Each consecutive term in the sum is separated by a difference of 6, so the [tex]n[/tex]-th term is [tex]4+6(n-1)=6n-2[/tex] for [tex]n\ge1[/tex]. The last term is 70, so there are [tex]6n-2=70\implies n=12[/tex] terms in the sum.
Now,
[tex]S=4+10+\cdots+64+70[/tex]
but also
[tex]S=70+64+\cdots+10+4[/tex]
Doubling the sum and grouping terms in the same position gives
[tex]2S=(4+70)+(10+64)+\cdots+(64+10)+(70+4)=12\cdot74[/tex]
[tex]\implies\boxed{S=444}[/tex]
A regression analysis can be used to determine a. if a linear relationship exists between two categorical variables. b. a cause-and-effect relationship between two variables. c. if the difference in the population means may be zero. d. if a linear relationship exists between two quantitative variables.
Answer: Option (d) is correct.
Step-by-step explanation:
Regression analysis can be used to determine if there is any linear relationship exists between the two quantitative variables.
There are one dependent variable and one or more than one independent variable in a single regression equation.
After running a simple regression, we get to know the relationship between the dependent variable and explanatory variable.
Various statistical software are used for running regression like STATA, E- Views, SPSS, etc.
Regression analysis is used primarily to determine if a linear relationship exists between two quantitative variables. While it can identify relationships between variables, it does not imply causation. It does not establish a relationship between two categorical variables or if the difference in the population means may be zero.
Explanation:Regression analysis is a statistical methodology often used in mathematics and related fields. Primarily, it is used to determine if a linear relationship exists between two quantitative variables.
For example, you could use regression analysis to see if there is a linear relationship between the age of a car (quantitative variable) and the distance it can travel on a tank of gas (another quantitative variable).
It's important to note however that while regression analysis can identify a correlation or relationship between variables, it does not necessarily imply causation, or a cause-and-effect relationship. There could be other related variables causing the effect.
For your options, regression analysis can't be used to determine if a linear relationship exists between two categorical variables or if the difference in the population means may be zero.
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You have 144 feet of fencing to enclose a rectangular region. What is the maximum area? a) 5184 square feet b) 1292 square feet c) 1296 square feet d) 20.736 square feet e) none
Final answer:
The maximum area that can be enclosed with 144 feet of fencing is when the enclosure is a square. Calculating the side length as 36 feet results in a maximum area of 1296 square feet.
Explanation:
Maximizing the Area of a Rectangular Region with a Given Perimeter:
To find the maximum area that can be enclosed with 144 feet of fencing in a rectangular shape, we can use the knowledge that for a given perimeter, a rectangle with equal length and width (a square) will have the maximum possible area. Let's denote the length of the rectangle as L and the width as W. Since the perimeter is twice the sum of the length and width, we have 2L + 2W = 144 feet. To form a square, which gives the maximum area, L equals W, making 4L = 144 feet or L = 36 feet. The maximum area is L squared, which is 36 feet by 36 feet, equaling 1296 square feet.
The maximum area that can be enclosed with 144 feet of fencing is 1296 square feet, which corresponds to option c) 1296 square feet.
To maximize the area enclosed by a fixed perimeter, we look to geometry, which tells us that of all the rectangles with a given perimeter, the square has the highest area.
Let's denote the perimeter of the square as P and the length of each side of the square as s. Since the square has four equal sides, we have:
P = 4s
We are given that P is 144 feet, so we can find the length of each side s by dividing the total perimeter by 4:
s = P/4 = 144/4 = 36 feet
The area A of a square is given by the formula A = s^2, where s is the length of a side of the square. We calculated above that s = 36 feet, so:
A = s^2 = (36 feet)^2 = 1296 square feet
This is the maximum area that can be enclosed by 144 feet of fencing when arranged in a square. Matching our result with the provided options, the correct answer is:
c) 1296 square feet
USE INDUCTIVE REASONING TO PREDICT THE NEXT LINE IN THE PATTERN. 11 X 14 = 154 11 X 15 = 165 11 X 16= 176 THE NEXT LINE IS__X__=
Answer:
11 * 17 = 187.
Step-by-step explanation:
11 * 14 = 154
11 * 15 = 165
11 * 16 = 176
So we have the series 154, 165, 176 which has a common difference of 11.
so the next line is 11*17 = 176 + 11 = 187.
