I'll use the integrating factor method for the first DE, and undetermined coefficients for the second one.
(a) Multiply both sides by exp(7t ):
exp(7t ) dx/dt + 7 exp(7t ) x = 5 exp(7t ) cos(2t )
The left side is now the derivative of a product:
d/dt [exp(7t ) x] = 5 exp(7t ) cos(2t )
Integrate both sides:
exp(7t ) x = 10/53 exp(7t ) sin(2t ) + 35/53 exp(7t ) cos(2t ) + C
Solve for x :
x = 10/53 sin(2t ) + 35/53 cos(2t ) + C exp(-7t )
(b) Solve the corresonding homogeneous DE:
d²x/dt ² + 6 dx/dt + 8x = 0
has characteristic equation
r ² + 6r + 8 = (r + 4) (r + 2) = 0
with roots at r = -4 and r = -2. So the characteristic solution is
x (char.) = C₁ exp(-4t ) + C₂ exp(-2t )
For the particular solution, assume an ansatz of the form
x (part.) = a cos(3t ) + b sin(3t )
with derivatives
dx/dt = -3a sin(3t ) + 3b cos(3t )
d²x/dt ² = -9a cos(3t ) - 9b sin(3t )
Substitute these into the non-homogeneous DE and solve for the coefficients:
(-9a cos(3t ) - 9b sin(3t ))
… + 6 (-3a sin(3t ) + 3b cos(3t ))
… + 8 (a cos(3t ) + b sin(3t ))
= (-a + 18b) cos(3t ) + (-18a - b) sin(3t ) = 5 sin(3t )
So we have
-a + 18b = 0
-18a - b = 5
==> a = -18/65 and b = -1/65
so that the particular solution is
x (part.) = -18/65 cos(3t ) - 1/65 sin(3t )
and thus the general solution is
x (gen.) = x (char.) + x (part.)
x = C₁ exp(-4t ) + C₂ exp(-2t ) - 18/65 cos(3t ) - 1/65 sin(3t )
please help will mark brainly!!!!! need done. PERSONAL FINANCE
Answer:
Step-by-step explanation:
Blank DVD's are sold in packages of 50 for $17.95 if your company will need 2700 blank divide these next year how much money must your budget for blank dvd's
Answer:
Step-by-step explanation:
50 X 240 = 2700. So you will need 240 packs of 50. They cost 17.95 each, so the multiply. 240 X 17.95 is 4,308. So, 4,308 is your answer.
The mean per capita consumption of milk per year is 131 liters with a variance of 841. If a sample of 132 people is randomly selected, what is the probability that the sample mean would be less than 133.5 liters
Answer:
0.8389 = 83.89% probability that the sample mean would be less than 133.5 liters.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The mean per capita consumption of milk per year is 131 liters with a variance of 841.
This means that [tex]\mu = 131, \sigma = \sqrt{841} = 29[/tex]
Sample of 132 people
This means that [tex]n = 132, s = \frac{29}{\sqrt{132}}[/tex]
What is the probability that the sample mean would be less than 133.5 liters?
This is the p-value of Z when X = 133.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{133.5 - 131}{\frac{29}{\sqrt{132}}}[/tex]
[tex]Z = 0.99[/tex]
[tex]Z = 0.99[/tex] has a p-value of 0.8389
0.8389 = 83.89% probability that the sample mean would be less than 133.5 liters.
There is a bag with only red marbles and blue marbles.
The probability of randomly choosing a red marble is
7/10th
There are 42 red marbles in the bag and each is equally likely to be chosen.
Work out how many marbles in total there must be.
Answer:
60 marbles in total
Step-by-step explanation:
Find how many marbles there are in total by dividing 42 by 0.7:
42/0.7
= 60
So, there are 60 marbles in total
A trough is 10 ft long and its ends have the shape of isosceles triangles that are 4 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 15 ft3/min, how fast is the water level rising when the water is 8 inches deep
Answer:
7.5 ft³/min
Step-by-step explanation:
Let x be the depth below the surface of the water. The height, h of the water is thus h = 10 - x.
