The obtained solution is in implicit form: erf(0.5x) = -4y - 32 + C, where C is an arbitrary constant.
To solve the given differential equation, we'll use the method of integrating factors. The equation can be rewritten as:
dx = xy + 8x + 2y + 16
Rearranging the terms:
dx - xy - 8x = 2y + 16
To find the integrating factor, we'll consider the coefficient of y, which is -x. Multiplying the entire equation by -1 will make it easier to work with:
-x dx + xy + 8x = -2y - 16
The integrating factor is defined as the exponential of the integral of the coefficient of y. In this case, the coefficient is -x, so the integrating factor is [tex]e^{\int -x dx}[/tex].
Integrating -x with respect to x gives us:
∫-x dx = [tex]-0.5x^2[/tex]
Therefore, the integrating factor is [tex]e^{(-0.5x^2)[/tex].
Now, multiply the original equation by the integrating factor:
[tex]e^{-0.5x^2} * (dx - xy - 8x) = e^{-0.5x^2} * (2y + 16)[/tex]
Using the product rule of differentiation on the left side:
[tex](e^{-0.5x^2} * dx) - (x * e^{-0.5x^2} * dx) - (8x * e^{-0.5x^2}) = 2y * e^{-0.5x^2} + 16 * e^{-0.5x^2}[/tex]
Simplifying the left side:
[tex]d(e^{-0.5x^2}) - (x * e^{-0.5x^2} * dx) - (8x * e^{-0.5x^2}) = 2y * e^{-0.5x^2} + 16 * e^{-0.5x^2}[/tex]
Now, integrating both sides with respect to x:
[tex]\int d(e^{-0.5x^2}) - \int x * e^{-0.5x^2} * dx - \int 8x * e^{-0.5x^2} dx = 2y * e^{-0.5x^2} + 16 * e^{-0.5x^2} dx\\[/tex]
The first term on the left side integrates to [tex]e^{-0.5x^2}[/tex]. The second term can be solved using integration by parts,
considering u = x and [tex]dv = e^{-0.5x^2} dx[/tex]:
[tex]\int x * e^{-0.5x^2} * dx = -0.5\int e^{-0.5x^2} * dx^2 = -0.5 * e^{-0.5x^2[/tex]
The third term can also be solved using integration by parts, considering u = 8x and [tex]dv = e^{-0.5x^2} dx[/tex]:
[tex]\int 8x * e^{-0.5x^2} * dx = -4\int x * e^{-0.5x^2} * dx = -4 * -0.5 * e^{-0.5x^2} = 2 * e^{-0.5x^2}\\[/tex]
Simplifying the right side:
[tex]\int 2y * e^{-0.5x^2} + 16 * e^{-0.5x^2} dx = \int (2y + 16) * e^{-0.5x^2} dx\\[/tex]
Now, let's combine the terms on both sides:
[tex]e^{-0.5x^2} - 0.5 * e^{-0.5x^2} - 2 * e^{-0.5x^2} = \int (2y + 16) * e^{-0.5x^2} dx[/tex]
Simplifying further:
e^{-0.5x^2} - 0.5 * e^{-0.5x^2} - 2 * e^{-0.5x^2} = \int (2y + 16) * e^{-0.5x^2} dx
Combining the terms on the left side:
[tex]-0.5 * e^{-0.5x^2} = \int (2y + 16) * e^{-0.5x^2} dx[/tex]
Now, we can integrate both sides:
[tex]-0.5 \int e^{-0.5x^2} dx = \int (2y + 16) * e^{-0.5x^2} dx[/tex]
The integral on the left side is a well-known integral involving the error function, erf(x):
[tex]-0.5 \int e^{-0.5x^2}dx = -0.5 \sqrt{\pi /2} * erf(0.5x)[/tex]
The integral on the right side is simply (2y + 16) times the integral of [tex]e^{-0.5x^2[/tex], which is [tex]\sqrt{ \pi /2}[/tex].
Putting it all together:
-0.5 √(π/2) * erf(0.5x) = (2y + 16) √(π/2) + C
Dividing both sides by -0.5 √(π/2) and simplifying:
erf(0.5x) = -4y - 32 + C
The error function erf(0.5x) is a known function that cannot be easily expressed in terms of elementary functions. Therefore, we have obtained a solution in implicit form:
erf(0.5x) = -4y - 32 + C
where C is an arbitrary constant.