A retiree receives $5120 a year interest from $40,000 placed in two bonds, one paying 14% and the other 12%. How much is invested in each bond? 3)
To find out how much is invested in each bond, set up a system of equations and solve for the values of x and y.
Explanation:To find out how much is invested in each bond, we can set up a system of equations.
Let x be the amount invested in the bond paying 14% interest, and y be the amount invested in the bond paying 12% interest.
We know that the retiree receives $5120 in interest each year. So, we have the equation:
x(0.14) + y(0.12) = 5120
Since the total amount invested is $40,000, we can also write the equation:
x + y = 40000
We can now solve this system of equations using any method, such as substitution or elimination, to find the values of x and y.
use logarithmic differentiation to find dy/dx
y=(lnx)^x
Try this suggested solution.
M1Q8.) Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate.
Rating plays on Broadway, Poor, good, or excellent would be a type of Ordinal measurement.
You can think or ordinal like order, which could be listing something from best to worst.
The answer is Ordinal.
Ordinal measurement can shown by name, group, or rank. Poor, good, and excellent shows the ratings of the play by "rank", in other words, by order. Thus proves our answer.
Best of Luck!
A car manufacturer is reducing the number of incidents with the transmission by issuing a voluntary recall. During week 3 of the recall, the manufacturer fixed 391 cars. In week 13, the manufacturer fixed 361 cars. Assume that the reduction in the number of cars each week is linear. Write an equation in function form to show the number of cars seen each week by the mechanic.
To find the equation in function form for the number of cars fixed each week by the mechanic, we can use the slope-intercept form of a linear equation. The equation is y = -3x + 400, where x represents the week number and y represents the number of cars fixed.
Explanation:To write an equation in function form to show the number of cars seen each week by the mechanic, we can let the variable x represent the week number and y represent the number of cars fixed. We know that the reduction in the number of cars each week is linear, so we can use the slope-intercept form of a linear equation, y = mx + b. To find the slope, we can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. Let's use the points (3, 391) and (13, 361) to find the slope. Plugging these values into the formula gives us m = (361 - 391) / (13 - 3) = -3. Therefore, the equation in function form is y = -3x + b. To find the y-intercept b, we can use one of the points on the line. Let's use the point (3, 391): 391 = -3(3) + b. Solving for b gives us b = 400. Therefore, the equation in function form is y = -3x + 400, where x represents the week number and y represents the number of cars fixed.
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The equation in function form to show the number of cars seen each week by the mechanic is y = -3x + 400, where x represents the week and y represents the number of cars fixed by the mechanic.
Explanation:To write an equation in function form to show the number of cars seen each week by the mechanic, we can use the given information that the reduction in the number of cars each week is linear. Let's assume the number of cars fixed in week 3 as y = 391 and in week 13 as y = 361. We can use the formula for the equation of a line, y = mx + b, where m is the slope and b is the y-intercept.
Using the slope formula, m = (y2 - y1) / (x2 - x1), where (x1, y1) = (3, 391) and (x2, y2) = (13, 361), we find m = (361 - 391) / (13 - 3) = -3.
Therefore, the equation in function form to show the number of cars seen each week by the mechanic is y = -3x + b. To find the y-intercept, we can substitute the coordinates of one of the points (x, y) = (3, 391) into the equation, 391 = -3(3) + b. Solving for b gives b = 400.
Thus, the equation in function form is y = -3x + 400, where x represents the week and y represents the number of cars fixed by the mechanic.
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A study to determine the sensitivity and specificity of a new test for macular degeneration is conducted on 2430 people. Macular degeneration occurs at a rate of 16.72%. Your sample has the same prevalence of macular degeneration. You find that 377 people with macular degeneration tested positive with the new test. You also have a total of 561 positive test results in your study. CALCULATE THE SPECIFICITY of this test.
Question 2 options:
A)
83.29%
B)
98.45%
C)
92.86%
D)
67.20%
E)
90.91%
According to the American Lung Association 7% of the population has lung disease. Of the people having lung disease 90% are smokers. Of the people not having lung disease 20% are smokers. What are the chances that a smoker has lung disease?
Answer:
The chances that a smoker has lung disease 25.30%.