Now, the volume of water V = Ah where A = area of isosceles base of trough = 1/2bh' where b = base of triangle = 4 ft and h' = height of triangle = 1 ft. So, A = 1/2 × 4 ft × 1 ft = 2 ft²
So, V = Ah = 2h = 2(10 - x)
The rate of change of volume is thus
dV/dt = d[2(10 - x)]/dt = -2dx/dt
Since dV/dt = 15 ft³/min,
dx/dt = -(dV/dt)/2 = -15 ft³/min ÷ 2 = -7.5 ft³/min
Since the height of the water is h = 10 - x, the rate at which the water level is rising is dh/dt = d[10 - x]/dt
= -dx/dt
= -(-7.5 ft³/min)
= 7.5 ft³/min
And the height at this point when x = 8 inches = 8 in × 1 ft/12 in = 0.67 ft is h = (10 - 0.67) ft = 9.33 ft
Can someone help me please..
Answer:
Quadratic formula
Step-by-step explanation:
The function is quadratic because it is a parabola. Exponential functions shoot either upwards or downwards rapidly, and it is clearly not linear due to it's curve. It also isn't piecewise because the function never stops or starts irregularly.
The length of a rectangle is six times it’s width. If the area of the rectangle is 486 cm^2, find the perimeter.
Answer:
54 cm is the perimeter I think
The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 minutes. (a) Find the probability that a customer has to wait more than 4 minutes. (Round your answer to three decimal places.)
Answer:
0.758
explaination
using poisson distribution
0.08208+0.2052+0.2565+0.2138
0 .758
The dress store is having a sale where all merchandise is 1/4 off. A woman buys $48 of merchandise at a sale price.
Answer:$36 depending on what question is i just assuming how much she has to pay
Step-by-step explanation:
48 divded by 4 is 12. $48-$12 is $36. The $12 is the 1/4 discount.
9. Mariah has 28 centimeters of reed
and 10 meters of reed for weaving
baskets. How many meters of reed
does she have? Write your answer as a
decimal and explain your answer.
One way to check on how representative a survey is of the population from which it was drawn is to compare various characteristics of the sample with the population characteristics. A typical variable used for this purpose is age. The 2010 GSS of the American adult population found a mean age 49.28 years and a standard deviation of 17.21 for its sample of 4,857 adults. Assume that we know from Census data that the mean age of all American adults is 37.2 years.
Required:
a. State the research and the null hypothesis setting for a two-tailed test.
b. Calculate the t statistics and test the null hypothesis setting alpha at .01. What did you find?
c. What is your decision about the null hypothesis? What does this tell us about how representative the sample is of the American adult population?
Answer:
a) See step by Step explanation
b) z(s) = 48.88
c) We reject H₀. The sample is not representative of American Adult Population
Step-by-step explanation:
From sample
sample mean . x = 49.28
sample standard deviationn s = 17.21
sample size n₁ = 4857
Population mean according to Census data
μ = 37.2
a) Test Hypothesis
Null Hypothesis . H₀ . x = μ = 37.2
Alternative Hypothesis Hₐ . x ≠ μ
b) We have sample size (4857) we can use normal distribution
z (c) for α = 0.01 α/2 . = 0.005 is from z-table . z(c) = 2.575
To calculate z(s) = ( x - μ ) / s /√n
z(s) = 12.08 * √4857 / 17.21
z(s) = 12.08* 69.64 / 17.21
z(s) = 48.88
z(s) > z(c)
We should reject H₀. The sample is not representative of American Adult population
the admission fee for a charity event is $7 for children and 10$ for adults. The event was attended by 700 people, and the total amount collected in admissions was $6,400.
Answer:
200 kids and 500 adults
Step-by-step explanation:
x+y=700
7x+10y=6,400
(200,500)
kids=200
adults=500
what is the distance in the image below?
The distance is:
5 + 3 = 8 units
Since the segment is completely horizontal we need not to use formula for computing the length of a segment in 2D euclidean space.
Instead we can simplify the problem to a single dimension, only considering the x-coordinates of the endpoints of the segment.
The x-coordinates are -5 and 3.
Subtracting and applying absolute value yields the answer,
[tex]\mathrm{abs}(-5-3)=\boxed{8}[/tex].
Hope this helps.
is y=3x^2-x-1 a function
Answer: Yes it is a function.
This is because any x input leads to exactly one y output.
