The general solutions for the given differential equations are:
1. y = (1 / sin x) + C
2. |y² - 2t²| = Ce^(2t)
3. y = Ce^t
To find the general solution for each of the given differential equations, we can apply different methods. For the equation y' + y² sin x = 0, we can separate variables and integrate to find the solution. For the equation t²y' + 2ty = y³, we can use a substitution to transform it into a separable differential equation. Finally, for the equation y' - y = 2te²t, we can use the integrating factor method to solve the equation.
1. For the equation y' + y² sin x = 0, we can separate variables and integrate both sides. Rearranging the equation, we have: dy / (y² sin x) = -dx. Integrating both sides gives: -1 / sin x = -x + C, where C is the constant of integration. Solving for y, we get: y = (1 / sin x) + C.
2. For the equation t²y' + 2ty = y³, we can use the substitution u = y² to transform it into a separable differential equation. Taking the derivative of u with respect to t gives: du/dt = 2yy'. Substituting the expression for y' and simplifying, we get: (1 / 2) du / (u - 2t²) = dt. Integrating both sides gives: (1 / 2) ln|u - 2t²| = t + C, where C is the constant of integration. Substituting back u = y², we have: (1 / 2) ln|y² - 2t²| = t + C. Taking the exponential of both sides and simplifying, we obtain: |y² - 2t²| = Ce^(2t), where C is the constant of integration.
3. For the equation y' - y = 2te²t, we can use the integrating factor method. The integrating factor is given by e^(-∫ dt) = e^(-t) since the coefficient of y' is -1. Multiplying both sides of the equation by the integrating factor, we have: e^(-t)y' - e^(-t)y = 2te^(t - t). Simplifying, we get: d / dt (e^(-t)y) = 0. Integrating both sides gives: e^(-t)y = C, where C is the constant of integration. Solving for y, we obtain: y = Ce^t, where C is the constant of integration.
In conclusion, the general solutions for the given differential equations are:
1. y = (1 / sin x) + C
2. |y² - 2t²| = Ce^(2t)
3. y = Ce^t
where C represents the constant of integration in each case.
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Help me with this Math question.
Answer:
D
Step-by-step explanation:
Just graph the equation and where they intercept is the solution.
Just use desmos graphing calculator if you don't know how to graph.
Solve the equation on the interval [0,2m). √2 cos x + 1=0
the solution to the equation on the interval [0, 2π) is x = 3π/4.
To solve the equation, we want to find the values of x that satisfy the equation within the given interval.
First, we isolate the cosine term by subtracting 1 from both sides:
√2 cos(x) = -1
Next, we divide both sides by √2:
cos(x) = -1/√2
To find the solutions, we need to determine the angles whose cosine value is equal to -1/√2. We can use the unit circle or reference angles to determine these angles.
In the interval [0, 2π), the angle that satisfies cos(x) = -1/√2 is x = 3π/4.
Therefore, the solution to the equation on the interval [0, 2π) is x = 3π/4.
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What is the solution to the equation below?
4w=2/3
A
6/3
B
8/3
C 2/12
D 4 2/3
THESE ARE FRACTIONS
Answer:
I think it is 2/12
Leo has a rectangular garden with a perimeter of 40 feet. The width of the garden is 8 feet. What is the length of the garden?
Answer:
5 feet
Step-by-step explanation:
find the equation for the angle then find what x equals
Answer:
x = 18
Step-by-step explanation:
∠ GFH and ∠ EFH form a right angle GFE , then
2x + 3x = 90 ← equation for the angle
5x = 90 ( divide both sides by 5 )
x = 18
A researcher wants to test the claim that the average lifespan for florescent lights is 1600 hours. A random sample of 100 fluorescent lights has a mean lifespan of 1580 hours, and a standard deviation of 100 hours. Is there evidence to support the claim at 5% level of significance?
Yes, there is evidence at a 5% level of significance to support the claim that the average lifespan for fluorescent lights is not 1600 hours.
How do we calculate?Null hypothesis: is defined as the average lifespan for fluorescent lights is 1600 hours.
Alternative hypothesis : is defined as average lifespan for fluorescent lights is not 1600 hours.
Sample size (n) = 100
Sample mean = 1580 hours
Sample standard deviation (s) = 100 hour
t = (Sample mean - μ) / (s / √n)
μ = hypothesized population mean
s= sample standard deviation
t = (1580 - 1600) / (100 / √100)
t = -20 / (100 / 10)
t = -20 / 10
t = -2
The significance level of 0.05 will be divided by 2 to get an alpha level of 0.025.