Step-by-step explanation:
Let L is the event of the lung disease and S is the event of being smoker,
According to the question,
The probability of lung disease, P(L) = 7 % = 0.07,
⇒ The probability of not having lung disease, P(L') = 100 % - 7 % = 93 % = 0.93,
The probability of the people having lung disease who are smokers,
P(L∩S) = 90% of 0.07 = 6.3% = 0.063,
The probability of the people not having lung disease who are smokers,
P(L'∩S) = 20% of 0.93 = 18.60% = 0.186,
Thus, the total probability of being smoker, P(S) = P(L∩S) + P(L'∩S) = 0.063 + 0.186 = 0.249,
Hence, the probability that a smoker has lung disease,
[tex]P(\frac{L}{S})=\frac{P(L\cap S)}{P(S)}[/tex]
[tex]=\frac{0.063}{0.249}[/tex]
[tex]=0.253012048193[/tex]
[tex]=25.3012048193\%[/tex]
[tex]\approx 25.30\%[/tex]
Final answer:
To find the chances that a smoker has lung disease, we need to use conditional probability. Assuming a total population of 100, the chances are 25.2%.
Explanation:
To find the chances that a smoker has lung disease, we need to use conditional probability. Let's assume the total population is 100. According to the American Lung Association, 7% of the population has lung disease, so the number of people with lung disease is 7.
Of these 7 people with lung disease, 90% are smokers. So, the number of smokers with lung disease is 7 * 0.9 = 6.3.
Out of the remaining people (100 - 7 = 93) without lung disease, 20% are smokers. So, the number of smokers without lung disease is 93 * 0.2 = 18.6.
Therefore, the total number of smokers is 6.3 + 18.6 = 24.9.
Hence, the chances that a smoker has lung disease is 6.3 / 24.9 = 0.252 (rounded to three decimal places) or 25.2% (rounded to the nearest percent).
The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 and a standard deviation of 5,000 miles. What is the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles? Round your answer to 4 decimal places.
Answer: 0.6731
Step-by-step explanation:
Given : Mean : [tex]\mu = 40,000\text{ miles}[/tex]
Standard deviation : [tex]\sigma = 5,000\text{ miles}[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x=36,000
[tex]z=\dfrac{36000-40000}{5000}=-0.8[/tex]
For x=46,000
[tex]z=\dfrac{46000-40000}{5000}=1.2[/tex]
The P-value : [tex]P(-0.8<z<1.2)=P(z<1.2)-P(z<-0.8)[/tex]
[tex]=0.8849303-0.2118554=0.6730749\approx0.6731[/tex]
Hence, the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles =0.6731
the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles is 0.6730 (rounded to four decimal places).
The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 miles and a standard deviation of 5,000 miles. To find the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles, we can calculate the z-scores for 36,000 and 46,000 and then use the standard normal distribution to find the probabilities for these z-scores.
To compute the z-scores:
For 36,000: z= (36,000 - 40,000) / 5,000 = -0.8For 46,000: z= (46,000 - 40,000) / 5,000 = 1.2We then look up the corresponding probabilities for these z-scores in a standard normal distribution table or use a calculator with normal distribution functions. The probability corresponding to z=-0.8 is approximately 0.2119, and the probability corresponding to z=1.2 is approximately 0.8849. The probability of the tire's life being between 36,000 and 46,000 is the difference: 0.8849 - 0.2119 = 0.6730.
Therefore, the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles is 0.6730 (rounded to four decimal places).
Find the probability that a person is not qualified if he or she was approved by the manager. certain job, 85% are qualified and 15% are not. The personnel manager claims that she approves qualified people 85% of the time; she approves unqualified people 40% of the time. The probability is 0.15 (Type an integer or decimal rounded to four decimal places as needed.)
Answer: The probability that a person is not qualified if a person was approved by the manager is 0.0766.
Step-by-step explanation:
Since we have given that
Probability that a person approves qualified = 0.85× 0.85 = 0.7225
Probability that a person does not approve qualified = 0.85 × 0.15 = 0.1275
Probability that a person approves unqualified = 0.40 × 0.15 = 0.06
Probability that a person does not approve unqualified = 0.60 × 0.15 = 0.009
so, using the conditional probability, we get that [tex]p(unqualified\mid approved)=\dfrac{0.06}{0.7225+0.06}=\dfrac{0.06}{0.7825}=0.0766[/tex]
Hence, the probability that a person is not qualified if a person was approved by the manager is 0.0766.
