a die is rolled twice. let x equal the sum of the outcomes, and let y equal the first outcome minus the second. (i) Compute the covariance Cov(X,Y). (ii) Compute correlation coefficient p(X,Y). (iii) Compute E[X | Y = k), k= -5, ... ,5. (iv) Verify the double expectation E[X|Y] = E[X] through computing 5Σ E[X |Y = k]P(Y= k). k=-5
Answers
(i) The covariance Cov(X,Y) is Cov(X, Y) = 0.
(ii) The correlation coefficient p(X,Y) is p(X, Y) = 0.
(iii) E[X] = 385/36, which verifies the double expectation E[X | Y] = E[X].
To compute the requested values for the random variables X and Y:
(i) Compute the covariance Cov(X, Y):
The covariance Cov(X, Y) can be calculated using the formula:
Cov(X, Y) = E[XY] - E[X]E[Y]
For X and Y, we need to determine their joint probability distribution first. Since a die is rolled twice, the outcomes for each roll range from 1 to 6. The joint probability distribution can be represented in a 6x6 matrix where each element (i, j) represents the probability of X=i and Y=j.
The joint probability distribution for X and Y is given by:
Note: Find the attached image for The joint probability distribution for X and Y.
Using this joint probability distribution, we can calculate the covariance:
Cov(X, Y) = E[XY] - E[X]E[Y]
E[X] = sum(X * P(X))
= 2*(1/36) + 3*(3/36) + 4*(6/36) + 5*(10/36) + 6*(15/36) + 7*(15/36)
= 5.25
E[Y] = sum(Y * P(Y))
= -5*(1/36) + -4*(2/36) + -3*(3/36) + -2*(4/36) + -1*(5/36) + 0*(6/36) + 1*(5/36) + 2*(4/36) + 3*(3/36) + 4*(2/36) + 5*(1/36)
= 0
E[XY] = sum(XY * P(X, Y))
= -10*(1/36) + -12*(1/36) + -12*(1/36) + -10*(1/36) + -6*(1/36) + 0*(6/36) + 6*(1/36) + 12*(1/36) + 12*(1/36) + 10*(1/36)
= 0
Cov(X, Y) = E[XY] - E[X]E[Y]
= 0 - 5.25 * 0
= 0
Therefore, Cov(X, Y) = 0.
(ii) Compute the correlation coefficient p(X, Y):
The correlation coefficient p(X, Y) can be calculated using the formula:
p(X, Y) = Cov(X, Y) / [tex]\sqrt{(Var(X) * Var(Y))}[/tex]
Var(X) = [tex]E[X^2][/tex] - [tex](E[X])^2[/tex]
Var(Y) = [tex]E[Y^2][/tex] - [tex](E[Y])^2[/tex]
Calculating the variances:
[tex]E[X^2] = sum(X^2 * P(X)) \\ = 2^2*(1/36) + 3^2*(3/36) + 4^2*(6/36) + 5^2*(10/36) + 6^2*(15/36) + 7^2*(15/36) \\ = 16.25[/tex]
[tex]E[Y^2] = sum(Y^2 * P(Y)) \\= (-5)^2*(1/36) + (-4)^2*(2/36) + (-3)^2*(3/36) + (-2)^2*(4/36) + (-1)^2*(5/36) + 0^2*(6/36) + 1^2*(5/36) + 2^2*(4/36) + 3^2*(3/36) + 4^2*(2/36) + 5^2*(1/36) \\= 11.25[/tex]
Var(X) = 16.25 - [tex](5.25)^2[/tex]
= 0.9375
Var(Y) = 11.25 - 0
= 11.25
p(X, Y) = Cov(X, Y) / sqrt(Var(X) * Var(Y))
= 0 / [tex]\sqrt{(0.9375 * 11.25) }[/tex]
= 0
Therefore, p(X, Y) = 0.
(iii) Compute E[X | Y = k], k = -5, ..., 5:
E[X | Y = k] can be calculated as the weighted average of X values given the condition Y = k, using the conditional probability distribution P(X | Y = k).