We make use of a t-distribution table to look up the critical t-value in the with degrees of freedom equal to n-1 = 99.
The critical t-value is 1.984 for a significance level of 0.025 and degrees of freedom = 99.
In conclusion, we have evidence to reject the null hypothesis because the absolute value of the test statistic is greater than the critical t-value.
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Given the velocity v = ds/dt and the initial position of a body moving along a coordinate line, find the body's position at time t. v = 9.8t + 15, s(0) = 20 s(t) =
The body's position at time t is given by the equation s(t) = [tex]4.9t^2 + 15t + 20[/tex].
To find the body's position at time t,
we need to integrate the velocity function with respect to time and apply the initial condition.
Given:
v = 9.8t + 15
s(0) = 20
First, integrate the velocity function with respect to time to obtain the position function:
∫v dt = ∫(9.8t + 15) dt
s(t) = [tex]4.9t^2 + 15t + C[/tex]
Next, we apply the initial condition s(0) = 20 to determine the value of the constant C:
s(0) =[tex]4.9(0)^2 + 15(0) + C[/tex]
20 = C
Now, we have the complete position function:
s(t) =[tex]4.9t^2 + 15t + 20[/tex]
In conclusion, To find the position of the body at time t,
we integrated the velocity function with respect to time,
applied the initial condition to determine the constant,
and obtained the position function s(t) = [tex]4.9t^2 + 15t + 20[/tex].
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Use the SIN to find x?
190m
Xm
1. Set Up
2. Operation
3. Answer
52.2
Answer:
sine=opposite/hypotenuse
sin52.2=x/150
sin0.790=x/150
x=118.5
Which form of a linear equation is defined by y = mx +b?
O A. Parallel form
O B. Standard form
O C. Slope-intercept form
O D. Point-slope form
The linear equation is defined by y = mx +b is called ''Slope-intercept form''.
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
The linear equation is defined by,
⇒ y = mx + b
Now, We know that;
The equation defined as;
⇒ y = mx + b
Where, 'm' is slope and 'b' is y - intercept.
Hence, The linear equation is defined by y = mx +b is called ''Slope-intercept form''.
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1. what linear function, y=f(x) has f(0) = 8 and f(7) = 14 ?
A linear function is a mathematical function that can be represented by a straight line when graphed on a Cartesian coordinate system. It is also known as a first-degree polynomial because the highest exponent of the variable is 1. The general form of a linear function is: f(x) = mx + b
To find a linear function given two points on the line, you need to use the point-slope formula.
The formula is `y - y1 = m(x - x1)`, where `(x1, y1)` is a point on the line and `m` is the slope of the line.
Given that f(0) = 8 and f(7) = 14, we can find the slope of the line:$$\frac{f(7) - f(0)}{7 - 0} = \frac{14 - 8}{7} = \frac{6}{7}$$Now, we can use the point-slope formula with the point `(0, 8)` and the slope `6/7`:$$y - 8 = \frac{6}{7}(x - 0)$$
Simplifying this equation, we get:$$y - 8 = \frac{6}{7}x$$$$y = \frac{6}{7}x + 8$$
Therefore, the linear function with `f(0) = 8` and `f(7) = 14` is `y = (6/7)x + 8`.
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A linear function, y = f(x) that has f(0) = 8 and f(7) = 14 is given by: f(x) = mx + b, Where m represents the slope of the line and b represents the y-intercept. The equation of the linear function is: y = f(x) = (6/7)x + 8.
To find the slope of the line, we use the formula:
m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) = (0, 8) and (x₂, y₂) = (7, 14).
Substituting into the formula:
m = (14 - 8) / (7 - 0)
m = 6 / 7
Therefore, the equation of the line is given by: f(x) = (6/7)x + b.
To find the value of b, we use the fact that f(0) = 8.
Substituting into the equation, we get: 8 = (6/7)(0) + b.
Simplifying, we get: b = 8.
Therefore, the equation of the linear function is: y = f(x) = (6/7)x + 8.
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what is 13/50 as a decimal and percent
decimal = 0.26
percent = 26%
To find the x-intercept, we let y = 0 and solve for x and to find y-intercept, we let x=0 and solve for y. Figure out the x-intercept and y-intercept in given equation of the line.