Determine the validity of the following argument. If one of the arguments is listed in the text, please name it: If n is a real number such that n > 2, then n^2 > 4. Suppose that n^2 <= 4. Then n ? 2. Which rule of inference, if any, is being used?
Answer:
Step-by-step explanation:this is confusing for me oof
Use the continuous compound interest formula to find the indicated value. A $94,000; P $78,870; r= 7.8%; t =? t= years (Do not round until the final answer. Then round to two decimal places as needed.) TrueCar Enter your answer in the answer box. Prerequisite Skills Test for Finite Mathema eer of the heart bypass. Learn more Customize Getting Ready for Finite Mathematics Test AC 29057 12
Answer:
time period (t) is 2.25 years
Step-by-step explanation:
Given data in question
amount (a) = $94000
principal (p) = $78870
rate (r) = 7.8 % = 0.078
to find out
time period (t)
solution
we know that continuous compound interest formula i.e.
amount = principal [tex]e^{rt}[/tex] ...............1
we will put all value a, p and r in equation 1
amount = principal [tex]e^{rt}[/tex]
94000 = 78870 [tex]e^{0.078t}[/tex]
[tex]e^{0.078t}[/tex] = 94000 / 78870
now we take ln both side
ln [tex]e^{0.078t}[/tex] ln (94000 / 78870)
0.078 t = ln 1.19183466
0.078 t = 0.175494
t = 0.175494 /0.078
t = 2.249923
so time period (t) is 2.25 years
In this complex interest problem, by organizing and substitifying the values into the continuous compound interest formula, we obtain t = ln(1.1911) / 0.078, which approximately equals 2.81 years when rounded to two decimal places.
Explanation:The continuous compound interest formula is A = Pe^(rt), where P is the principal amount, r is the interest rate, t is the time in years, and A is the amount of money accumulated after n years, including interest.
In this case, we have A = $94,000, P = $78,870, r = 7.8% = 0.078, and we need to solve for t. Therefore, let's substitute the given values into the formula to solve for t:
$94,000 = $78,870 * e^(0.078t),
Solving this equation for 't' involves isolating 't' on one side. First, divide both sides by $78,870:
e^(0.078t) = $94,000 / $78,870 = 1.1911.
Take the natural logarithm (ln) of both sides:
0.078t = ln(1.1911),
Finally, divide both sides by 0.078 to solve for 't':
t = ln(1.1911) / 0.078.
With a calculator, this results in approximately t = 2.81 years when rounded to two decimal places.
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help please quickly!!
Answer:
m∠XQZ = 72°
Step-by-step explanation:
We presume the "if ..." condition is intended to mean that ray QY lies between rays QX and QZ. Then by angle addition, you get
∠XQY + ∠YQZ = ∠XQZ
(4a +8) + (28) = (8a)
36 = 4a . . . . . . . . . . . . . subtract 4a
72 = 8a = ∠XQZ . . . . multiply by 2
Of the films Empty Set Studios released last year, 60% were comedies and the rest were horror films. 75% of the comedies were profitable, but 75% of the horror moves were unprofitable. If the studio made a total of 40 films, and broke even on none of them, how many of their films were profitable?
Answer: There are 22.5 films were profitable.
Step-by-step explanation:
Since we have given that
Number of total films = 40
Percentage of comedies = 60%
Number of comedies is given by
[tex]0.6\times 40\\\\=24[/tex]
Percentage of horror films = 40%
Number of horror films is given by
[tex]0.4\times 40\\\\=16[/tex]
Percentage of comedies were profitable = 75%
Number of profitable comedies is given by
[tex]0.75\times 24=18[/tex]
Percentage of horror were unprofitable = 75%
Percentage of horror were profitable = 25%
Number of profitable horror films is given by
[tex]0.25\times 18\\\\=4.5[/tex]
So, Total number of profitable films were
[tex]18+4.5=22.5[/tex]
Hence, there are 22.5 films were profitable.
Write an equation of the circle with center (-4, -9) and diameter 10.
Answer:
[tex](x+4)^2+(y+9)^2=25[/tex]
Step-by-step explanation:
The equation of a circle with center (h,k) and radius r is
[tex](x-h)^2+(y-k)^2=r^2[/tex].
