\frac{\sigma}{\sqrt{n} } }} }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean replacement time = 3.3 years
[tex]\sigma[/tex] = standard deviation = 0.6 years
n = sample of laptops = 50
The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.
Now, Probability that the 50 randomly selected laptops will have a mean replacement time of 3.1 years or less is given by = P(M [tex]\leq[/tex] 3.1 years)
P(M [tex]\leq[/tex] 3.1 years) = P( [tex]\frac{ M-\mu}{\frac{\sigma}{\sqrt{n} } }} }[/tex] [tex]\leq[/tex] [tex]\frac{ 3.1-3.3}{\frac{0.6}{\sqrt{50} } }} }[/tex] ) = P(Z [tex]\leq[/tex] -2.357) = 1 - P(Z [tex]\leq[/tex] 2.357)
= 1 - 0.99078 = 0.0092 or 0.92%
So, in the z table the P(Z [tex]\leq[/tex] x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 2.357 in the z table which will lie between x = 2.35 and x = 2.36 which has an area of 0.99078.
Hence, the required probability is 0.0092 or 0.92%.
Now, based on the result above; Yes, the computer store has been given laptops of lower than average quality because the probability of this data is unlikely to have occurred by chance alone as the probability of happening the given event is very low as 0.92%.
Final answer:
To find the probability, we need to standardize the sample mean using the z-score formula and then use a standard normal distribution table or a calculator to find the probability.
Explanation:
To find the probability that the mean replacement time of 50 randomly selected laptops is 3.1 years or less, we can use the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
We know that the population mean is 3.3 years, the population standard deviation is 0.6 years, and the sample size is 50. To find the probability, we need to standardize the sample mean using the z-score formula and then use a standard normal distribution table or a calculator to find the probability.
The formula for the z-score is:
z = (x - μ) / (σ / √n)
Substituting the given values:
z = (3.1 - 3.3) / (0.6 / √50)
Calculating the z-score:
z = -0.2 / (0.6 / 7.0711)
z ≈ -0.2 / 0.0848
z ≈ -2.359
Using a standard normal distribution table or a calculator, we find that the probability of obtaining a z-score less than -2.359 is approximately 0.0093. Therefore, the probability that 50 randomly selected laptops will have a mean replacement time of 3.1 years or less is approximately 0.0093, or 0.93%.
Based on this probability, it does not appear that the computer store has been given laptops of lower-than-average quality. The probability of obtaining this data by chance alone is low enough to suggest that it is unlikely to have occurred by chance alone.