The particular solution that satisfies the given condition is s = (16/3)t³ + (9/2)t² - 6t + 120
To find the particular solution of the differential equation ds/dt = 16t² + 9t - 6 with the condition s = 120 when t = 0, we need to integrate the right-hand side of the equation with respect to t and then solve for the constant of integration using the given condition.
First, let's integrate the right-hand side of the equation:
∫(ds/dt) dt = ∫(16t² + 9t - 6) dt
Integrating term by term, we get:
s = (16/3)t³ + (9/2)t² - 6t + C
Now, we can use the given condition s = 120 when t = 0 to determine the value of the constant of integration C:
120 = (16/3)(0)³ + (9/2)(0)² - 6(0) + C
120 = C
Therefore, the particular solution that satisfies the given condition is:
s = (16/3)t³ + (9/2)t² - 6t + 120
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Give the general solution of the linear system x+y-2z = 0 2x + 2y3z = 1 3x + 3y + z = 7.
Answer:
The general solution to the given linear system is x = 3z - 1, y = -z + 1, where z is a free variable. This means that the solution consists of infinitely many points that lie on a straight line in three-dimensional space.
To solve the linear system, we can use the method of elimination or Gaussian elimination. Here, we'll use Gaussian elimination to find the general solution.
We start by writing the augmented matrix of the system:
[1 1 -2 | 0]
[2 2 3 | 1]
[3 3 1 | 7]
To simplify the matrix, we perform row operations to create zeros in the first column below the first entry. We subtract twice the first row from the second row and subtract three times the first row from the third row:
[1 1 -2 | 0]
[0 0 7 | 1]
[0 0 7 | 7]
Next, we divide the second and third rows by 7 to create leading ones:
[1 1 -2 | 0]
[0 0 1 | 1/7]
[0 0 1 | 1]
Now, we perform row operations to create zeros in the second column below the second entry. We subtract the third row from the second row:
[1 1 -2 | 0]
[0 0 1 | 1/7]
[0 0 0 | 0]
From the last row, we can see that 0z = 0, which means that z is a free variable. We can assign a parameter to z, say t, and solve for x and y in terms of t. From the first row, we have x + y - 2z = 0. Plugging in the values for x and y, we get x = 3z - 1 and y = -z + 1. Therefore, the general solution to the linear system is x = 3z - 1, y = -z + 1, where z is a free variable.
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The projection matrix is P = A(AT A)-1A". If A is invertible, what is e? Choose the best answer, e.g., if the answer is 2/4, the best answer is 1/2. The value of e varies based on A. Oe=b - Pb e = 0 Oe=AtAb
The value of e varies based on A. Oe=b - Pb e = 0 Oe=AtAb would be (AT A)-1 AT b.
The given projection matrix is P = A(AT A)-1A".
We have been asked to find the value of e if A is invertible. Let's proceed further and solve this problem. First, we need to find the product of A and its transpose, i.e., AT A.A.T.A = [a11 a12 ... a1n] [a21 a22 ... a2n] ... [an1 an2 ... ann] = [Σ(ai1)(aj1) Σ(ai1)(aj2) ... Σ(ai1)(ajn)] [Σ(ai2)(aj1) Σ(ai2)(aj2) ... Σ(ai2)(ajn)] ... [Σ(ain)(aj1) Σ(ain)(aj2) ... Σ(ain)(ajn)]
The inverse of AT A is (AT A)-1. Thus, (AT A)-1 AT A = I.Where I is the identity matrix. So we get P = A(AT A)-1 A".
Now, the value of e can be calculated as: Oe = b - Pe = b - A(AT A)-1 A" b = A x (AT A)-1 x AT b
This is the expression for the solution of the least square problem and if A is invertible, we can find the solution by directly calculating A-1 x b which is nothing but e. Thus, the value of e is e = A-1b.
Substituting the given expression of e, we get e = (AT A)-1 AT b.
Thus, the correct answer is e = (AT A)-1 AT b.
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Hector has a toy train that is 68 centimeters long. He puts a 23 centimeters caboose at the end. How long is the train with the caboose?
Answer: 91 centimeters
Step-by-step explanation:
Answer:
91
Step-by-step explanation:
68+23=91
Find the SUM of the perfect square roots that fall between the square root of 26 and the square root of 70.