You are given (h,k)=(-4,-9) and the radius=(diameter)/2=10/2=5.
Plug in the information and you will have your equation:
[tex](x-(-4))^2+(y-(-9))^2=(5)^2[/tex].
Simplify:
A national random sample of 654 women aged 20 – 29 years was taken, and each woman's body mass index (BMI) was measured. The sample data had mean BMI ????¯=26.8 and standard deviation ????=7.42 . What is the 95% ???? confidence interval (????,????) for the mean BMI of all young women?
Answer:
The required interval is (26.23 , 27.37)
Step-by-step explanation:
The mean is = 26.8
The standard deviation is = 7.42
n = 654
At 95% confidence interval, the z score is 1.96
To find the desired interval we will calculate as:
[tex]26.8+1.96(\frac{7.42}{\sqrt{654} } )[/tex]
And [tex]26.8-1.96(\frac{7.42}{\sqrt{654} } )[/tex]
[tex]26.8+0.57[/tex] and [tex]26.8-0.57[/tex]
= 27.37 and 26.23
So, the required interval is (26.23 , 27.37)
The 95% confidence interval for the mean BMI of all young women, given a sample of 654 women with a mean BMI of 26.8 and a standard deviation of 7.42 is approximately (26.23, 27.37). This is calculated by finding the standard error and then using that to find the interval around the sample mean which covers 95% of the probable values for the population mean.
Explanation:The question is asking you to calculate the 95% confidence interval for the mean BMI of all young women, given a sample size of 654 women with a mean BMI of 26.8 and a standard deviation of 7.42.
In order to do this, we first need to calculate the standard error, which is the standard deviation divided by the square root of the number of observations. This gives us 7.42 divided by the square root of 654, or about 0.29032.
The 95% confidence interval is calculated by taking the mean and adding and subtracting the standard error multiplied by the relevant z-score for a 95% confidence interval, which is 1.96 (from the standard normal distribution table). So we take 26.8 plus and minus 1.96 times 0.29032, giving us a 95% confidence interval of approximately (26.23, 27.37).
So, we can say with 95% confidence that the mean BMI for all young women is between 26.23 and 27.37.
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If you have an 18% solution, how many milligrams is in each milliliter of solution?
A. 18 mg
B. 180 mg
C. 1.8 mg
D. 1800 mg
Answer:
B. 180 mg
Step-by-step explanation:
In order to answer the given problem we need to be aware that:
1000 milligrams = 1 milliliter
The above means that in 1 milliliter a 100% solution means 1000 milligrams. Because we have 18% solution, then:
(1000 milligrams / 1 milliliter) * 18% =
(1000 milligrams / 1 milliliter) * (18/100) =
(1000*18/100) milligrams/milliliter =
180 milligrams/milliliter
In conclusion, the answer is B. 180 mg.
Find the inverse of the matrices if they Exist. Use the algorithm introduced in the section. [1 0 -2 -3 1 4 2 -3 4]
Answer:
\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right]
Step-by-step explanation:
Given is a matrix 3x3 as
[tex]\left[\begin{array}{ccc}1&0&2\\-3&1&4\\2&-3&4\end{array}\right][/tex]
|A| =2 hence inverse exists.
Cofactors are 16 20 7
6 8 3
2 2 1
Hence inverse =
[tex]\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right][/tex]
A 4:1 scale drawing of a bearing is shown on an A-size print. Using a ruler, you measure the inside diameter of the part on the paper and you get 1.50 inches. What is the actual part size in inches? A. 6.0 B. 1.5 C.4.0 D 375
Answer:
A. 6.0
Step-by-step explanation:
We have been given that the the inside diameter of the part on the paper is 1.50 inches.
We have been given that scale is 4:1 for actual size to drawing side.
[tex]\text{Scale}=\frac{\text{Actual size}}{\text{Map size}}=\frac{4}{1}[/tex]
Upon substituting 1.50 in our given proportion, we will get:
[tex]\frac{\text{Actual size}}{1.50\text{ inches}}=\frac{4}{1}[/tex]
[tex]\frac{\text{Actual size}}{1.50\text{inches}}\times 1.50\text{ inches}=\frac{4}{1}\times 1.50\text{ inches}[/tex]
[tex]\text{Actual size}=4\times 1.50\text{ inches}[/tex]
[tex]\text{Actual size}=6.0\text{ inches}[/tex]
Therefore, the actual size is 6.0 inches and option A is the correct choice.
