The general solution of the given differential equations are:
y = c₁e^(x/4) + c₂xe^(x/4) (for 16y''-8y'+y=0)
y = c₁e^x + c₂e^(-2x) (for y"+y'-2y=0)
y = c₁e^x + c₂e^(-2x) + (1/2)x
(for y"+y'-2y=x²)
Given differential equations are:
16y''-8y'+y=0
y"+y'-2y=0
y"+y'-2y = x²
To find the general solution to the given differential equations, we will solve these equations one by one.
(i) 16y'' - 8y' + y = 0
The characteristic equation is:
16m² - 8m + 1 = 0
Solving this quadratic equation, we get m = 1/4, 1/4
Hence, the general solution of the given differential equation is:
y = c₁e^(x/4) + c₂xe^(x/4)..................................................(1)
(ii) y" + y' - 2y = 0
The characteristic equation is:
m² + m - 2 = 0
Solving this quadratic equation, we get m = 1, -2
Hence, the general solution of the given differential equation is:
y = c₁e^x + c₂e^(-2x)..................................................(2)
(iii) y" + y' - 2y = x²
The characteristic equation is:
m² + m - 2 = 0
Solving this quadratic equation, we get m = 1, -2.
The complementary function (CF) of this differential equation is:
y = c₁e^x + c₂e^(-2x)..................................................(3)
Now, we will find the particular integral (PI). Let's assume that the PI of the differential equation is of the form:
y = Ax² + Bx + C
Substituting the value of y in the given differential equation, we get:
2A - 4A + 2Ax² + 4Ax - 2Ax² = x²
Equating the coefficients of x², x, and the constant terms on both sides, we get:
2A - 2A = 1,
4A - 4A = 0, and
2A = 0
Solving these equations, we get
A = 1/2,
B = 0, and
C = 0
Hence, the particular integral of the given differential equation is:
y = (1/2)x²..................................................(4)
The general solution of the given differential equation is the sum of CF and PI.
Hence, the general solution is:
y = c₁e^x + c₂e^(-2x) + (1/2)x²..................................................(5)
Conclusion: Therefore, the general solution of the given differential equations are:
y = c₁e^(x/4) + c₂xe^(x/4) (for 16y''-8y'+y=0)
y = c₁e^x + c₂e^(-2x) (for y"+y'-2y=0)
y = c₁e^x + c₂e^(-2x) + (1/2)x
(for y"+y'-2y=x²)
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The particular solution is: y = -1/2 x². The general solution is: y = c1 e^(-2x) + c2 e^(x) - 1/2 x²
The general solution of the given differential equations are:
Given differential equation: 16y'' - 8y' + y = 0
The auxiliary equation is: 16m² - 8m + 1 = 0
On solving the above quadratic equation, we get:
m = 1/4, 1/4
∴ General solution of the given differential equation is:
y = c1 e^(x/4) + c2 x e^(x/4)
Given differential equation: y" + y' - 2y = 0
The auxiliary equation is: m² + m - 2 = 0
On solving the above quadratic equation, we get:
m = -2, 1
∴ General solution of the given differential equation is:
y = c1 e^(-2x) + c2 e^(x)
Given differential equation: y" + y' - 2y = x²
The auxiliary equation is: m² + m - 2 = 0
On solving the above quadratic equation, we get:m = -2, 1
∴ The complementary solution is:y = c1 e^(-2x) + c2 e^(x)
Now we have to find the particular solution, let us assume the particular solution of the given differential equation:
y = ax² + bx + c
We will use the method of undetermined coefficients.
Substituting y in the differential equation:y" + y' - 2y = x²a(2) + 2a + b - 2ax² - 2bx - 2c = x²
Comparing the coefficients of x² on both sides, we get:-2a = 1
∴ a = -1/2
Comparing the coefficients of x on both sides, we get:-2b = 0 ∴ b = 0
Comparing the constant terms on both sides, we get:2c = 0 ∴ c = 0
Thus, the particular solution is: y = -1/2 x²
Now, the general solution is: y = c1 e^(-2x) + c2 e^(x) - 1/2 x²
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predict the total packing cost for 25,000 orders, weighing 40,000 pounds, with 4,000 fragile items. round regression intercept to whole dollar and coefficients to two decimal places (nearest cent). enter the final answer rounded to the nearest dollar.