Answer:
yo is your math class teacher named wicker? I think I'm ur classmate lol
if given a radius of 6mm, what is the diameter?
if given a diameter of 22ft, what is the radius?
Answer:
If the radius is 6mm, the diameter is 12mm.
If the diameter is 22 ft, the radius is 11 ft.
Step-by-step explanation:
1st one) When you try to find the diameter of any circle, and you already have the radius, you need to multiply it by two (double it)
6 x 2 = 12mm
2nd one) Then to do the opposite, you need to divide the diameter by 2.
22/2 = 11 ft
Hope this helped :)
5. Find the circumference of a circle with a radius
of 15 feet.
Answer:
94.25 feet.
Step-by-step explanation:
At a blood drive, 4 donors with type 0 + blood, 4 donors with type A+ blood, and 3 donors with type B + blood are in line. In how many distinguishable ways can the donors be in line? The donors can be in ____ line in different ways.
The number of ways to arrange the 11 donors in line is 11!. 11! = 39,916,800.
The donors can be in line in different ways.
To calculate the number of distinguishable ways, we can use the concept of permutations. Since all the donors are distinct (different blood types), we need to find the total number of permutations of these donors.
The total number of donors is 4 (type O+), 4 (type A+), and 3 (type B+), giving a total of 11 donors.
The number of ways to arrange these donors in line can be calculated using the formula for permutations. The formula for permutations of n objects taken all at a time is n!.
Therefore, the number of ways to arrange the 11 donors in line is 11!.
Calculating 11!, we get:
11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 39,916,800.
Hence, the donors can be in line in 39,916,800 different ways.
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Given a population with standard deviation 8. how large a random sample should you take so that the probablity is 0.8664 that the sample mean is within 0.8 of the population mean
.
we need a sample size of at least 248 to have a probability of 0.8664 that the sample mean is within 0.8 of the population mean.
Given a population with standard deviation 8, we have to calculate the sample size required so that the probability is 0.8664 that the sample mean is within 0.8 of the population mean.To solve the problem, we have to use the formula as follows:$$n = \frac{z^2\sigma^2}{d^2}$$
Where, n = sample sizeσ = population standard deviation d = precision level z = z-score
So, z can be found using the standard normal table. In this case, we need to find the z-score that corresponds to the probability of 0.8664 plus half of the remaining probability of 1 - 0.8664, which is equal to 0.0668.Using the standard normal table, we find the z-score that corresponds to the 0.9334 probability, which is 1.48 (approximately).Now, we can substitute all the values into the formula and solve for n.$$n = \frac{z^2\sigma^2}{d^2}$$$$n = \frac{(1.48)^2 \cdot 8^2}{(0.8)^2}$$$$n = 247.15$$
Therefore, we need a sample size of at least 248 to have a probability of 0.8664 that the sample mean is within 0.8 of the population mean.
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The needed sample size is given as follows:
n = 250.
How to use the normal distribution?We first must use the z-score formula, as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
In which:
X is the measure.[tex]\mu[/tex] is the population mean.[tex]\sigma[/tex] is the population standard deviation.The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, and can be positive(above the mean) or negative(below the mean).
The z-score table is used to obtain the p-value of the z-score, and it represents the percentile of the measure represented by X in the distribution.
By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation given by the equation presented as follows: [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
The p-value of the z-score in this problem is given as follows, considering the symmetry of the normal distribution:
0.5 + 0.8864/2 = 0.9432.
Hence the z-score is given as follows:
z = 1.58.
Then the sample size is obtained as follows:
[tex]1.58 = \frac{0.8}{\frac{8}{\sqrt{n}}}[/tex]
[tex]\sqrt{n} = 15.8[/tex]
n = 15.8²
n = 250.
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HELPP ME PLSSSS NO BOTS OR I WILL REPORT YOUU!!
Answer:
True
Step-by-step explanation:
It pases vertical line test but does not have an inverse
Can someone explain why the statement is true. Will Mark brainliest.
Answer:
The triangle is isosceles.
Step-by-step explanation:
This means that one angle's sine is the same as the other's cosine.
Verify the equation: (tan x - 1)/(tan x + 1) = (1 - cot x)/(1 + cot x)
Answer:
True.