A dart is thrown upward with an initial velocity of 66 ft/s at an angle of elevation of 54°. Consider the position of the dart at any time t. Neglect air resistance. (Assume t is in seconds.) Find parametric equations that model the problem situation.
Final answer:
The parametric equations that model the problem situation in this case are x(t) = v0x * t, [tex]y(t) = v0y * t - (1/2) * g * t^2,[/tex], and vy(t) = v0y - g * t.
Explanation:
To find parametric equations that model the problem situation, we need to consider the horizontal and vertical components of the motion separately.
Horizontal Component:
The horizontal velocity remains constant throughout the motion. Therefore, the horizontal position can be given by the equation:
x(t) = v0x * t
where x(t) is the horizontal position at time t and v0x is the initial horizontal velocity.
Vertical Component:
The vertical position depends on the initial velocity, acceleration due to gravity, and time. We can use the following equations:
[tex]y(t) = v0y * t - (1/2) * g * t^2,[/tex]
vy(t) = v0y - g * t
where y(t) is the vertical position at time t, v0y is the initial vertical velocity, g is the acceleration due to gravity (approximately 32 ft/s2), and vy(t) is the vertical velocity at time t.
Given the differential Equation y'+2y=2e^x ;solve this equation using the integration factor; solve for y to get the general solution.
[tex]y'+2y=2e^x\Longrightarrow y'=2e^x-2y[/tex]
If [tex]f'(x)=g(x)[/tex] then [tex]y=\int{g(x)dx}[/tex]
So we extract,
[tex]y=\int{2e^x-2x}dx[/tex]
Which becomes,
[tex]y=2e^x-x^2+C[/tex]
Hope this helps.
r3t40
Furthermore, each license plate string must contain exactly 8 distinct characters (including the space character). For example, CMSC250 is not a valid license plate string, but ’CM8Z 2S0’ and ’BIGCARSZ’ are. How many license plate strings are possible?
Answer:
1,556,675,366,400
Step-by-step explanation:
There are 37 possible characters, of which 8 can be chosen. Order matters, so the number is ...
37P8 = 1,556,675,366,400
_____
This number includes 84,144,614,400 strings in which the space character is either first or last. Such strings may be ruled invalid because they are indistinguishable from 7-character strings.
_____
nPk = n!/(n-k)! . . . . the number of permutations of n things taken k at a time
The 37 allowed characters are the 26 letters of the alphabet, 10 digits, and 1 space character.
Solve one of the following two non-homogeneous differential equations using whatever technique your prefer. Put an "X" through the equation you would not like me to grade. If you do not technique you prefer. Put an "X" through the equation you would not like me to grade. If you do not put an "X" through one of the equations, I will grade whichever problem I prefer to grade. a) y" - 4y + 4y = 6xe^2x b) y" + 9y = 5 cos x - 7 sin x
Answer:
a.[tex]y(x)=c_1e^{2x}+c-2xe^{2x}+x^3e^{2x}[/tex]
b[tex]y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x[/tex]
Step-by-step explanation:
1.[tex]y''-4y'+4y=6x e^{2x}[/tex]
Auxillary equation
[tex]D^2-4D+ 4=0[/tex]
[tex](D-2)(D-2)=0[/tex]
D=2,2
Then complementary solution =[tex] C_1e^{2x}+C_2xe^{2x}[/tex]
Particular solution [tex]=\frac{6 xe^{2x}}{(D-2)^2}[/tex]
D is replace by D+2 then we get
P.I=[tex]\frac{6xe^{2x}}{0}[/tex]
P.I=[tex]\frac{e^{ax}}{D+a} \cdot .V[/tex]
where V is a function of x
P.I=[tex]\frac}x^3e^{2x}[/tex]
By integrating two times
Hence, the general solution
[tex]y(x)=c_1e^{2x}+c-2xe^{2x}+x^3e^{2x}[/tex]
b.y''+9y=5 cos x-7 sin x
Auxillary equation
[tex]D^2+9=0[/tex]
D=[tex]\pm 3i[/tex]
[tex]C.F=c_1 cos 3x+ c_2sin 3x[/tex]
P.I=[tex]\frac{5 cos x-7 sin x}{D^2+9}[/tex]
P.I=[tex]\frac{sin ax}{D^2+bD +C}[/tex]
Then D square is replace by -a square
[tex] D^2 [/tex] is replace by - then we get
P.I=-5 cos x+7 sin x
The general solution
[tex]y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x[/tex]
Find the following: F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k Exercise Find the curl and the divergence of the vector field.