The graph passes the vertical line test. It is impossible to draw a single vertical line through more than one point on the parabolic curve.
x^2-9x+20 the factor of this trinomial are(____)(___)
Answer:
(x-4) (x-5)
Step-by-step explanation:
* means multiply
first you figure out what 2 numbers
when added make 9
when multiplied make 20
those are 4 and 5
(x 4) (x 5)
in this case
-4 -5 make -9
-4 * -5 make 20
(x-4) (x-5)
Answer:
(x-4)(x-5)
Step-by-step explanation:
Firstly you need to use the second equation formula to get the value of x.
x= [tex]\frac{-(-9)+- \sqrt{(-9)^{2}-4*1*20 } }{2*1}[/tex]
x= [tex]\frac{9+- \sqrt{81-80 } }{2}[/tex]
x= [tex]\frac{9+-1}{2}[/tex]
so,
x=[tex]\frac{9+1}{2}[/tex] x=5
and
x=[tex]\frac{9-1}{2}[/tex] x=4
When writing the factor, we have to change signs of 5 and 4. So it will be -5 and -4.
That's why the awnser is (x-4)(x-5)
Hope it helps!
Let f(x)
2x + 8, g(x) = x2 + 2x – 8, and h(x) = 3x – 6.
Perform the indicated operation. (Simplify as far as possible.)
(h · f)(3) =
Answer:
36
Step-by-step explanation:
(h · f)(x) = h(f(x))
h(f(x)) = h(2x+8)
h(f(x))= 3(2x+8) - 6
h(f(x)) = 6x + 24 - 6
h(f(x))= 6x + 18
If x = 3
h(f(x))= 6(3) + 18
h(f(x))= 18 + 18
h(f(x))= 36
Hence (h · f)(3) = 36
Which function has the following characteristics?
- A vertical asymptote at x=3
- A horizontal asymptote at y=2
- Domain: {x ≠ ±3}
A. y= (2x-8) / (x-3)
B. y= (2x^2 - 8) / (x^2 - 9)
C. y= (x^2 - 9) / (x^2 - 4)
D. y= (2x^2 - 18) / (x^2 - 4)
The function has the characteristics is (b) y= (2x^2 - 8) / (x^2 - 9)
How to determine the function?The features are given as:
A vertical asymptote at x=3A horizontal asymptote at y=2Domain: {x ≠ ±3}The function that has the above features is (b).
This is proved as follows:
y= (2x^2 - 8) / (x^2 - 9)
Set the denominator not equal to 0, to determine the domain
x^2 - 9 ≠ 0
Add 9 to both sides
x^2 ≠ 9
Take the square roots
x ≠ ±3 --- domain
Replace ≠ with =
x = ±3 --- vertical asymptote
Set the numerator to 0
2x^2 - 8 = 0
Divide through by 2
x^2 - 4 = 0
This gives
x^2 = 4
Take the square roots
x = 2 ---- horizontal asymptote
Hence, the function has the characteristics is (b) y= (2x^2 - 8) / (x^2 - 9)
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Area of composite shapes ?
Answer: 58
Step-by-step explanation: you add them all together
Its 108 the other answer is the perimeter not the area.
What is the correct definition for sec theta?
Answer:
D Is the correct answer Thats was too easy
Answer:
sec(θ) = hypotenuse / adjacent.
Step-by-step explanation:
sec theta= cos -1 theta
whitch numbre produces a rational number when multiplied by 1/3 ?
Answer:
Step-by-step explanation:
multiplication of two rational numbers produce a rational number.
12x + 1 - 2(y + 2) = 12x - ______ - 2y
Answer:
-3
Step-by-step explanation:
12x + 1 - 2(y + 2)
=> 12x + 1 - 2y - 4
=> 12x - 3 - 2y
Answer:
-3
Step-by-step explanation:
12x+1-2y-4
12x+1-2y-4
12x-2y-3
Two positive integers are 3 units apart on a number line. Their product is 108.
Which equation can be used to solve for m, the greater integer?
m(m – 3) = 108
m(m + 3) = 108
(m + 3)(m – 3) = 108
(m – 12)(m – 9) = 108
Answer:
m(m – 3) = 108
The correct equation can be used to solve for m, the greater integer is,
⇒ m (m - 3) = 108
We have to given that,
Two positive integers are 3 units apart on a number line.