Step-by-step explanation:
given equation: (tan x - 1)/(tan x + 1) = (1 - cot x)/(1 + cot x)
1. manipulate the right side by using trigonometric identities
(tan(x) - 1)/(tan(x) + 1) = (-cos(x) + sin(x))/(cos(x) + sin(x))
2. manipulate the right side by using trigonometric identities
(-cos(x) + sin(x))/(cos(x) + sin(x)) = (-cos(x) + sin(x))/(cos(x) + sin(x))
Both sides of the equation are now equal -> (tan x - 1)/(tan x + 1) = (1 - cot x)/(1 + cot x) is true.
Marcus changed jobs after college. His old salary was $48000 per year. Now his new salary is 37% more per year. What is his new salary?
Answer:
65,760
Step-by-step explanation:
move decimal over two places to make a percent a decimal so .37
then multiply that times the 48000 to get 17,760 and then add that to the original 48000 to get 65,760
how can you write a quadratic function in
standard form, given its vertex form?
If a test is worth 90 points and I got 12/15 questions right then what would my grade be?
Answer: 80
Step-by-step explanation:
Particle size is a very important property when working with paints. Take 13 measurements of a population of paint cans that have a population standard deviation of 200 angstroms, and find a sample mean of 3978.1 angstroms, construct a 98% confidence interval for the average size of particles in the population. and then answer the following;
confidence coefficient
a.2.09
b.1.65
c.1.96
D.2.33
The confidence coefficient for a 98% confidence interval is 2.33, indicating the number of standard deviations away from the mean.
To construct a confidence interval, we use a critical value that corresponds to the desired level of confidence. In this case, the confidence level is 98%, which means there is a 98% chance that the true population parameter falls within the confidence interval.
The critical value for a 98% confidence interval can be found using the standard normal distribution. Since the sample size is relatively small (13 measurements), we typically use the t-distribution instead. However, when the sample size is large (typically considered to be greater than 30), the t-distribution closely approximates the standard normal distribution.
For a 98% confidence level, the critical value is 2.33. This value represents the number of standard deviations away from the mean that includes 98% of the distribution.
Therefore, the correct answer is (D) 2.33 as the confidence coefficient for a 98% confidence interval.
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plz just give the eqution.
Answer:
16 = 3x
Step-by-step explanation:
It is an equilateral triangle. The formula for the perimeter of an equilateral triangle is P = 3a.
3X IS ANSWER AND IT IS SIMPLE BECAUSE P= 3X
The expression represents a polynomial with terms. The constant term is , the leading term is , and the leading coefficient is .
There is no polynomial attached ; considering the hypothetival
Answer:
x³ = leading term.
2 = leading Coefficient
1 = constant term.
Step-by-step explanation:
Considering an hypothetical situation ;
For any polynomial such as 2x³ + 3x + 1
The polynomial is represented as : ax³ + bx + c
Where a = leading Coefficient ;
x³ = leading term
c = constant term
Therefore ;
The leading term in this hypothetical question is :
x³ = leading term.
2 = leading Coefficient
1 = constant term.
IQs are known to be normally distributed with mean 100 and standard deviation 15. (a) What percentage of people have an IQ lower than 91? ? (b) Fill in the blank. 75% of the population have an IQ that is greater than Problem #7(a): Enter your answer as a percentage, correct to 2 decimals, without the % sign. e.g., 28.31 Problem #7(b): answer correct to 2 decimals
75% of the population has an IQ greater than 89.95.
(a)What percentage of people have an IQ lower than 91?The given distribution is the normal distribution, with the mean 100 and standard deviation 15. It is required to calculate the percentage of people having an IQ score lower than 91.
To calculate the percentage of people having an IQ score lower than 91, standardize the given IQ score of 91 using the formula of z-score.z=(x−μ)/σwherez is the standardized score,x is the raw score,μ is the mean, andσ is the standard deviation.
The values can be substituted as follows.z=(91−100)/15=−0.6Now, find the probability of having a z-score less than or equal to -0.6 using the standard normal distribution table.
The value in the table is 0.2743, which means the probability of having a z-score less than or equal to -0.6 is 0.2743.Thus, 27.43% of people have an IQ score lower than 91.
(a) 27.43% of people have an IQ lower than 91.(b)Fill in the blank. 75% of the population have an IQ that is greater than X.
In order to find X, the z-score can be calculated using the formula of z-score.z=(x−μ)/σwherez is the standardized score,x is the raw score,μ is the mean, andσ is the standard deviation.