6x + 2y = 12
not bots with links or so help me i will
Answer:
X-intercept: (2, 0) Y-intercept: (0, 6)Step-by-step explanation:
X-intercept: y=0:
6x +2×0 = 12
6x = 12
x = 2
Y-intercept: x=0:
6×0 + 2y = 12
2y = 12
y = 6
3. if u get this right i’ll give u brainliest
Answer:
(-3,2)
Step-by-step explanation:
3/4 of eight is six so go to two and then find the x, as x wasn't on the numbers it is not a multiple of two, so three was the closest number
may be wrong
) What is the GCF of 36 and 60?
4
6
12
18
Answer:
12
Step-by-step explanation:
List out the factors of each
36: 1,2,3,4,6,9,12,18,36
60: 1,2,3,4,5,6,10,12,15,20,30,60
the highest number they both have in common is 12
Please Help Quick I am being timed
A. 2
B. 3
C. 4
D. 5
Answer:
c.Step-by-step explanation:
As I figured out, any number that is not 0, meaning any number with a value is a Significant Digit. For example, in 3003 there's only 2 significant digits. The easiest way to figure these things out is adding the digits.
I hope this helps :D
i’d appreciate if someone would help me here:-)
Answer:-4
Step-by-step explanation:
please help me i need this. will get 100 points
The probability that a plant produces between 15 and 19 strawberries is given as follows:
47.5%.
How to calculate a probability?The two parameters that are needed to calculate a probability are listed as follows:
Number of desired outcomes in the context of a problem or experiment.Number of total outcomes in the context of a problem or experiment.Then the probability is then calculated as the division of the number of desired outcomes by the number of total outcomes.
The desired areas for this problem are given as follows:
Between 15 and 17: 34%.Between 17 and 19%: 13.5%.Hence the probability is given as follows:
34 + 13.5 = 47.5%.
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In ΔFGH, f = 930 inches, g = 520 inches and ∠H=169°. Find ∠G, to the nearest degree.
Answer:
Thats so hard oh my gosh!
Step-by-step explanation:
Answer:4 degrees
Step-by-step explanation:
Flip a coin 100 times. Find the expected number of heads, with its uncertainty (the typical fluctuation).
Roll a die 100 times. Find the expected number of times getting a '6', with its uncertainty.
Roll a die 100 times. Find the expected number times not getting a '6', with its uncertainty.
For each of the three cases:
Express the result (best value ± uncertainty) with uncertainty rounded to one significant digit.
Base your calculations, first, on the Binomial distribution.
Repeat the calculation but now based on the Poisson distribution.
Discuss the appropriateness of the Poisson distribution for these three cases. That is...
Does it seem to give a good approximation? Why should this be so?
Even if a good approximation, the Poisson cannot be quite right. Why not?
Hints: The Poisson approximates the binomial for n large and p small but allows infinite successes.
The typical fluctuation is based on the square root of the variance, which is equal to 5.
The typical fluctuation is based on the square root of the variance, which is equal to 3.7.
The typical fluctuation is based on the square root of the variance, which is equal to 3.7.
The Poisson distribution does not account for the finite probability of zero events occurring, and it can not handle problems where the expected number of events is very large, as it assumes an infinite number of successes, which is not the case in real life.
1) Binomial distribution
Binomial distribution is the probability distribution of obtaining exactly r successes in n independent trials with two possible outcomes of a given event.
The best value for the expected number of heads when a coin is flipped 100 times is 50.0, with the uncertainty being 5.0.
The typical fluctuation is based on the square root of the variance, which is equal to 100 x 0.5 x (1-0.5) = 25, which gives the fluctuation to be 5 (the square root of 25).
The Poisson distribution is an excellent approximation for binomial distributions, especially when n is large and p is small.
2) Binomial distribution
The best value for the expected number of times getting a '6' when a die is rolled 100 times is 16.7, with an uncertainty of 4.1.
The typical fluctuation is based on the square root of the variance, which is equal to 100 x (1/6) x (5/6) = 13.9, which gives the fluctuation to be 3.7 (the square root of 13.9)..
3) Binomial distribution
The expected number of times not getting a '6' when a die is rolled 100 times is 83.3, with an uncertainty of 4.1.
The typical fluctuation is based on the square root of the variance, which is equal to 100 x (5/6) x (1/6) = 13.9, which gives the fluctuation to be 3.7 (the square root of 13.9).
Poisson distribution
The Poisson distribution is a good approximation for the binomial distribution because n is large and p is small. Furthermore, the Poisson distribution can be used to predict the probability of a specific number of events occurring in a specific amount of time, which makes it ideal for modeling the number of radioactive decays or the number of phone calls to a call center.