[tex]\vec F(x,y,z)=e^{xy}\sin z\,\vec\jmath+y\tan^{-1}\dfrac xz\,\vec k[/tex]
Divergence is easier to compute:
[tex]\mathrm{div}\vec F=\dfrac{\partial(e^{xy}\sin z)}{\partial y}+\dfrac{\partial\left(y\tan^{-1}\frac xz\right)}{\partial z}[/tex]
[tex]\mathrm{div}\vec F=xe^{xy}\sin z-\dfrac{xy}{x^2+z^2}[/tex]
Curl is a bit more tedious. Denote by [tex]D_t[/tex] the differential operator, namely the derivative with respect to the variable [tex]t[/tex]. Then
[tex]\mathrm{curl}\vec F=\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\D_x&D_y&D_z\\0&e^{xy}\sin z&y\tan^{-1}\frac xz\end{vmatrix}[/tex]
[tex]\mathrm{curl}\vec F=\left(D_y\left[y\tan^{-1}\dfrac xz\right]-D_z\left[e^{xy}\sin z\right]\right)\,\vec\imath-D_x\left[y\tan^{-1}\dfrac xz\right]\,\vec\jmath+D_x\left[e^{xy}\sin z}\right]\,\vec k[/tex]
[tex]\mathrm{curl}\vec F=\left(\tan^{-1}\dfrac xz-e^{xy}\cos z\right)\,\vec\imath-\dfrac{yz}{x^2+z^2}\,\vec\jmath+ye^{xy}\sin z\,\vec k[/tex]
To find the curl and divergence of a given vector field, you first identify the vector's components. The curl is calculated using a determinant and the divergence is obtained by computing a dot product of the gradient operator with the vector field.
Explanation:In order to find the curl and the divergence of the vector field F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k, we first need to identify its components. The components are as follows: e^(xy) sin z, and y tan^−1(x/z).
The Curl of a vector field F in three dimensions is typically denoted as ∇ × F or curl F, where '∇' is the del operator. In Cartesian coordinates, this can be calculated using a determinant that involves the unit vectors î, ĵ, and k, the gradient operator, and the components of F.
The Divergence of a vector field F in three dimensions, typically denoted as ∇ . F or div F, is obtained by computing a dot product of the gradient operator with the vector field. This can also be calculated using Cartesian coordinates.
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Write an equation of the circle with center (9, -3) and radius 8.
Answer:
(x-9)^2 + (y+3)^2 = 8^2
or
(x-9)^2 + (y+3)^2 = 64
Step-by-step explanation:
An equation for a circle can be written as
(x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r is the radius
(x-9)^2 + (y- -3)^2 = 8^2
(x-9)^2 + (y+3)^2 = 8^2
or
(x-9)^2 + (y+3)^2 = 64
M1Q1.) Which plot represents a stemplot of the data?
Answer:
answer 1
Step-by-step explanation:
Since there's only 1 value in the 90-99 groups it is either 1 or 3.
Furthermore, there are only 2 values starting with 10.
This only fits with answer 1
The answer is A.
A stem plot shows every number with the tens place and above being on the left side while the ones place is on the right side.
For example,
9 is in the tens place while 4 is in the ones place. So, this would look like:
9 | 4
There is only one number with a "9" in the tens place and there is only two numbers with "10" in the tens place. This only sastisfys the first option.
Best of Luck!
Lane French has a bad credit rating and went to a local cash center. He took out a $100 loan payable in two weeks at $113.50. What is the percent of interest paid on this loan?
Answer: Percent of interest paid on this loan annually = 351% p.a
Step-by-step explanation:
Given that,
principal amount = $100(loan)
time period = 14 days
interest amount (SI) = $13.50
we have to calculate the rate of interest (i),
Simple interest(SI) = principal amount × rate of interest (i) × time period
13.50 = 100 × i × [tex]\frac{14}{365}[/tex]
i = [tex]\frac{4927.5}{1400}[/tex]
i = 3.51
i = 351% p.a.