And, Their product is 108.
Let us assume that,
In a number line, first point is m
Then, Second point is, m - 3
So, We get;
The correct equation can be used to solve for m, the greater integer is,
⇒ m (m - 3) = 108
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How and what is the value of X?
Answer:
9 =x
Step-by-step explanation:
The angles are vertical angles and vertical angles are equal
56 = 6x+2
Subtract 2 from each side
56-2 = 6x+2-2
54 = 6x
Divide each side by 6
54/6 = 6x/6
9 =x
2. What facts are needed to solve the problem?
Answer:
firstly we have to identify the problems, understand carefully and chose the best way to solve problems.
Write the following using algebraic notation, using the letter x for any
unknown numbers:
I think of a number, double it, then add fifteen.
You do X2 + 15 and that will be your answer.
By the way, the 2 is a power and is meant to be smaller on top of the X.
13. Given that
[tex] {x}^{2} + {y}^{2} + 10y + 16 = 0[/tex]
and
[tex] {(x - 3)}^{2} + {y}^{2} = 1[/tex] are two circles on the same plane. Find:
a) the coordinates of the center and the radius for each circle.
b) the equation of the straight line joining the center of both circles.
step by step explanation:
[tex]\mathfrak{x}^{2}+{y}^{2}+16=0[/tex]
=[x2+16=0x26]
=[2x{y}^2{16}~0]
=[4×{y}^0{16}]
=[32x{y}^x]
3x+4 number of terms
9514 1404 393
Answer:
2
Step-by-step explanation:
In this expression, the terms are the parts of the sum. They are 3x and 4. There are 2 terms.
i need help with these questions. anyone down to help me ?please
9514 1404 393
Answer:
A: less than 2 hoursB: 2 to 5 hoursC: more than 5 hoursStep-by-step explanation:
The attached graph shows the various company costs for x number of hours. The graph nearest the x-axis represents the lowest cost.
We can see that cost is lowest using Company A for 2 hours or less, and Company C for 5 hours or more. For times between those, Company B has the lowest charges.
Of course, the equation for charges in each case is the sum of the service fee and the product of hourly charge and number of hours (x).
__
I find the graphing calculator to be the most efficient tool for solving these. The alternative is to compare the equations pairwise to see which gives lower rates. With a little practice, you learn that the "break even hours" will be the difference in service fees divided by the difference in hourly cost.
For example A will cost the same as B when the $20 service fee and the $10/hour cost difference are the same: for 2 hours. A and C will cost the same when the $45 service fee and the $15/hour cost difference are the same, after 3 hours. B and C will cost the same when the $25 difference in service fees and the $5/hour cost difference are the same, after 5 hours.
So B is cheaper above 2 hours, and C is cheaper than that above 5 hours. With no service fee, A is cheaper for small numbers of hours (<2).
a bag contain 3 black balls and 2 white balls.
1. A ball is taken from the black and then replaced, a second is taken. what is the probabilities that.
(a) there are both black,
(b)one is black one is white,
(c) at lease one is black,
(d) at most one is one is black.
2. find out if all the balls are chosen without replacement.
please kindly solve with explanation. thank you.