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Part A
An economist has measured people's annual salary (in thousands of dollars) and their years of relevant job experience, thinking that a linear relationship between them might exist.
Let the proposed regression relationship between Salary and experience be as follows: E(Salary) = beta subscript 0 space plus space beta subscript 1 space cross times Years of Experience
and assume the output from running the regression is as follows:
Call:
lm(formula = Salary ~ Year, data = Income)
Residuals:
Min 1Q Median 3Q Max
-53.650 -20.256 0.127 18.423 65.596
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 31.8387 8.5565 3.721 0.00033***
Years 2.8205 0.3302 8.543 1.74e-13 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 25.98 on 98 degrees of freedom
Multiple R-squared: 0.4268, Adjusted R-squared: 0.421
F-statistic: 72.98 on 1 and 98 DF, p-value: 1.737e-13
---
Residual standard error: 8.044 on 445 degrees of freedom
Multiple R-squared: 0.6914, Adjusted R-squared: 0.6886
F-statistic: 249.2 on 4 and 445 DF, p-value: < 2.2e-16
If we wished to conduct a hypothesis test as to whether there is a linear relationship between salary and years of experience, what are the correct null and alternate hypotheses?
Answers:
a.
H subscript 0 : space beta subscript 0 space equals space 0 H subscript 1 : space beta subscript 0 greater than space 0
b.
H subscript 0 : space beta subscript 0 space equals space 0 H subscript 1 : space beta subscript 0 space end subscript not equal to space 0
c.
H subscript 0 space : thin space beta subscript 1 space equals space 0 H subscript 1 : space beta subscript 1 space end subscript space not equal to space 0
d.
H subscript 0 : space beta subscript 1 space equals space 0 H subscript 1 : space beta subscript 1 space greater than space 0
Part B
Using the output in Q1, what is the correct p-value for the test in Q1?
Answers:
a.
0.00033
b.
0.000000000000174
c.
1.74e-13
d.
0.00000393
Part C
What is the fitted regression model from this output in Q1?
Answers:
a.
E(Salary) = 31.8387 + 2.8205 x Years of Experience
b.
E( Years of Experience ) = 2.8205 + 31.8387 x Salary
c.
E( Years of Experience ) = 31.8387 + 2.8205 x Salary
d.
E(Salary) = 2.8205 + 31.8387 x Years of Experience
Part D
Which of the following is a correct statement regarding r squared ?
Answers:
a.
r squared space equals space 0.4268 meaning that Years of Experience explains 42.68 percent sign of the variability in Salary.
b.
r squared space equals space 0.00033 meaning that Years of Experience explains 0.033 percent sign of the variability in Salary.
c.
r squared space equals space 0.00033 and because 0.00033 space less than space 0.05 we reject H subscript 0 and accept H subscript 1 at the 5% level of significance, ie we conclude there is a significant linear relationship between Salary and Years of Experience.
d.
r squared space equals space 0.4268 and because 0.4268 space greater than space 0.05 we do not reject H subscript 0 at the 5% level of significance, ie we conclude there is no significant linear relationship between Salary and Years of Experience.
The correct statement regarding r squared is:
r squared equals 0.4268 meaning that Years of Experience explains 42.68 percent of the variability in Salary.
Part A: The correct null and alternate hypotheses are:
H₀: β₁=0;
H₁: β₁≠0.
Part B: The correct p-value for the test in Q1 is 1.74e-13.
Part C: The fitted regression model from this output in Q1 is:
E(Salary) = 31.8387 + 2.8205 x Years of Experience.
Part D: The correct statement regarding r squared is:
r squared equals 0.4268 meaning that Years of Experience explains 42.68 percent of the variability in Salary.
Explanation: The output shows a multiple linear regression model:
Salary=β0+β1x
Years of Experience + ϵ.β0 is the intercept and represents the expected mean salary for an individual with 0 years of experience.
β1 is the slope and represents the expected change in salary due to one year increase in experience.
ϵ is the error term (deviation from the expected salary).
The correct null and alternate hypotheses are:
H₀: β₁=0 (there is no linear relationship between salary and years of experience).
H₁: β₁≠0 (there is a linear relationship between salary and years of experience).
The correct p-value for the test in Q1 is 1.74e-13, which is much smaller than the significance level of 0.05.
Thus, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that there is a linear relationship between salary and years of experience.
The fitted regression model from this output in Q1 is:
E(Salary) = 31.8387 + 2.8205 x Years of Experience.