Final answer:
The student paid a 13.5% interest on the $100 loan from the local cash center.
Explanation:
Percent of interest paid:
Initial loan amount: $100
Amount to be repaid: $113.50
Interest paid: $113.50 - $100 = $13.50
Percent interest paid = (Interest paid / Initial loan amount) * 100%
Percent interest paid = ($13.50 / $100) * 100% = 13.5%
At time t = 0 a car has a velocity of 16 m/s. It slows down with an acceleration given by –0.50t, in m/s^2 for t in seconds. By the time it stops it has traveled: A) 15 m B) 31 m C) 62 m D) 85 m E) 100 m
The car's velocity at time [tex]t[/tex] is
[tex]v(t)=16\dfrac{\rm m}{\rm s}+\displaystyle\int_0^t\left(\left(-0.50\frac{\rm m}{\mathrm s^2}\right)u\right)\,\mathrm du=16\dfrac{\rm m}{\rm s}+\left(-0.25\dfrac{\rm m}{\mathrm s^2}\right)t^2[/tex]
It comes to rest at
[tex]v(t)=0\implies16\dfrac{\rm m}{\rm s}=\left(0.25\dfrac{\rm m}{\mathrm s^2}\right)t^2\implies t=8.0\,\mathrm s[/tex]
Its velocity over this period is positive, so that the total distance the car travels is
[tex]\displaystyle\int_0^{8.0}v(t)\,\mathrm dt=\left(16\dfrac{\rm m}{\rm s}\right)(8.0\,\mathrm s)+\frac13\left(-0.25\dfrac{\rm m}{\mathrm s^2}\right)(8.0\,\mathrm s)^3=\boxed{85\,\mathrm m}[/tex]
so the answer is D.
The car takes 32 seconds to stop from its initial velocity of 16m/s. During this period, the car has traveled a distance of 256 meters, which is not an option in the given choices.
Explanation:The initial velocity of the car is given as 16 m/s and the acceleration is given as -0.5t m/s^2. We know that the car slows down until it stops, which means its final velocity is 0 m/s.
Firstly, we need to find out the time it takes for the car to stop. That could be calculated with the equation 'v = u + at', where v is the final velocity, u is the initial velocity, a is the acceleration and t is time. Since we know v = 0 m/s, u = 16 m/s and a = -0.5t m/s^2, we could set the equation to find the time to be '0 = 16 - 0.5t'. Solving this equation gives t=32 seconds.
Second, to find the distance traveled by the car during this time, we use the equation 's = ut + 0.5at^2', where s is the distance, u is the initial velocity, a is the acceleration and t is time. Substituting the known values into this equation, we get 's = 16(32) + 0.5*(-0.5*32)*(32)', which simplifies to s = 512 - 256 = 256 meters. Hence, the answer is not in the options given.
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1 kilogram (kg) is about 2.2 times as heavy as 1 pound (lb). Suppose the function f determines Emanuel's weight (in lbs), f ( t ) , given the number of days t since the beginning of 2017. The function g determines Emanuel's weight (in kg), g ( t ) , given the number of days t since the beginning of 2017. a Suppose f(35) 171. What is the value of g(35)?b. Write a formula for g using the function f.
Final answer:
To convert Emanuel's weight from pounds to kilograms on the 35th day, multiply 171 lbs by 0.4536 to get 77.52 kg. The general formula for converting function f(t) to g(t) is g(t) = f(t) × 0.4536.
Explanation:
If Emanuel's weight in pounds on the 35th day since the beginning of 2017 is 171 lbs ( f(35) = 171 ), we can find his weight in kilograms ( g(35) ) using the conversion factor from pounds to kilograms. Since 1 pound is equivalent to approximately 0.4536 kilograms on Earth, we can calculate g(35) by multiplying Emanuel's weight in pounds by this conversion factor:
g(35) = 171 lbs × 0.4536 kg/lb
This results in g(35) = 77.5156 kg. When we round this to significant figures based on the given conversion fact of pounds to kilograms (which is inexact and has 4 significant figures), Emanuel's weight would be g(35) = 77.52 kg (to 4 SFs).
The formula for g using the function f is:
g(t) = f(t) × 0.4536