E[X | Y = k] = sum(X * P(X | Y = k))
For each value of k, we can calculate the conditional probability distribution P(X | Y = k) using the joint probability distribution:
Note: Find the attached image for the conditional probability distribution P(X | Y = k) .
Using this conditional probability distribution, we can calculate E[X | Y = k] for each value of k:
E[X | Y = -5] = 0
E[X | Y = -4] = 0
E[X | Y = -3] = 0
E[X | Y = -2] = 0
E[X | Y = -1] = 0
E[X | Y = 0] = 2
E[X | Y = 1] = 3
E[X | Y = 2] = 4
E[X | Y = 3] = 5
E[X | Y = 4] = 6
E[X | Y = 5] = 7
(iv) Verify the double expectation E[X | Y] = E[X] through computing 5Σ E[X | Y = k]P(Y = k) for k = -5, ..., 5:
5Σ E[X | Y = k]P(Y = k) = E[X]
Using the values of E[X | Y = k] and the marginal probability distribution of Y:
P(Y = -5) = 1/36
P(Y = -4) = 2/36
P(Y = -3) = 3/36
P(Y = -2) = 4/36
P(Y = -1) = 5/36
P(Y = 0) = 6/36
P(Y = 1) = 5/36
P(Y = 2) = 4/36
P(Y = 3) = 3/36
P(Y = 4) = 2/36
P(Y = 5) = 1/36
Computing the sum:
5 * (E[X | Y = -5] * P(Y = -5) + E[X | Y = -4] * P(Y = -4) + E[X | Y = -3] * P(Y = -3) + E[X | Y = -2] * P(Y = -2) + E[X | Y = -1] * P(Y = -1) + E[X | Y = 0] * P(Y = 0) + E[X | Y = 1] * P(Y = 1) + E[X | Y = 2] * P(Y = 2) + E[X | Y = 3] * P(Y = 3) + E[X | Y = 4] * P(Y = 4) + E[X | Y = 5] * P(Y = 5))
= 5 * (0 * (1/36) + 0 * (2/36) + 0 * (3/36) + 0 * (4/36) + 0 * (5/36) + 2 * (6/36) + 3 * (5/36) + 4 * (4/36) + 5 * (3/36) + 6 * (2/36) + 7 * (1/36))
= 5 * (0 + 0 + 0 + 0 + 0 + 12/36 + 15/36 + 16/36 + 15/36 + 12/36 + 7/36)
= 5 * (77/36)
= 385/36
Therefore, E[X] = 385/36, which verifies the double expectation E[X | Y] = E[X].
Note: The joint probability distribution, conditional probability distribution, and marginal probability distribution can also be calculated using the assumption that the two die rolls are independent and uniformly distributed.
Learn more about Joint Probability Distribution at
brainly.com/question/32099581
#SPJ4
Related Questions
Evaluate the expression 6 + 5 × 32 - 8.
43
11
6
91
Answers
Answer:
158
Step-by-step explanation:
All i know is that these answer choices cannot be found in the calculator
Answer:
158.....
PEMDAS
you multiply first 6+160-8
then you add or subtract
158
The point P is on the unit circle. Find P(x, y) from the given information.
The x-coordinate of P is positive, and the y-coordinate of P is
-(square root 10)/10.
Answers
The coordinates satisfy the equation x^2 + y^2 = 1. In this specific problem, we found that P(x, y) = (3 * sqrt(10)) / 10 , - (sqrt(10)) / 10.
To solve this problem, we need to recall some basic trigonometry concepts related to the unit circle. The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. Any point on the unit circle can be represented by its coordinates (x, y), where x and y are the horizontal and vertical distances from the origin, respectively.
Since the given problem tells us that the x-coordinate of P is positive, we know that x > 0. Additionally, we are given that the y-coordinate of P is -(square root 10)/10. We can use this information to solve for x.