The coefficient of determination, or R-squared, is a statistical measure that shows how well the regression model fits the observed data.
It is the proportion of the variance in the dependent variable that is explained by the independent variable(s).
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If Y is inversely proportional to x and y=4 when x= 100, what is the value of y when x=250
Answer:
y=10
Step-by-step explanation:
[tex]\frac{4}{100}[/tex]=[tex]\frac{y}{250}[/tex]
cross-multiply, 4*250=100y
isolate the variable and solve for y, 1000=100y
divide 100 on both sides, 10=y
A cylinder has a volume of 2,309.07 cubic cm and a height of 15 cm. What is the
radius of the cylinder? Use 3.14 for st in your calculations and round to the
nearest whole number.
cm
Answer:
7
Step-by-step explanation:
the answer I got was 7 hope this helped
A fresh food distributor receives orders from 100 customers daily. Assume that the quantities ordered by customers, in kg, are independent continuous random variables uniformly distributed over the interval (0, 9). Assuming that the distributor only has the capacity to ship 477 kg of products daily, calculate the probability that all orders are fulfilled on a day chosen at random. Indicate the result to at least four decimal places.
The probability that all orders are fulfilled on a day chosen at random is approximately 1. Answer: 1.0000 (rounded off to at least four decimal places).
The quantities ordered by customers are independent continuous random variables, and they are uniformly distributed over the interval (0, 9).
The fresh food distributor only has the capacity to ship 477 kg of products daily, and the distributor receives orders from 100 customers daily.
The probability that all orders are fulfilled on a day chosen at random is given by;P(all orders fulfilled) = P(X1 + X2 + ... + X100 < 477)
where X is the quantity ordered by each customer. Since X is a continuous random variable, we can use the probability density function of a uniform distribution to calculate the probability density function of X as;f(x) = 1/9, 0 < x < 9
Hence, the probability that all orders are fulfilled on a day chosen at random is given by;
P(all orders fulfilled) = P(X1 + X2 + ... + X100 < 477)= P[(X1/9) + (X2/9) + ... + (X100/9) < (477/9)]= P[U < (53 + 1/3)], where U ~ Uniform(0, 1)
Now, using the central limit theorem, we can approximate the distribution of U by a normal distribution with mean μ = 1/2 and variance σ^2 = 1/12 such that;Z = (U - μ) / σ ~ N(0, 1)
Hence, P[U < (53 + 1/3)] = P[Z < (53 + 1/3 - μ) / σ]= P[Z < (53 + 1/3 - 1/2) / sqrt(1/12)]≈ P[Z < 9.6067]≈ 1
Thus, the probability that all orders are fulfilled on a day chosen at random is approximately 1. Answer: 1.0000 (rounded off to at least four decimal places).
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what is 4836 divided by 9735829 plus 28369 times 284383?
Answer:
2.28871.0005^14
Step-by-step explanation:
Answer:
i got 8067661468.26
Step-by-step explanation:
Jim , a meteorologist for local television XYZ, would like to report the rainfall. The following are the rainfall measurements (in inches) for to-day’s date for 14 randomly chosen past years. The data are as follows: 0.47 0.27 0.13 0.54 0.08 1.05 0.34 0.26 0.42 0.17 0.50 0.86 0.01 2.5 NOTE: answers should be given to 2 decimal places REQUIRED 1. Find the RANGE of the data provided and what measure does this show 2. Calculate the sample variance . Why is variance not written in? 3. Calculate the sample standard deviation. What information does this measure give to the users? Why is it necessary to compute standard deviation instead of stopping at the variance calculation? 4. Find the COEFFICIENT OF VARIATION 5. Use Z-score to find if there is any outliers in the series
Where the above is given,
Range -2.49 inchesSample variance - 0.5599Sample standard deviation - 0.7483Coefficient of variation - 7497.32% Z14 = 3.3999, which is greater than 2, suggesting that the data point 2.50 inches may be considered an outlier based on the Z-Score criterion.What is the explanation for the above ?Range
Range = Maximum value - Minimum value
Range = 2.50 - 0.01
= 2.49 inches
Sample Variance
Step 1 - Calculate the mean (average) of the data.
Mean = (0.47 + 0.27 +0.13 + 0.54 + 0.08 + 1.05 + 0.34 + 0.26 + 0.42 + 0.17 + 0.50 + 0.86 + 0.01 + 2.5) / 14
= 1.36 /14
= 0.0971
Step 2 - Calculate the squared deviation from the mean for each data point.