The z-score for the given problem can be calculated as follows:z = (x - μ)/σ (standardized score formula)z = (x - 100)/15 (values substituted)To find the value of x for which 75% of the population have an IQ greater than x, we need to determine the z-score that corresponds to the 25th percentile.
This is because 75% of the population is above the 25th percentile and below the 100th percentile.Using a standard normal distribution table, we can find the z-score that corresponds to the 25th percentile. The z-score is approximately -0.67.
Now that we have the z-score, we can solve for x as follows.-0.67 = (x - 100)/15 (substitute z-score)-10.05 = x - 100 (multiply both sides by 15)-89.95 = x
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Rudy makes 4 baskets out of every 10 attempts. At this rate, how many baskets would he score in a season when he attempted 75 shots?
What is the area of one of the triangular faces? in 2 4 in. 3 in. 7 in. 7 in. 5 in. 4 in. 5 in.
Answer:
55
Step-by-step explanation:
Two angles are supplementary. One angle measures 132 degrees. Find the measure of the other angle.
Answer:
48 degrees
Step-by-step explanation:
Supplementary angles add up to 180 degrees.
If one angle is 132 degrees, the other must be 48 degrees.
if you had 4 quarters and 8 nickels how much money would you have?
Answer:
$1.40
Step-by-step explanation:
4 quarters
1 quarter = 25 cents
Therefore, 4 quarters = 1/25 x 4/x
multiply 4 by 25, since 1 quarter = 25 cents
4 x 25 = 100
4 quarters = 100 cents
which equals $1
8 nickels
1 nickle = 5 cents
Therefore, 8 nickles = 1/5 x 8/x
multipy 8 by 5, since 1 nickel = 5cents
8 x 5 = 40
8 nickels = 40 cents
So add both 40 cents and 100 cents, which equals 140 cents.
But you still have to change the cents to dollars.
Which is 100 cents = $1
Add 40 cents
= $1.40
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Answer:
Only the mean increased.
Step-by-step explanation:
The mean is the total divided by the number of scores, and since adding 21 will increase the total score significantly, the mean increases.
The median is still 7.
Pablo saves $85 per month for 6 months. Then he deposits the money in an account that earns 2.1% simple interest. How much interest will he earn over 4 years? (no links just answers)
Answer:
$42.84
Step-by-step explanation:
P = 85 * 6 = 510
Formula:
I = Prt
Given:
P = 510
r = 2.1% or 0.021
t = 4
Work:
I = Prt
I = 510(0.021)(4)
I = 42.84
Please help.
Is algebra.
Answer:
#4 is B
#5 is also B
Step-by-step explanation:
i big brain
Answer:
4. b) [tex]4x^2-20xy+25y^2[/tex]
5. b) [tex]x^2+14x+49[/tex]
Step-by-step explanation:
4. [tex](2x-5y)^2[/tex]
First, one must rewrite the exponential equation as a multiplication problem,
[tex](2x-5y)(2x-5y)[/tex]
Now distribute, multiply every term in one of the parenthesis by every term in the other parenthesis,
[tex]=(2x-5y)(2x-5y)\\\\=(2x)(2x)+(2x)(-5y)+(-5y)(2x)+(-5y)(-5y)[/tex]
Now simplify the given expression,
[tex]=(2x)(2x)+(2x)(-5y)+(-5y)(2x)+(-5y)(-5y)\\\\=4x^2-10xy-10xy+25y^2[/tex]
Combine like terms,
[tex]=4x^2-20xy+25y^2[/tex]
5.[tex](x+7)^2[/tex]
To solve this problem, one should follow the same series of steps as they did to solve the last expression. First, rewrite the exponential expression as a multiplication problem.
[tex](x+7)(x+7)[/tex]
Now distribute, multiply every term in one of the parenthesis by every term in the other parenthesis,
[tex]=(x)(x)+(7)(x)+(7)(x)+(7)(7)[/tex]
Simplify the expression,
[tex]=(x)(x)+(7)(x)+(7)(x)+(7)(7)\\\\=x^2 + 7x + 7x + 49[/tex]
Finally, combine like terms,
[tex]=x^2+14x+49[/tex]
Does the following improper integral converge or diverge? Show your reasoning. 1 те dax (b) Apply an appropriate trigonometric substitution to confirm that san 4V1 - 22 d. = = (c) Find the general solution to the following diff ential equation. dy (22+-2) dc 3, 7-2, 1
(a)The power is 1/2, which is less than 1 the improper integral ∫(1 / √(x)) dx from a to infinity diverges.
b)The value of the integral of ∫(1 / √(x)) dx from 1 to infinity is ln(√2 + 1).
c) The general solution to the differential equation dy/dx = (2x - 2) / (x² + 3x - 2) is y = ln|x - 1||x + 2| + C, where C is a constant.
To determine if the improper integral converges or diverges, to evaluate the integral:
∫(1 / √(x)) dx from a to infinity
This integral represents the area under the curve of the function 1/√(x) from x = a to x = infinity.
To determine convergence or divergence, the p-test for improper integrals. For the p-test, the power of x in the denominator, which is 1/2.
If the power is greater than 1, the integral converges. If the power is less than or equal to 1, the integral diverges.
To confirm the result using a trigonometric substitution, let's substitute x = tan²(t):
√(x) = √(tan²(t)) = tan(t)
dx = 2tan(t)sec²(t) dt
substitute these values into the integral:
∫(1 / √(x)) dx = ∫(1 / tan(t))(2tan(t)sec²(t)) dt
= ∫2sec(t) dt
To determine the limits of integration. Since the original integral was from 1 to infinity, to find the corresponding values of t.
When x = 1, tan²(t) = 1, which implies tan(t) = ±1. the positive value because dealing with positive values of x.
tan(t) = 1 when t = π/4
The integral with the appropriate limits:
∫(1 / √(x)) dx = ∫2sec(t) dt from t = 0 to t = π/4
Evaluating the integral:
∫2sec(t) dt = 2ln|sec(t) + tan(t)| from t = 0 to t = π/4
Plugging in the limits:
2ln|sec(π/4) + tan(π/4)| - 2ln|sec(0) + tan(0)|
ln(√2 + 1) - ln(1)
ln(√2 + 1)
The given differential equation is:
dy/dx = (2x - 2) / (x^2 + 3x - 2)
To find the general solution, by factoring the denominator:
dy/dx = (2x - 2) / [(x - 1)(x + 2)]
decompose the fraction into partial fractions:
dy/dx = A/(x - 1) + B/(x + 2)
To find the values of A and B, both sides of the equation by the denominator (x - 1)(x + 2):
2x - 2 = A(x + 2) + B(x - 1)
Expanding the right side and collecting like terms:
2x - 2 = Ax + 2A + Bx - B
Matching the coefficients of x and the constant terms on both sides, the following system of equations:
A + B = 2 (coefficient of x)
2A - B = -2 (constant term)
Solving this system of equations, A = 1 and B = 1.
Substituting these values back into the partial fraction decomposition:
dy/dx = 1/(x - 1) + 1/(x + 2)
integrate both sides with respect to x:
∫ dy = ∫ (1/(x - 1) + 1/(x + 2)) dx
Integrating each term separately:
y = ln|x - 1| + ln|x + 2| + C
Combining the logarithmic terms using properties of logarithms:
y = ln|x - 1||x + 2| + C
This is the general solution to the given differential equation.
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find the area of the following figure round to the nearest
Answer:
lets divide the figure into two parts.
triangle base=2ft+2ft+6ft
triangle base=10ft
area of triangle = 1/2×base×height
area of triangle = 1/2×10ft×12ft
area of triangle =60ft²
area of square=side²
area of square=(6ft)²
area of square=36ft²
The table below shows the score
Analisa and Luke earned on four science
projects.
Science Project Scores
Project Analisa Luke
95 90
2
81 84
3
76 95
4
88 91
5
2
?
Analisa and Luke worked on a fifth
science project together. They each
carned the same score on the project.
When the fifth score is included in the
table, Analisa's mean score does not
change
Which of the following statements
describes how Luke's mean score
changes when the fifth score is included
in the table?
Answer:
It increases by 2.5
Step-by-step explanation:
yeah
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Answer:
A would be the answer I would choose
What is the Surface Area of the Triangular Prism below?
Answer:
c ) 2480 cm²
Step-by-step explanation:
Surface Area of a Triangular prism =
S = bh + lb + 2ls
b = 30cm
h = 8cm
s = 17 cm
l = 35cm
The surface area = 30cm × 8 cm + 35 × 30 cm + 2(35 × 17)
= 240 cm² + 1050 cm² + 1190 cm²
= 2480 cm²
Option c is the correct option