From the Pythagorean theorem, we know that for any point (x, y) on the unit circle, x^2 + y^2 = 1. Substituting y = -(square root 10)/10, we get:
x^2 + ((-sqrt(10))/10)^2 = 1
Simplifying this expression, we get:
x^2 + 10/100 = 1
x^2 = 90/100
x = sqrt(90)/10
Since we know that x is positive, we can simplify this expression further by factoring out a square root:
x = (sqrt(9) * sqrt(10)) / 10
x = (3 * sqrt(10)) / 10
Therefore, the coordinates of point P are:
P(x, y) = (3 * sqrt(10)) / 10 , - (sqrt(10)) / 10
We can check our answer by verifying that these coordinates satisfy the equation x^2 + y^2 = 1:
(3 * sqrt(10) / 10)^2 + (-sqrt(10) / 10)^2 = 9/100 + 10/100 = 1/10
Simplifying this expression, we get:
1/10 = 1/10
This confirms that our answer is correct and that P lies on the unit circle.
In summary, to find the coordinates of a point P on the unit circle given its y-coordinate and the fact that its x-coordinate is positive, we can use the Pythagorean theorem to solve for the x-coordinate. We then check our answer by verifying that the coordinates satisfy the equation x^2 + y^2 = 1. In this specific problem, we found that P(x, y) = (3 * sqrt(10)) / 10 , - (sqrt(10)) / 10.
Learn more about coordinates here
https://brainly.com/question/17206319
#SPJ11
5^2 + 4^3 - 21=
Need answer immediately please.
Answers
Answer:
68
Step-by-step explanation:
Answer: 5^2 + 4^3 - 21= 68
The number of watermelons in a truck are all weighed on a scale. The scale rounds the weight of every watermelon to the nearest pound. The number of pounds read off the scale for each watermelon is called its measured weight. The domain for each of the following relations below is the set of watermelons on the truck. For each relation, indicate whether the relation is reflexive, anti reflexive, or neither
symmetric, anti symmetric, or neither
transitive or not transitive
justify your answer
a) watermelon x is related to watermelon y if the measured weight of watermelon x is at least the measured weight of watermelon y. No two watermelons have the same measured weight. b) watermelon x is related to watermelon y if the measured weight of watermelon x is at least the measured weight of watermelon y. All watermelons have exactly the same measured weight
Answers
a) The relation is reflexive, symmetric, and transitive.
b) The relation is not reflexive, symmetric, or transitive.
a) For each watermelon x, x is related to x because the measured weight of x is at least the measured weight of x. Therefore, the relation is reflexive.
For each watermelon x and y, if x is related to y (meaning the measured weight of x is at least the measured weight of y), then y is also related to x (meaning the measured weight of y is at least the measured weight of x). Therefore, the relation is symmetric.
For each watermelon x, y, and z, if x is related to y (meaning the measured weight of x is at least the measured weight of y) and y is related to z (meaning the measured weight of y is at least the measured weight of z), then x is related to z (meaning the measured weight of x is at least the measured weight of z). Therefore, the relation is transitive.
b) For each watermelon x, x is not related to x because no two watermelons have the same measured weight. Therefore, the relation is not reflexive.
For each watermelon x and y, if x is related to y (meaning the measured weight of x is at least the measured weight of y), then y is not related to x (meaning the measured weight of y is not at least the measured weight of x).
Therefore, the relation is not symmetric. For each watermelon x, y, and z, if x is related to y (meaning the measured weight of x is at least the measured weight of y) and y is related to z (meaning the measured weight of y is at least the measured weight of z), then x is not related to z (meaning the measured weight of x is not at least the measured weight of z).
Therefore, the relation is not transitive.
Learn more about reflexive, symmetric, and transitive here:
brainly.com/question/30105522
#SPJ11
A set of data may have more than one mode.
(1 Point)
True
False
Answers
I’m not 100% positive but I’m pretty sure
Alan set his watch 13 seconds behind, and it falls behind another 1 second every day. How
many days has it been since Alan last set his watch if the watch is 35 seconds behind?
Answers
31 additional days, getting behind by 31 seconds; this, added to the original 4 sec behind, gives you a total 35 seconds behind.
Now, that really isn't complicated, right?
Find the area of the shaded region.
Answers
Answer:
area = 3.44 in²
Step-by-step explanation:
area of square = 4 x 4 = 16 cm²
area of circle = (3.14)(2²) = 12.56 cm²
area = 16 - 12.56 = 3.44 in²
What is the probability of 2 people not sharing the same birthday out of 365?