The predicted total packing cost for 25,000 orders is $150,800
To predict the total packing cost for 25,000 orders, to use the information provided and apply regression analysis. Let's assume we have a linear regression model with the following variables:
X: Number of orders
Y: Packing cost
Based on the given information, the following data:
X (Number of orders) = 25,000
Total weight of orders = 40,000 pounds
Number of fragile items = 4,000
Now, let's assume a regression equation in the form: Y = b0 + b1 × X + b2 ×Weight + b3 × Fragile
Where:
b0 is the regression intercept (rounded to the nearest whole dollar)
b1, b2, and b3 are coefficients (rounded to two decimal places or nearest cent)
Weight is the total weight of the orders (40,000 pounds)
Fragile is the number of fragile items (4,000)
Since the exact regression equation and coefficients, let's assume some hypothetical values:
b0 (intercept) = $50 (rounded)
b1 (coefficient for number of orders) = $2.75 (rounded to two decimal places or nearest cent)
b2 (coefficient for weight) = $0.05 (rounded to two decimal places or nearest cent)
b3 (coefficient for fragile items) = $20 (rounded to two decimal places or nearest cent)
calculate the predicted packing cost for 25,000 orders:
Y = b0 + b1 × X + b2 × Weight + b3 × Fragile
Y = 50 + 2.75 × 25,000 + 0.05 × 40,000 + 20 × 4,000
Y = 50 + 68,750 + 2,000 + 80,000
Y = 150,800
Keep in mind that the actual values of the regression intercept and coefficients might be different, but this is a hypothetical calculation based on the information provided.
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A group of 800 students wants to eat lunch in the cafeteria. if each table at in the cafeteria seats 8 students, how many tables will the students need?
The number of tables that will be required to seat all students present at the cafeteria is 100.
By applying simple logic, the answer to this question can be obtained.
First, let us state all the information given in the question.
No. of students in the whole group = 800
Amount of students that each table can accommodate is 8 students.
So, the number of tables required can be defined as:
No. of Tables = (Total no. of students)/(No. of students for each table)
This means,
N = 800/8
N = 100 tables.
So, with the availability of a minimum of 100 tables in the cafeteria, all the students can be comfortably seated.
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A manufacturer of yeast finds that the culture grows exponentially at the rate of 13% per hour . a) if the initial mass is 3.7 , what mass will be present after: 7 hours and then 2 days
After 7 hours, the mass of yeast will be approximately 9.718 grams. After 2 days (48 hours), the mass of yeast will be approximately 128.041 grams.
To calculate the mass of yeast after a certain time using exponential growth, we can use the formula:
[tex]M = M_0 * e^{(rt)}[/tex]
Where:
M is the final mass
M0 is the initial mass
e is the base of the natural logarithm (approximately 2.71828)
r is the growth rate (expressed as a decimal)
t is the time in hours
Let's calculate the mass of yeast after 7 hours:
M = 3.7 (initial mass)
r = 13% per hour
= 0.13
t = 7 hours
[tex]M = 3.7 * e^{(0.13 * 7)}[/tex]
Using a calculator, we can find that [tex]e^{(0.13 * 7)[/tex] is approximately 2.628.
M ≈ 3.7 * 2.628
≈ 9.718 grams
Now, let's calculate the mass of yeast after 2 days (48 hours):
M = 3.7 (initial mass)
r = 13% per hour
= 0.13
t = 48 hours
[tex]M = 3.7 * e^{(0.13 * 48)][/tex]
Using a calculator, we can find that [tex]e^{(0.13 * 48)}[/tex] is approximately 34.630.
M ≈ 3.7 * 34.630
≈ 128.041 grams
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a) After 7 hours, the mass will be approximately 7.8272.
b) After 2 days, the mass will be approximately 69.1614.
The growth of the yeast culture is exponential at a rate of 13% per hour.
To find the mass present after a certain time, we can use the formula for exponential growth:
Final mass = Initial mass × [tex](1 + growth ~rate)^{(number~ of~ hours)}[/tex]
a) After 7 hours:
Final mass = 3.7 ×[tex](1 + 0.13)^7[/tex]
To calculate this, we can plug in the values into a calculator or use the exponent rules:
Final mass = 3.7 × [tex](1.13)^{7}[/tex] ≈ 7.8272
Therefore, the mass present after 7 hours will be approximately 7.8272.
b) After 2 days:
Since there are 24 hours in a day, 2 days will be equivalent to 2 × 24 = 48 hours.