Deviation from mean = Data point - Mean
Squared deviation = (Deviation from mean)²
Sum of squared deviations = (0.3739 + 0.0091 + 0.7307 + 0.1638 + 0.9469 + 0.104 + 0.0145 + 0.0081 + 0.0971 + 0.3037 + 0.0601 + 0.2601 + 1.0679 + 3.1391)
= 7.2799
Sample Variance = Sum of squared deviations / (n - 1)
= 7.2799/ (14 - 1)
= 0.5599
Sample Standard Deviation:
Sample Standard Deviation = √(Sample Variance)
= √0.5599
= 0.7483
Coefficient of Variation:
Coefficient of Variation = (Sample Standard Deviation / Mean) * 100
= (7.2799/ 0.0971) * 100
= 7497.32
Z-Scores for each data point
Z1= (0.47 - 0.0971)/ 0.7212= 0.5318
Z2= (0.27 - 0.0971)/ 0.7212= 0.9983
Z3= (0.13 - 0.0971)/ 0.7212= 0.0456
Z4= (0.54 - 0.0971)/ 0.7212= 0.6189
Z5= (0.08 - 0.0971)/ 0.7212= -0.0251
Z6= (1.05 - 0.0971)/ 0.7212= 1.2914
Z7= (0.34 - 0.0971)/ 0.7212= 0.4574
Z8= (0.26 - 0.0971)/ 0.7212= 0.2251
Z9= (0.42 - 0.0971)/ 0.7212= 0.4969
Z10= (0.17 - 0.0971)/ 0.7212= 0.1002
Z11= (0.50 - 0.0971)/ 0.7212= 0.6506
Z12= (0.86 - 0.0971)/ 0.7212= 1.0242
Z13= (0.01 - 0.0971)/ 0.7212= -0.1208
Z14= (2.50 - 0.0971)/ 0.7212= 3.3999
To determine if there are any outliers, we can compare the absolute values of the Z-Scores to a certain threshold , commonly considered as 2.
If the absolute value of a Z-Score is greater than 2, it indicates that the corresponding data point is an outlier.
In this case, Z14 = 3.3999, which is greater than 2 , suggesting that the data point 2.50 inches may be considered an outlier based on the Z-Score criterion.
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In a certain chemical, the ratio of zinc to copper is 3 to 14. Ajar of the chemical contains
546 grams of copper. How many grams of zinc does it contain?
It contains grams of zinc
Answer:
Given -
In a certain chemical, the ratio of zinc to copper is 4 to 13.
The chemical contains 546 grams of copper.
Prove -
How many grams of zinc does it chemical contain.
Answer
suppose that scalar multiple of the zinc and copper be y .
As given
In a certain chemical, the ratio of zinc to copper is 4 to 13.
The chemical contains 546 grams of copper.
Than equation is
13y = 546
13y = 546
y = 42
Zinc contain in the certain chemical = 4y
= 4 42
= 168 grams
Therefore 168 grams of zinc contain in a certain chemical .Step-by-step explanation:
Compute in the ancient Egyptian way: (b) 55÷6 (a) 26 ÷ 20 (c) 71 21 ÷ (d) 25 18 (e) 52 ÷ 68 (f) 13 36
Ancient Egyptian way of computation, division was performed using a method called "repeated subtraction." (a) 26 ÷ 20 = 1 remainder 6.
(b) 55 ÷ 6 = 9 remainder 19.
(c) 71 21 ÷ = 50 remainder 29.
(d) 25 18 ÷ = 7.
(e) 52 ÷ 68 = 0 remainder 52.
(f) 13 36 ÷ = 0 remainder -23.
In the ancient Egyptian way of computation, division was performed using a method called "repeated subtraction." Here's how it would be applied to the given divisions:
(a) 26 ÷ 20:
To divide 26 by 20, we repeatedly subtract 20 from 26 until we cannot subtract anymore. The number of times we subtract is the quotient.
26 - 20 = 6
6 - 20 = -14 (cannot subtract anymore)
Therefore, 26 ÷ 20 = 1 remainder 6.
(b) 55 ÷ 6:
Using the same method, we repeatedly subtract 6 from 55 until we cannot subtract anymore.
55 - 6 = 49
49 - 6 = 43
43 - 6 = 37
37 - 6 = 31
31 - 6 = 25
25 - 6 = 19 (cannot subtract anymore)
Therefore, 55 ÷ 6 = 9 remainder 19.