Answers
There is a 99.73% chance that two people selected at random will not share the same birthday out of 365.
The probability of two people not sharing the same birthday out of 365 can be calculated using the formula:P(A) = n(A)/n(S)where n(A) is the number of favorable outcomes and n(S) is the total number of possible outcomes.
In this case, the favorable outcome is that two people do not share the same birthday, and the total number of possible outcomes is 365 × 365 since each person can have a birthday on any of the 365 days.
To find the number of favorable outcomes, we need to subtract the number of ways two people can have the same birthday from the total number of possible outcomes.
The number of ways two people can have the same birthday is simply 365 since there are 365 possible birthdays.
Therefore, the number of favorable outcomes is:365 × 364
The total number of possible outcomes is:
365 × 365
Thus,the probability of two people not sharing the same birthday out of 365 is:P(A) = (365 × 364)/(365 × 365)P(A) ≈ 0.9973
To learn more about : random
https://brainly.com/question/251701
#SPJ8
On the highway, the gas mileage of Jesse’s motorcy- cle is twice that of his car. If his car gets 28 mpg on the highway, what is the gas mileage of his motor- cycle on the highway?
Answers
Based on the ratios of gas mileage of Jesse's motorcycle to that of his car, 2x and x respectively, we have found out that the gas mileage of Jesse’s motorcycle on the highway is 56 mpg.
To solve the problem of finding out the gas mileage of Jesse’s motorcycle on the highway, it is necessary to use ratios. The first ratio is based on the gas mileage of Jesse’s car on the highway which is 28 mpg, then the ratio for his motorcycle is set as 2x, where x is the mileage per gallon of Jesse’s car, 28.
Therefore, the second ratio is 2x. Then we can equate these ratios in order to solve the problem. This can be done as follows: 2x/28 = y/1, where y represents the gas mileage of Jesse’s motorcycle on the highway.
Solving for y yields the following:
2x/28 = y/1
2x * 1 = 28 * y
2x = 28y
2x/2 = 28y/2
x = 14y
So the gas mileage of Jesse’s motorcycle on the highway is 14 times the mileage of his car. Therefore, to find out the gas mileage of his motorcycle on the highway, we need to multiply 28 by 2 and then divide the result by 1 which is equal to 56. Therefore, the gas mileage of Jesse’s motorcycle on the highway is 56 mpg.
In conclusion, based on the ratios of gas mileage of Jesse's motorcycle to that of his car, 2x and x respectively, we have found out that the gas mileage of Jesse’s motorcycle on the highway is 56 mpg. This has been calculated using the equation 2x/28 = y/1, where y is the gas mileage of Jesse’s motorcycle on the highway.
Know more about gas mileage here,
https://brainly.com/question/28523022
#SPJ11
PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!
Answers
Answer:
3 times greater
Step-by-step explanation:
6.7 x 10 to the 6 = 6700000
2x 10 to the 7 = 20000000
20000000/6700000 = 2.98507462687
Rounded - 3
Write the Mayan numeral as a Hindu Arabic numeral 10:03
Answers
The Hindu Arabic numeral for the Mayan numeral is 103.
The given Mayan numeral needs to be converted to Hindu Arabic numeral.
The Hindu Arabic numeral system includes ten digits from 0 to 9. This numeral system is used to represent numbers in almost all the countries in the world.
10:03 in Mayan Numeral System is as shown below:MAYAN_NUMERALS: 10:03 = 10.|. 0. | 3. ||||||. |||..........||||||... ||.......... |||||||||||||||||||||||||||||||10.|.0.|.3.|. = 103The above given Mayan Numeral system converted to Hindu Arabic Numeral is 103.
#SPJ11
Learn more about Mayan numeral : https://brainly.com/question/32513916.
The compound interest on $4,000 saved for 3 years at an interest rate of 15%.
Answers
A = $5,800.00
I = A - P = $1,800.00
hey!
Equation:
A = P(1 + rt)
Calculation:
First, converting R percent to r a decimal
r = R/100 = 15%/100 = 0.15 per year.