Answer:
Step-by-step explanation:
Total number of balls = 3 + 2 = 5
1)
a)
[tex]Probability \ of \ taking \ 2 \ black \ ball \ with \ replacement\\\\ = \frac{3C_1}{5C_1} \times \frac{3C_1}{5C_1} =\frac{3}{5} \times \frac{3}{5} = \frac{9}{25}\\\\[/tex]
b)
[tex]Probability \ of \ one \ black \ and \ one\ white \ with \ replacement \\\\= \frac{3C_1}{5C_1} \times \frac{2C_1}{5C_1} = \frac{3}{5} \times \frac{2}{5} = \frac{6}{25}[/tex]
c)
Probability of at least one black( means BB or BW or WB)
[tex]=\frac{3}{5} \times \frac{3}{5} + \frac{3}{5} \times \frac{2}{5} + \frac{2}{5} \times \frac{3}{5} \\\\= \frac{9}{25} + \frac{6}{25} + \frac{6}{25}\\\\= \frac{21}{25}[/tex]
d)
Probability of at most one black ( means WW or WB or BW)
[tex]=\frac{2}{5} \times \frac{2}{5} + \frac{3}{5} \times \frac{2}{5} \times \frac{2}{5} + \frac{3}{5}\\\\= \frac{4}{25} + \frac{6}{25} + \frac{6}{25}\\\\=\frac{16}{25}[/tex]
2)
a) Probability both black without replacement
[tex]=\frac{3}{5} \times \frac{2}{4}\\\\=\frac{6}{20}\\\\=\frac{3}{10}[/tex]
b) Probability of one black and one white
[tex]=\frac{3}{5} \times \frac{2}{4}\\\\=\frac{6}{20}\\\\=\frac{3}{10}[/tex]
c) Probability of at least one black ( BB or BW or WB)
[tex]=\frac{3}{5} \times \frac{2}{4} + \frac{3}{5} \times \frac{2}{4} + \frac{2}{5} \times \frac{3}{4}\\\\=\frac{6}{20} + \frac{6}{20} + \frac{6}{20} \\\\=\frac{18}{20} \\\\=\frac{9}{10}[/tex]
d) Probability of at most one black ( BW or WW or WB)
[tex]=\frac{3}{5} \times \frac{2}{4} + \frac{2}{5} \times \frac{1}{4} + \frac{2}{5} \times \frac{3}{4}\\\\=\frac{6}{20} + \frac{2}{20} + \frac{6}{20} \\\\=\frac{14}{20}\\\\=\frac{7}{10}[/tex]
A forestry researcher wants to estimate the average height of trees in a forest near Atlanta, Georgia. She takes a random sample of 18 trees from this forest. The researcher found that the average height was 4.8 meters with a standard deviation of 0.55 meters. Assume that the distribution of the heights of these trees is normal. For this sample what is the margin of error for her 99% confidence interval
Answer:
The margin of error for her confdence interval is of 0.3757.
Step-by-step explanation:
We have the standard deviation for the sample, which means that the t-distribution is used to solve this question.
The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So
df = 18 - 1 = 17
99% confidence interval
Now, we have to find a value of T, which is found looking at the t table, with 17 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.99}{2} = 0.995[/tex]. So we have T = 2.8982
The margin of error is:
[tex]M = T\frac{s}{\sqrt{n}}[/tex]
In which s is the standard deviation of the sample and n is the size of the sample.
Standard deviation of 0.55 meters.
This means that [tex]s = 0.55[/tex]
What is the margin of error for her 99% confidence interval?
[tex]M = T\frac{s}{\sqrt{n}}[/tex]
[tex]M = 2.8982\frac{0.55}{\sqrt{18}}[/tex]
[tex]M = 0.3757[/tex]
The margin of error for her confdence interval is of 0.3757.
Margin of error is the distance between the mean and the limit of confidence intervals. The margin of error for the given condition is 3.28 approximately.
What is the margin of error for small samples?Suppose that we have:
Sample size n < 30
Sample standard deviation = sPopulation standard deviation = [tex]\sigma[/tex]Level of significance = [tex]\alpha[/tex]Degree of freedom = n-1Then the margin of error(MOE) is obtained as
Case 1: Population standard deviation is knownMargin of Error = [tex]MOE = T_{c}\dfrac{\sigma}{\sqrt{n}}[/tex]
Case 2: Population standard deviation is unknown[tex]MOE = T_{c}\dfrac{s}{\sqrt{n}}[/tex]
where [tex]T_{c}[/tex] is critical value of the test statistic at level of significance
For the given case, taking the random variable X to be tracking the height of trees in the sample taken of trees from the considered forest.
Then, by the given data, we get:
[tex]\overline{x} = 4.8[/tex], [tex]s = 4.8[/tex], n = 18
The degree of freedom is n-1 = 17
Level of significance = 100% - 99% = 1% = 0.01
The critical value of t at level of significance 0.01 with degree of freedom 17 is obtained as T = 2.90 (from the t critical values table)
Thus, margin of error for 99% confidence interval for considered case is:
[tex]MOE = T_{c}\dfrac{s}{\sqrt{n}}\\\\MOE = 2.9 \times \dfrac{4.8}{\sqrt{18}} \approx 3.28[/tex]
Thus, the margin of error for the given condition is 3.28 approximately.
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