Final mass = 3.7 × [tex](1 + 0.13)^{48}[/tex]
Again, we can use a calculator or simplify using the exponent rules:
Final mass = 3.7 ×[tex](1.13)^{48}[/tex] ≈ 69.1614
Therefore, the mass present after 2 days will be approximately 69.1614.
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An article states that false-positives in polygraph tests (i.e., tests in which an individual fails even though he or she is telling the truth) are relatively common and occur about 15% of the time. Suppose that such a test is given to 10 trustworthy individuals. (Round all answers to four decimal places.)
(a) What is the probability that all 10 pass?
P(X = 10) =
(b) What is the probability that more than 2 fail, even though all are trustworthy?
P (more than 2 fail, even though all are trustworthy) =
(c) The article indicated that 400 FBI agents were required to take a polygraph test. Consider the random variable x = number of the 400 tested who fail. If all 400 agents tested are trustworthy, what are the mean and standard deviation of x?
Mean = 3
Standard deviation = 4
(a) To find the probability that all 10 trustworthy individuals pass the polygraph test,
we can use the binomial probability formula:
P(X = 10) = C(10, 10) * (0.15)^10 * (1 - 0.15)^(10 - 10)
Calculating the values:
C(10, 10) = 1 (since choosing all 10 out of 10 is only one possibility)
(0.15)^10 ≈ 0.0000000778
(1 - 0.15)^(10 - 10) = 1 (anything raised to the power of 0 is 1)
P(X = 10) ≈ 1 * 0.0000000778 * 1 ≈ 0.0000000778
The probability that all 10 trustworthy individuals pass the polygraph test is approximately 0.0000000778.
(b) To find the probability that more than 2 trustworthy individuals fail the test, we need to calculate the probability of exactly 0, 1, and 2 individuals failing and subtract it from 1 (to find the complementary probability).
P(more than 2 fail, even though all are trustworthy) = 1 - P(X = 0) - P(X = 1) - P(X = 2)
Using the binomial probability formula:
P(X = 0) = C(10, 0) * (0.15)^0 * (1 - 0.15)^(10 - 0)
P(X = 1) = C(10, 1) * (0.15)^1 * (1 - 0.15)^(10 - 1)
P(X = 2) = C(10, 2) * (0.15)^2 * (1 - 0.15)^(10 - 2)
Calculating the values:
C(10, 0) = 1
C(10, 1) = 10
C(10, 2) = 45
(0.15)^0 = 1
(0.15)^1 = 0.15
(0.15)^2 ≈ 0.0225
(1 - 0.15)^(10 - 0) = 0.85^10 ≈ 0.1967
(1 - 0.15)^(10 - 1) = 0.85^9 ≈ 0.2209
(1 - 0.15)^(10 - 2) = 0.85^8 ≈ 0.2476
P(more than 2 fail, even though all are trustworthy) = 1 - 1 * 0.1967 - 10 * 0.15 * 0.2209 - 45 * 0.0225 * 0.2476 ≈ 0.0004
The probability that more than 2 trustworthy individuals fail the polygraph test, even though all are trustworthy, is approximately 0.0004.
(c) The mean (expected value) of a binomial distribution is given by μ = np, where n is the number of trials (400 agents tested) and p is the probability of success (the probability of failing for a trustworthy agent, which is 0.15).
Mean = μ = np = 400 * 0.15 = 60
The standard deviation of a binomial distribution is given by σ = sqrt(np(1-p)).
Standard deviation = σ = sqrt(400 * 0.15 * (1 - 0.15)) ≈ 4
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How does the number 32.4 change when you multiply it by 10 to the power of 2 ? select all that apply.
a). the digit 2 increases in value from 2 ones to 2 hundreds.
b). each place is multiplied by 1,000
c). the digit 3 shifts 2 places to the left, from the tens place to the thousands place.
The Options (a) and (c) apply to the question, i.e. the digit 2 increases in value from 2 ones to 2 hundred, and, the digit 3 shifts 2 places to the left, from the tens place to the thousands place.
32.4×10²=32.4×100=3240
Hence, digit 2 moves from one's place to a hundred's. (a) satisfied
And similarly, digit 3 moves from ten's place to thousand's place. Now, 1000=10³=10²×10.
Hence, it shifts 2 places to the left.
Therefore, (c) is satisfied.
As for (b), where the statement: Each place is multiplied by 1,000; the statement does not hold true since each digit is shifted 2 places, which indicates multiplied by 10²=100, not 1000.
Hence (a) and (c) applies to our question.
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