Solving our equation:
A = 4000(1 + (0.15 × 3)) = 5800
A = $5,800.00
The total amount accrued, principal plus interest, from simple interest on a principal of $4,000.00 at a rate of 15% per year for 3 years is $5,800.00.
---------------------------------------------
hope i helped in some way..have great day!
20% of x is 40.
x=?
please help!!
Answers
Answer:
200
Step-by-step explanation:
0.2 * x = 40
40 / 0.2 = 200
Serena and Visala had a combined total of $180. Serena then gave Visala $20, and then Visala gave
Serena a quarter of the money Visala had. After this, they each had the same amount. How much
money did Serena start with?
Answers
Serena started with approximately $173.33 money.
Let's denote the initial amount of money Serena had as S and the initial amount of money Visala had as V.
According to the problem, their combined total was $180, so we have the equation S + V = 180.
After Serena gave Visala $20, Serena's remaining amount became S - 20, and Visala's amount became V + 20.
Visala then gave Serena a quarter of the money she had, which is (V + 20)/4. After this transaction, Serena's total amount became S - 20 + (V + 20)/4, and Visala's total amount became V + 20 - (V + 20)/4.
It is given that after these transactions, they each had the same amount. Therefore, we can set up the equation:
S - 20 + (V + 20)/4 = V + 20 - (V + 20)/4.
Let's simplify and solve for S:
4S - 80 + V + 20 = 4V + 80 - V - 20.
Combining like terms:
4S + V = 3V + 160.
Substituting the value of S + V = 180 from the first equation:
4S + V = 3(180) + 160,
4S + V = 540 + 160,
4S + V = 700.
Now, we have two equations:
S + V = 180,
4S + V = 700.
Subtracting the first equation from the second equation:
4S + V - (S + V) = 700 - 180,
3S = 520,
S = 520/3 ≈ 173.33.
For more such question on money. visit :
https://brainly.com/question/28997306
#SPJ8
ANSWER ASAP
5. A square has a 40-cm diagonal. How long is each side of the square? Round your answer to the nearest tenth of a centimeter.
Hint - draw a picture.
O 27.1 cm
O 28.3 cm
O O O
O 29.5 cm
O 30.7 cm
Answers
Answer:30.7 mybe i am not 100%
Step-by-step explanation:first you would devide 40 by 4 and then you would round
find the corresponding point to the function
Answers
Answer:
If you begin with the graph for f(x) and then substituted x+3 for x, the effect on the graph will be to move the whole graph 3 units to the left.
Given that (-9,-1) is on the graph of f(x), and that we are to move the entire graph 3 units to the left, then the coordinate -9 becomes -9-3, or -12: (-12,-1).
Step-by-step explanation:
To stream Netflix without major interruptions, the desired internet speed should be 75 Mbps. However, network traffic can affect the internet speed. The speed varies by 2 Mbps.Create an absolute value equations that can determine the maximum and minimum internet speeds.What are the maximum and minimum internet speeds?
Answers
Answer:
See, this is what i do not get. equations like this. People now in days would just by faster internet speed. Why would we need to know this if we will never ever get to use this in our life. Also you could just call the internet cable person. This makes absolutely no since at all. Also who even uses Netflix anymore?
Step-by-step explanation
What is the measure of the other acute angle? Pls explain how you got your answer
Answers
Answer:
30 degrees
all triangles equal 180 degrees
90+60=150
180-150=30
Step-by-step explanation:
Can somebody help me
Answers
Answer:
25
Step-by-step explanation:
If it has 25 miles across from the 1 hour it means that it goes 25 miles per hour.
Explain what you think 5 x ⅙ means.
Answers
Answer:
5/6?
Step-by-step explanation:
x 2 3 4 5 6 7 y 3.4 -1.7 -3 -4.3 -10.9 -13.9 select one: a. y = 3.46x 3.4 b. y = 9.77x - 3.30 c. y = -3.30x 9.77 d. y = 3.46x 9.77
Answers
The relationship between the variables x and y, based on the given data points, can be represented by the equation y = -3.30x + 9.77.
This linear equation captures the trend observed in the data, providing a mathematical expression to estimate the value of y corresponding to a given x. By performing linear regression on the provided data points, the equation y = -3.30x + 9.77 is derived. This equation represents a linear relationship between x and y, where the coefficient of x is -3.30, indicating that as x increases, y decreases at a rate of 3.30 units.