(c) 71 21 ÷:
To divide 71 21 by a number, we first convert it to a whole number by multiplying the fraction part by the denominator.
71 21 = 71 + (21/100) = 71 + 21/100
Now, we can perform division using repeated subtraction.
71 - 21 = 50
50 - 21 = 29 (cannot subtract anymore)
Therefore, 71 21 ÷ = 50 remainder 29.
(d) 25 18 ÷:
Similar to the previous case, we convert 25 18 to a whole number.
25 18 = 25 + (18/100) = 25 + 18/100
Performing division:
25 - 18 = 7
Therefore, 25 18 ÷ = 7.
(e) 52 ÷ 68:
Since 52 is smaller than 68, the quotient is 0.
Therefore, 52 ÷ 68 = 0 remainder 52.
(f) 13 36 ÷:
Converting to a whole number:
13 36 = 13 + (36/100) = 13 + 36/100
Performing division:
13 - 36 = -23 (cannot subtract anymore)
Therefore, 13 36 ÷ = 0 remainder -23.
Please note that the ancient Egyptian method of division is not as efficient as modern division methods and may not produce exact decimal results.
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Solve the system of inequalities graphically:
x−2y≤3,3x+4y≥12,x≥0,y≥1
The solution to the system of inequalities is the region above the x-axis and to the right of the line y = 1, shaded in green.
Given system of inequalities is: x - 2y ≤ 3 ...(1)3x + 4y ≥ 12 ...(2)x ≥ 0 ...(3)y ≥ 1 ...(4)
We graph the lines x - 2y = 3 and 3x + 4y = 12 and shade the appropriate regions.
Let's start with the line x - 2y = 3.
We rewrite this as y = (1/2)x - 3/2 and plot the line as shown below: graph{(1/2)x - 3/2 [-10, 10, -5, 5]}
Now we determine which side of the line we want to shade.
Since the inequality is of the form ≤, we shade below the line y = (1/2)x - 3/2 (including the line itself) as shown below: graph {(1/2)x - 3/2 [-10, 10, -5, 5](-10,-5)--(10,0)}
Next, we graph the line 3x + 4y = 12. We rewrite this as y = (-3/4)x + 3 and plot the line as shown below: graph{(-3/4)x + 3 [-10, 10, -5, 5]}
We determine which side of the line we want to shade. Since the inequality is of the form ≥, we shade above the line y = (-3/4)x + 3 (including the line itself) as shown below: graph{(-3/4)x + 3 [-10, 10, -5, 5](-10,4)--(10,0)}
Finally, we shade the region that satisfies x ≥ 0 and y ≥ 1.
This is the region above the x-axis and to the right of the line y = 1 as shown below: graph{(-3/4)x + 3 [-10, 10, -5, 5](-10,4)--(10,0)(0,1)--(10,1)[above]}
The shaded region is the region that satisfies all three inequalities.
Thus, the solution to the system of inequalities is the region above the x-axis and to the right of the line y = 1, shaded in green.
We graph the lines x - 2y = 3 and 3x + 4y = 12 and shade the appropriate regions.
Let's start with the line x - 2y = 3. We rewrite this as y = (1/2)x - 3/2 and plot the line.
Now we determine which side of the line we want to shade. Since the inequality is of the form ≤, we shade below the line y = (1/2)x - 3/2 (including the line itself).
Next, we graph the line 3x + 4y = 12. We rewrite this as y = (-3/4)x + 3 and plot the line. We determine which side of the line we want to shade.
Since the inequality is of the form ≥, we shade above the line y = (-3/4)x + 3 (including the line itself).
Finally, we shade the region that satisfies x ≥ 0 and y ≥ 1.
This is the region above the x-axis and to the right of the line y = 1. The shaded region is the region that satisfies all three inequalities.
Thus, the solution to the system of inequalities is the region above the x-axis and to the right of the line y = 1, shaded in green.
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(22) + (3x) = 4
solve for x
Answer:
-6
Step-by-step explanation:
Step One: The goal is to isolate the x. So first, we would do 4-22, which is -18. The equation is: 3x=-18.
Step Two: Lastly, we need to divide by three to completely isolate the x. This is -6.