However, the Poisson distribution does not account for the finite probability of zero events occurring, and it can not handle problems where the expected number of events is very large, as it assumes an infinite number of successes, which is not the case in real life.
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Identify the volume of a cone with diameter 18 cm and height 15 cm.
a. V = 3817 cm^(3)
b. V = 1272.3 cm^(3)
c. V = 1908.5 cm^(3)
d. V = 1424.1 cm^(3)
The volume of a cone with diameter 18 cm and height 15 cm is b. V = 1272.3 cm^(3).
To calculate the volume of a cone, we use the formula:
V = (1/3) * π * r^2 * h
where V is the volume, π is the mathematical constant approximately equal to 3.14159, r is the radius of the cone's base, and h is the height of the cone.
Given that the diameter of the cone is 18 cm, we can calculate the radius by dividing the diameter by 2:
r = 18 cm / 2 = 9 cm
Substituting the values into the volume formula:
V = (1/3) * π * 9^2 * 15
Calculating:
V ≈ 1272.3 cm^3
Therefore, the volume of the cone is approximately 1272.3 cm^3.
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A manufacturer claims that the lifetime of a certain type of battery has a population mean of μ = 40 hours with a standard deviation of a = 5 hours. Assume the manufactures claim is true and let a represent the mean lifetime of the batteries in a simple random sample of size n = 100. Find the mean of the sampling distribution of , μ = Find the standard deviation of the sampling distribution of , I What is P(40.6)? Round to the nearest thousandths (3 decimal places) The area this probability represents is (choose: right/left/two) tailed. Suppose another random sample of 100 batteries gives = 39.1 hours. Is this unusually short? (yes/no) Because P(≤39.1) = Round to the nearest thousandths (3 decimal places) The area this probability represents is
It should be noted that the probability of obtaining a sample mean of 39.1 hours or less is quite low (0.035), it can be considered unusually short.
How to calculate the probabilityThe mean of the sampling distribution of the sample mean, μ, is equal to the population mean, which is μ = 40 hours.
The standard deviation of the sampling distribution is σ(μ) = 5 / √100
= 5 / 10
= 0.5 hours.
Plugging in the values, we get (40.6 - 40) / 0.5
= 0.6 / 0.5
= 1.2.
Looking up the z-score of 1.2 in the standard normal distribution table (or using a calculator), we find that the probability is approximately 0.884.
Now, let's calculate P(≤39.1). Similarly, we calculate the z-score as (x - μ) / σ(μ), where x = 39.1 hours. Plugging in the values, we get (39.1 - 40) / 0.5
= -0.9 / 0.5
= -1.8.
Using the z-score table or a calculator, we find that the probability is approximately 0.035.
This probability represents the area under the curve to the left of -1.8, which is a left-tailed probability.
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Can someone solve for the question mark?
6x + ? = 10
Write -8 7/8
as a decimal number,
Answer:
-8.875
Step-by-step explanation:
8 7/8 > -71/8 > -8.875
Find the value of x in the picture below(round to nearest test)
Answer:
x = 13
Step-by-step explanation:
Using Pythagoras' identity in the right triangle.
The square on the hypotenuse is equal to the sum of the squares on the other 2 sides, that is
x² = 5² + 12² = 25 + 144 = 169 ( take the square root of both sides )
x = [tex]\sqrt{169}[/tex] = 13
Asha owns a car-wash and is trying to decide whether or not to purchase a vending machine so customers
can buy coffee while they wait. She'll get the machine if she's convinced that more than 30% of her
customers would buy coffee. She plans on taking a random sample of n customers and asking them
whether or not they would buy coffee from the machine, and she'll then do a significance test using
a = 0.05 to see if the sample proportion who say "yes" is significantly greater than 30%.
Suppose that in reality, it is actually 33% of her customers that would buy coffee.
Which of the changes below would result in the highest power for her test?
Answer: D
Step-by-step explanation:
She uses a sample size of n=200, and 50% of all customers would actually buy coffee.
The change that would result in the highest power for her test is D. increase the significance level to α = 0.01 and use a sample size of n = 300.
What is significance level?The significance level simply means the probability of rejecting the null hypothesis when it is true.
It should be noted that the power of the test depends on 4 factors which are the sample size, standard deviation, effect size, and significance level.
In the given scenario, the larger sample size will increase the power of the test and a higher significance level will result in a higher power.
Therefore, the test is done with the largest sample size and highest significance level of 0.10
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Solve the system of equations below.
One penny is 1% of a dollar
$ 0.25 is 25% of a dollars . $1.25 is 12.5% dollars
Answer:
$1.25 is NOT 12.5% dollars. It is 125% of a dollar.
Step-by-step explanation:
We have 1 and 1/4 dollars. 1 and 1/4 as a percentage is 125%
Solve for x
5=2x-3
Enter your answer in the box
x = __
Step-by-step explanation:
✧ [tex] \underline{ \underline{ \large{ \tt{G \: I \: V \: E \: N \: \:E \: Q \: U \: A \: T \: I \: O \: N}}}} : [/tex]
5 = 2x - 3❀ [tex] \underline{ \underline{ \large{ \tt{ \: T \: O \: \: F\: I\: N \: D}}} }: [/tex]
Value of x☄ [tex] \underline{ \underline{ \large{ \tt{S \: O \: L\: U \: T \: I\: O \: N}}}} : [/tex]
♨ [tex] \large{ \sf{5 = 2x - 3}}[/tex]
~Swap the sides of the equation :
⇾ [tex] \large{ \sf{2x - 3 = 5}}[/tex]
~We want to remove the 3 first. Since the original equation is -3 , we are going to use the opposite operation and add 3 to the both sides :
⇾ [tex] \large{ \sf{2x - 3 + 3 = 5 + 3}}[/tex]
⇾ [tex] \large{ \sf{2x \: \cancel{ - 3} \: \cancel{ + 3}}} = 8[/tex]
⇾ [tex] \large{ \sf{2x = 8}}[/tex]
~Now , We need to think about how to remove the coefficient 2. Since the opposite of multiplication is division , we are going to divide both sides of the equation by 2 :
⇾ [tex] \large{ \sf{ \frac{2x}{2} = \frac{8}{2}}} [/tex]
⇾ [tex] \boxed{ \large{ \sf{x = 4}}}[/tex]
☯ [tex] \underline{ \underline{ \large{ \tt{C \: H \:E \: C \: K}}}}: [/tex]
☪ [tex] \large{ \tt{L \: H \: S : \: 5 }}[/tex]
[tex] \large{ \tt{R \: H \: S \ \: : \: 2x - 3 = 2 \times 4 - 3 = 8 - 3 = \underline{ \tt{5}}}}[/tex] [ Plug the value of x ]
☥ Since this is a true statement , our answer ( x = 4 ) is correct. Yay! We got our answer :)
♕ [tex] \large{ \boxed{ \underline{ \large{ \tt{Our \: Final \: Answer : \boxed{ \underline{ \bold{ \text{x = 5}}}}}}}}}[/tex]
[tex] \underline{ \underline{ \text{H \: O\: P \: E \: \: I \: \: H \: E \: L \: P \: E \: D}}}[/tex] !! ♡
[tex] \underline{ \underline{ \text{H \: A \: V \: E \: A \: W \: O \: N \: D \: E \: R \: F \: U \: L \: D \: A\: Y \: / \: N\: I \: G\: H \: T}}}[/tex] !! ツ
☃ [tex] \underline{ \underline{ \tt{C \:A \: R \: R \: Y \: \: O \:N \: \: L \: E \: A \: R\: N \: I \: N \: G}}} [/tex] !!✎
▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁
For the following systems, draw a direction field and plot some representative trajectories. Using your graph, give the type and stability of the origin as a critical point. You may need to look at the eigenvalues to be sure. 3 5 3 -2 4 2 2 -2 1 x, b. X'= X, X 5 3 1 -5 4 1 4 2 2 a. X' c. X'= -63) — —
Plot direction fields and trajectories. Analyze eigenvalues to determine stability and type of critical point.
For system (a):
The direction field and trajectories should be plotted based on the given matrix:
[3 5] [x]
[3 -2] * [y]
To determine the type and stability of the origin as a critical point, we can analyze the eigenvalues of the matrix. The eigenvalues are found by solving the characteristic equation:
det(A - λI) = 0,
where A is the given matrix and λ is the eigenvalue.
For system (b):
The direction field and trajectories should be plotted based on the given matrix:
[1 -5] [x]
[4 1] * [y]
To determine the type and stability of the origin as a critical point, we can again analyze the eigenvalues of the matrix.
For system (c):
The direction field and trajectories should be plotted based on the given matrix:
[-6 3] [x]
[ -4 -2] * [y]
To determine the type and stability of the origin as a critical point, we once again analyze the eigenvalues of the matrix.
Analyzing the eigenvalues will allow us to determine if the critical point is a stable node, unstable node, saddle point, or any other type of critical point.
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Solve the system of equations using substitution
y=x+1
x+y = 7