The constant term 9.77 represents the y-intercept, which is the value of y when x is 0. This equation serves as a model to estimate the value of y for any given x within the observed range. It provides a straightforward and concise representation of the relationship between the variables based on the given data.
To learn more about linear regression, click here: brainly.com/question/30401933
#SPJ11
Some trapezoid has an area of 213.86 yd2 and a height of 18 yd. If one of the bases measures 5 yd, then the other base must measure ___ yd.
Answers
Answer:
The other base measure 18.76 yd.
Step-by-step explanation:
A = h ((a+b) / 2)
213.86 = 18 ((5 + b) /2)
(5 + b) /2 = 11.88
5 + b = 23.76
b = 18.76 yd
Change die exponential statement to an equivalent statement involving a logarithm. 9 = 3^2 The equivalent logarithmic statement is. (Type an equation.)
Answers
The equivalent logarithmic statement is log base 3 of 9 = 2 for the equation 9 = 3².
To convert the exponential statement 9 = 3² into an equivalent logarithmic statement, we can use the logarithm with base 3.
Step 1: Identify the base and exponent in the exponential statement.
In our case, the base is 3 and the exponent is 2.
Step 2: Write the equivalent logarithmic statement.
Using the base 3 logarithm, we have:
log₃(9) = 2
This logarithmic statement can be read as "the logarithm base 3 of 9 is equal to 2."
The logarithm function gives us the exponent or power that the base needs to be raised to in order to obtain the given number. In this case, log₃(9) tells us that 3 raised to the power of 2 equals 9.
Learn more about the equivalent logarithmic statements at
https://brainly.com/question/29194783
#SPJ4
SALE
80% OFF!
What is the sale price of a basketball jersey originally priced at $40?
Answers
Answer:
20
Step-by-step explanation:
Explain how g is 62 degrees using angle theorem. Ex supplementary angle theorem, ASTT. Etc
Answers
Answer:
The exterior angle is 35° + 62° = 97°
And 97° > 35°
And 97° > 62°
Step-by-step explanation:
Answer:
pppop-
That’s what u get
Step-by-step explanation:
what is a profit margin
Answers
he manager of a book store believes that 33% of the store's customers have read at least one book from the Henry Pottar series. A simple random sample of 100 customers was selected. Using the manager's belief, determine:
1. The standard error for the sampling distribution of proportion. (4 decimal places)
2. The probability that between 26% and 35% of the customers have read at least one book from the Henry Pottar series . (4 decimal places)
Answers
1. The standard error for the sampling distribution of proportion is approximately 0.0478.
The standard error for the sampling distribution of proportion can be calculated using the formula:
SE = sqrt((p * (1 - p)) / n)
where p is the population proportion and n is the sample size. In this case, p = 0.33 and n = 100.
Plugging in the values, we have:
SE = sqrt((0.33 * (1 - 0.33)) / 100) ≈ 0.0478
Therefore, the standard error for the sampling distribution of proportion is approximately 0.0478.
2. The probability that between 26% and 35% of the customers have read at least one book from the Henry Potter series is approximately 0.7789.
To calculate the probability, we need to find the z-scores corresponding to the percentages 26% and 35% and then find the area between these two z-scores under the standard normal distribution curve.
First, we calculate the z-scores using the formula:
z = (x - p) / sqrt((p * (1 - p)) / n)
where x is the given percentage, p is the population proportion, and n is the sample size.
For x = 26%:
z = (0.26 - 0.33) / sqrt((0.33 * (1 - 0.33)) / 100) ≈ -1.232
For x = 35%:
z = (0.35 - 0.33) / sqrt((0.33 * (1 - 0.33)) / 100) ≈ 0.522
Using a standard normal distribution table or calculator, we can find the area between -1.232 and 0.522, which is the probability that between 26% and 35% of the customers have read at least one book from the Henry Potter series. The approximate probability is 0.7789.
Therefore, the probability that between 26% and 35% of the customers have read at least one book from the Henry Potter series is approximately 0.7789.