Which expression is equivalent to 1.5a + 2.4 (a + 0.5b) - 0.2b?
a. 3.9a + b
b. 2.5a + 0.7b
c. 3.9a + 0.3b
d. 2.5a + 0.3b + 2.4
The answer is A, 3.9a + b
Step-by-step explanation:
youre welcome
2. How does the graph of the following function compare to the quadratic parent function? * (1 Point) 8 (x) = x2 + 5 Moves up 5 Moves down 5 Moves to the left 5 Moves to the right 5
Answer:
b
Step-by-step explanation:
oi did the quiz f. 6373737
a) The difference of the age of two sisters is 5 years and the product of their age is
24. Find the age of the two sisters.
Answer:
3 years old and 8 years old
Step-by-step explanation:
8-3=5
3x8=24
plz mark me as brainliest
2/6, 5/12, 3/7, and 4/10. List least to most
Use digits to write the value of the 4 in this number.
842,963
Answer:
40,000
Step by step explanation:
842,963
840,000
40,000
What is the area of a rectangle with side lengths 2/5 feet and 4/6 feet?
Answer:
[tex]\frac{4}{15}[/tex] (4/15)
Step-by-step explanation:
[tex]\frac{4}{6}=\frac{2}{3}[/tex]
[tex]\frac{2}{5}*\frac{2}{3};[/tex]
1- Multiply the numerators:
[tex]2*2=4[/tex]
2- Multiply the denominators:
[tex]5*3=15[/tex]
3- Thus:
[tex]\frac{2}{5}*\frac{2}{3}= \frac{4}{15}[/tex]
Hope this helps ;)
The odds against a certain football team winning the championship are 70 : 1. a) Determine the probability that the team wins the championship. b) Determine the probability that the team does not win the championship.
a) The probability that the team wins the championship is approximately 0.0141 or 1.41%. b) The probability that the team does not win the championship is approximately 0.9859 or 98.59%.
a) To determine the probability that the team wins the championship, we need to convert the odds against winning into a probability.
The odds against winning are given as 70:1. This means that for every 70 unfavorable outcomes (losing), there is 1 favorable outcome (winning).
To calculate the probability of winning, we divide 1 by the sum of the favorable and unfavorable outcomes:
Probability of winning = 1 / (70 + 1) = 1 / 71 ≈ 0.0141 (or 1.41%)
Therefore, the probability that the team wins the championship is approximately 0.0141 or 1.41%.
b) The probability of not winning the championship is equal to 1 minus the probability of winning:
Probability of not winning = 1 - 0.0141 = 0.9859 (or 98.59%)
Therefore, the probability that the team does not win the championship is approximately 0.9859 or 98.59%.
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sales tax: 68% shirts: $35 pants: $27 shoes: $44 what is the total cost
Express (–1+iV3) and (-1 – iV3) in the exponential form to show that:
2nnt (-1+iV3)n +(-1 – iV3)n = 2n+1cos 3
The proof of 2ⁿ (-1 + i√3)ⁿ + (-1 - i√3)ⁿ can be expressed as 2ⁿ⁺¹cos(πn/3) is proved below.
To express (-1 + i√3) and (-1 - i√3) in exponential form, we can use Euler's formula, which states that [tex]e^{(i\theta)[/tex] = cos(θ) + isin(θ).
Let's start with (-1 + i√3):
(-1 + i√3) = 2 x (cos(π) + i x sin(π/3))
Now, let's simplify (-1 - i√3):
(-1 - i√3) = 2 (cos(π) - isin(π/3))
Therefore, we have:
(-1 + i√3) = 2 e^(iπ/3)
(-1 - i√3) = 2 e^(-iπ/3)
Now, let's substitute these exponential forms into the expression:
2ⁿ (-1 + i√3)^n + (-1 - i√3)^n
= 2ⁿ(2 e^(iπ/3))^n + (2 e^(-iπ/3))^n
= 2ⁿ⁺¹ e^(iπn/3) + 2^(n+1) e^(-iπn/3)
Using Euler's formula again, we know that [tex]e^{(i\theta)} + e^{(-i\theta)[/tex] = 2cos(θ).
Therefore, we can rewrite the expression as:
2ⁿ⁺¹ (cos(πn/3) + cos(-πn/3))
= 2ⁿ⁺¹(cos(πn/3) + cos(πn/3))
= 2ⁿ⁺¹ 2 cos(πn/3)
= 2ⁿ⁺¹cos(πn/3)
So, we have shown that 2ⁿ (-1 + i√3)ⁿ + (-1 - i√3)ⁿ can be expressed as 2ⁿ⁺¹cos(πn/3).
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A guy wire supporting a radio tower is attached to the tower 128 feet above the ground. The wire makes a 45 degree angle with the ground. How long is the guy wire
Answer:
181 feet
Step-by-step explanation:
From the the diagram attached ,
Sinθ = a/b..................... Equation 1
Where θ = angle to the horizontal, a = Height of the tower, b = length of the wire.
make b the subject of the equation
b = a/sinθ................. Equation 2
Given: a = 128 feet, θ = 45°
Substitute these values into equation 2
b = 128/sin45°
b = 128/0.7071
b = 181 feet
b = 181 feet
Find the distance between the points (–8,10) and (4,10).
Answer:
12
Step-by-step explanation:
√(x2 - x1)² + (y2 - y1)²
√[4 - (-8)]² + (10 - 10)²
√(12)² + (0)²
√144 + 0
√144
=12
Let A and B be disjoint compact subspaces of a Hausdorff space X. Show that there exist disjoint open sets U and V, with ACU and BCV.
In Hausdorff-space "X", if A and B are disjoint "compact-subspaces", then there is disjoint "open-sets" U and V such that A is contained in U and B is contained in V, this is because by Hausdorff-Property, the existence of disjoint open neighborhoods for any two "distinct-points".
To prove the existence of disjoint "open-sets" U and V with A⊂U and B⊂V, where A and B are "compact-subspaces" of "Hausdorff-space" X,
Step (1) : A and B are disjoint compact-subspaces, we use Hausdorff property to find "open-sets" Uₐ and [tex]U_{b}[/tex] such that "A⊂Uₐ" and "B⊂[tex]U_{b}[/tex]", and "Uₐ∩[tex]U_{b}[/tex] = ∅". This can be done for every pair of points in A and B, respectively, because X is Hausdorff.
Step (2) : We consider, set U = ⋃ Uₐ, where "union" is taken over all of Uₐ for each-point in A. U is = union of "open-sets", hence open.
Step (3) : We consider set V = ⋃ [tex]U_{b}[/tex], where union is taken over for all [tex]U_{b}[/tex] for "every-point" in B. V is also a union of open-sets and so, open.
Step (4) : We claim that U and V are disjoint. Suppose there exists a point x in U∩V. Then x must be in Uₐ for some point a in A and also in [tex]U_{b}[/tex] for some point b in B. Since A and B are disjoint, a and b are different points. However, this contradicts the fact that Uₐ and [tex]U_{b}[/tex] are disjoint open sets.
Therefore, U and V are disjoint open sets with A⊂U and B⊂V.
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The given question is incomplete, the complete question is
Let A and B be disjoint compact subspaces of a Hausdorff space X. Show that there exist disjoint open sets U and V, with A⊂U and B⊂V.
Consider the derivation of the quadratic formula below. What is the missing radicand
in Step 6?
Answer:
[tex]\frac{b^2 - 4ac}{4a^2}[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required:
Complete step 6
At step 5, we have:
[tex](x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{4ac}{4a^2}[/tex]
Take LCM
[tex](x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}[/tex]
Take square roots of both sides to get step 6
[tex]x + \frac{b}{2a} = \±\sqrt{\frac{b^2 - 4ac}{4a^2}}[/tex]
Hence, the missing radicand is: [tex]\frac{b^2 - 4ac}{4a^2}[/tex]
Consider the following IVP: u"(t) + u'(t) - 12u (t)=0 (1) u (0) = 60 and u'(0) = 56. Show that u(t)=c₁₁e² + c ₂² -4 satisifes ODE (1) and find the values of c ER and C₂ ER such that the solution satisfies the given initial values.
The values of c₁ and c₂ that satisfy the initial values u(0) = 60 and u'(0) = 56 are:
c₁ = 148 / (3e²)
c₂ = (20 - 148/9)e⁴
The given solution, u(t) = c₁e² + c₂e⁻⁴, indeed satisfies the given ordinary differential equation (ODE) u"(t) + u'(t) - 12u(t) = 0. To find the values of c₁ and c₂ such that the solution satisfies the initial values u(0) = 60 and u'(0) = 56, we substitute these values into the solution.
First, let's find u(0) by substituting t = 0 into the solution:
u(0) = c₁e² + c₂e⁻⁴
Since u(0) = 60, we have:
60 = c₁e² + c₂e⁻⁴ (Equation 2)
Next, let's find u'(0) by differentiating the solution with respect to t and substituting t = 0:
u'(t) = 2c₁e² - 4c₂e⁻⁴
u'(0) = 2c₁e² - 4c₂e⁻⁴
Since u'(0) = 56, we have:
56 = 2c₁e² - 4c₂e⁻⁴ (Equation 3)
Now we have a system of two equations (Equations 2 and 3) with two unknowns (c₁ and c₂). We can solve this system to find the values of c₁ and c₂.
To do that, let's first divide Equation 3 by 2:
28 = c₁e² - 2c₂e⁻⁴
Next, let's multiply Equation 2 by 2:
120 = 2c₁e² + 2c₂e⁻⁴
Adding the two equations, we get:
148 = 3c₁e²
Dividing both sides by 3e², we find:
c₁ = 148 / (3e²)
Substituting this value of c₁ back into Equation 2, we can solve for c₂:
60 = (148 / (3e²))e² + c₂e⁻⁴
60 = 148/3 + c₂e⁻⁴
60 - 148/3 = c₂e⁻⁴
20 - 148/9 = c₂e⁻⁴
c₂ = (20 - 148/9)e⁴
Therefore, the values of c₁ and c₂ that satisfy the initial values u(0) = 60 and u'(0) = 56 are:
c₁ = 148 / (3e²)
c₂ = (20 - 148/9)e⁴
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solve for x to the nearest
Answer:
in right angled triangleBCD
BC=√{DC²-BC²)=√{10²-6²)=8
again in right angled triangle ABC
AB=√(BC²-AC²)
x=√(8²-7²)=3.87
Help with question 3 please I’ll give brainlest
Answer: b
Step-by-step explanation: When you rotate it around the axis (think of it as a pole) it will become a cylinder with radius 5
Simplify 4(3n + 2b + k)
a. 12nbk
b. 12n + 2bk
c. 12n + 8b + 4k
d. 24bkn
Answer:
C
Step-by-step explanation:
4(3n+2b+k)
12n+8b+4k
So it would be C.
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hope it helps
Function f is a quadratic function passing through the points (-4,0),(0,–12) and (3,0). Function g is modeled by the graph. Over which interval are both functions negative?
Answer:
Open points on 1 and 3 with ray connecting them
Step-by-step explanation:
Because it's right.
Answer:
Step-by-step explanation:
Bacteria triples every 3 hours. If there are 450 bacteria at t=0
min how many after 200 min
Therefore, the number of bacteria after 200 minutes can be found by multiplying the initial number of bacteria (450) by the tripling factor (3) raised to the power of 2 (for the full cycles) and multiplying by the remaining fraction of the tripling factor for the partial cycle.
Since the bacteria triples every 3 hours, we can calculate the number of tripling cycles that have occurred in 200 minutes. Since 3 hours is equivalent to 180 minutes, there are 2 full cycles in 200 minutes. To calculate the remaining fraction of the tripling cycle, we divide the remaining time (20 minutes) by the length of a single cycle (180 minutes). The fraction is 20/180, which simplifies to 1/9.
Now, we can calculate the number of bacteria after 200 minutes. We start with the initial number of bacteria, which is 450, and multiply it by the tripling factor (3) raised to the power of the number of full cycles (2). This accounts for the full cycles. Then, we multiply this result by the remaining fraction of the tripling factor (1/9) to account for the partial cycle.
Therefore, the number of bacteria after 200 minutes can be calculated as follows:
Number of bacteria = 450 * (3^2) * (1/9) = 450 * 9 * (1/9) = 450
Hence, after 200 minutes, there will still be 450 bacteria.
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What is the value of x? sin(x+37)°=cos(2x+8)° Enter your answer in the box. x =
The answer is x = 15.
15 or 20.33 are the possible values of the x.
What is algebraic Expression?Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.
We know that sin(x+37)°=cos(90°-(x+37)°) and cos(2x+8)°=sin(90°-(2x+8)°)
So we have sin(x+37)°=cos(2x+8)° becomes sin(x+37)°=sin(82°-2x)
For the above equation to be true, either of the following must hold:
x+37 = 82 - 2x (since the sin function is periodic)
x+37 = 180 - (82-2x)
Solving the first equation for x gives:
3x = 45
x = 15
Solving the second equation for x gives:
3x = 61
x = 20.33 (rounded to two decimal places)
Therefore, the possible values of x are 15 or 20.33 (rounded to two decimal places).
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