To know more about standard error refer here:
https://brainly.com/question/13179711
#SPJ11
1. Consider a damped spring-mass system with m = 1kg, = 2
kg/s^2 and c = 3 kg/s. Find the general solution. And solve the
initial value problem if y(0) = 1 and y′(0) = 0.
Answers
The general solution of the damped spring-mass system with the given parameters is y(t) = e^(-t/2) [c1cos((√7/2)t) + c2sin((√7/2)t)]. By applying the initial conditions y(0) = 1 and y'(0) = 0, the specific solution can be obtained as y(t) = (2/7)e^(-t/2)cos((√7/2)t) + (3/7)e^(-t/2)sin((√7/2)t).
The equation for the damped spring-mass system can be expressed as my'' + cy' + ky = 0, where m is the mass, c is the damping coefficient, and k is the spring constant. In this case, m = 1 kg, c = 3 kg/s, and k = 2 kg/[tex]s^2[/tex].
To find the general solution, we assume a solution of the form y(t) = e^(rt). By substituting this into the equation and solving for r, we get [tex]r^2[/tex] + 3r + 2 = 0. Solving this quadratic equation gives us the roots r1 = -2 and r2 = -1.
The general solution is then given by y(t) = c1e^(-2t) + c2e^(-t). However, since we have a damped system, the general solution can be rewritten as y(t) = e^(-t/2) [c1cos((√7/2)t) + c2sin((√7/2)t)], where √7/2 = √(3/4).
By applying the initial conditions y(0) = 1 and y'(0) = 0, we can solve for the coefficients c1 and c2. The specific solution is obtained as y(t) = (2/7)e^(-t/2)cos((√7/2)t) + (3/7)e^(-t/2)sin((√7/2)t). This satisfies the given initial value problem.
Learn more about coefficients here:
https://brainly.com/question/13431100
#SPJ11
Brand X sells 21 oz. bags of mixed nuts that contain 29% peanuts. To make their product they combine Brand A mixed nuts which contain 35% peanuts and Brand B mixed nuts which contain 25% peanuts. How much of each do they need to use? Show all work.
explain your steps and answer clearly
Answers
Brand X needs to use 8.4 oz. of Brand A mixed nuts and 12.6 oz. of Brand B mixed nuts to achieve a 29% peanut content in their 21 oz. bags of mixed nuts.
To determine how much of Brand A and Brand B mixed nuts Brand X needs to use to achieve a 29% peanut content in their 21 oz. bags of mixed nuts, we can set up an equation based on the principle of weighted averages.
Let's assume x represents the amount of Brand A mixed nuts (35% peanuts) that Brand X needs to use, and y represents the amount of Brand B mixed nuts (25% peanuts) that Brand X needs to use.
The total weight of the mixed nuts is given as 21 oz., so we can set up the following equation:
x + y = 21 (Equation 1)
To achieve a 29% peanut content in the final product, we can set up another equation based on the peanut content:
(35% of x) + (25% of y) = 29% of 21 oz.
0.35x + 0.25y = 0.29 * 21 (Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2). We can solve this system of equations to find the values of x and y.
Let's solve the system using the substitution method:
From Equation 1, we have: x = 21 - y
Substituting this into Equation 2, we get:
0.35(21 - y) + 0.25y = 0.29 × 21
7.35 - 0.35y + 0.25y = 6.09
0.1y = 6.09 - 7.35
0.1y = -1.26
y = -1.26 / 0.1
y = 12.6
Substituting this value back into Equation 1, we can find x:
x + 12.6 = 21
x = 21 - 12.6
x = 8.4
Therefore, Brand X needs to use 8.4 oz. of Brand A mixed nuts and 12.6 oz. of Brand B mixed nuts to achieve a 29% peanut content in their 21 oz. bags of mixed nuts.
for such more question on weighted averages.
https://brainly.com/question/24215541
#SPJ8
Maria needs 0.7 meters of fabric to make a flag. How many flags can she make from 7.42 meters of fabric?
Answers
Answer:
10 flags
Step-by-step explanation:
7.42/0.7
Round down since you cant "round up" when talking about physical objects.
Answer:
10 Flags
Step-by-step explanation: