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(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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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

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

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 :)

Answer:

yo is your math class teacher named wicker? I think I'm ur classmate lol

Answer:

94.25 feet.

Step-by-step explanation:

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.

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

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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Answer:

True

Step-by-step explanation:

It pases vertical line test but does not have an inverse

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.

Answer: 80

Step-by-step explanation:

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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The approximation of the integral I is   -5.1941 using the composite Trapezoidal rule with n = 4.

We need to divide the interval [0, 2] into subintervals and apply the Trapezoidal rule to each subinterval.

The formula for the composite Trapezoidal rule is given by:

I = (h/2) × [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]

Where:

h = (b - a) / n is the subinterval width

f(xi) is the value of the function at each subinterval point

In this case, n = 4, a = 0, and b = 2. So, h = (2 - 0) / 4 = 0.5.

Now, let's calculate the approximation:

[tex]f\left(x_0\right)\:=\:f\left(0\right)\:=\:\left(0\:-\:3\right)e^{\left(0^2\right)}\:=\:-3[/tex]

[tex]f\left(x_1\right)\:=\:f\left(0.5\right)\:=\:\left(0.5\:-\:3\right)e^{\left(0.5^2\right)}\:=-2.535[/tex]

[tex]f\left(x_2\right)\:=\:f\left(1\right)\:=\:\left(1\:-\:3\right)e^{\left(1^2\right)}\:=\:-1.716[/tex]

[tex]f\left(x_3\right)\:=\:f\left(1.5\right)\:=\:\left(1.5\:-\:3\right)e^{\left(1.5^2\right)}\:=\:-1.051[/tex]

[tex]f\left(x_4\right)\:=\:f\left(2\right)\:=\:\left(2\:-\:3\right)e^{\left(2^2\right)}\:=\:-0.065[/tex]

Now we can plug these values into the composite Trapezoidal rule formula:

I = (0.5/2) × [-3 + 2(-2.535) + 2(-1.716) + 2(-1.051) + (-0.065)]

= (0.25)× [-3 - 5.07 - 3.432 - 2.102 - 0.065]

= -5.1941

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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

By using the method of variation of parameters to solve a nonhomogeneous DE with W = -3 W2 = e 112 and W = er, we have

Given: W1=-3, W2=e^t and W3=er.The general solution of the non-homogeneous differential equation, y" + p(t) y' + q(t) y = g(t) , where p(t) and q(t) are functions of t and g(t) is non-zero function is given by;{eq}y = y_c + y_p {/eq}Where {eq}y_c {/eq} is complementary function and {eq}y_p {/eq} is particular function obtained by using variation of parameters.The solution is as follows:The given differential equation is{eq}y''+3y'+2y=-3e^{-t}+e^{t}+re^t{/eq}Characteristic equation is{eq}m^2+3m+2=0{/eq}Solving above equation gives us, {eq}m=-1,-2{/eq}Therefore, complementary function {eq}y_c=c_1e^{-t}+c_2e^{-2t} {/eq}Now, we find the particular solution by using the method of variation of parameters.Let {eq}y_p=u_1e^{-t}+u_2e^{-2t}{/eq}be a particular solution where {eq}u_1{/eq} and {eq}u_2{/eq} are functions of {eq}t.{/eq}Here W is a Wronskian and is given as:{eq}W=\begin{vmatrix}W_1&W_2\\W_1'&W_2'\\\end{vmatrix}=\begin{vmatrix}-3&e^t\\-1&e^t\\\end{vmatrix}=2e^{2t}+3e^{t}{/eq}Now, we find {eq}u_1{/eq} and {eq}u_2{/eq} as follows:{eq}u_1=\frac{-\int W_2 g(t) dt}{W}=\frac{-\int e^t(-3e^{-t}+e^{t}+re^t)dt}{2e^{2t}+3e^{t}}=-\frac{r}{5}-\frac{7}{10}+\frac{3}{10}e^{t}{/eq}Similarly,{eq}u_2=\frac{\int W_1 g(t) dt}{W}=\frac{\int -3e^{-t}(-3e^{-t}+e^{t}+re^t)dt}{2e^{2t}+3e^{t}}=-\frac{r}{5}-\frac{1}{10}+\frac{3}{10}e^{-2t}{/eq}Hence, the general solution of the differential equation is {eq}y=y_c+y_p=c_1e^{-t}+c_2e^{-2t}-\frac{r}{5}-\frac{7}{10}+\frac{3}{10}e^{t}-\frac{r}{5}-\frac{1}{10}+\frac{3}{10}e^{-2t}{/eq}So, Option D, {eq}-4u_2=0,~u_1=-\frac{r}{5}-\frac{7}{10}+\frac{3}{10}e^{t},~u_2=-\frac{r}{5}-\frac{1}{10}+\frac{3}{10}e^{-2t}{/eq} is correct.

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The last one as we can see that as the wight increases, the miles per gallon decreases.

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

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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Answer:

55

Step-by-step explanation:

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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Answer: 91 centimeters

Step-by-step explanation:

Answer:

91

Step-by-step explanation:

68+23=91

An isometry on R^n must be regular and the set of isometries forms a group under matrix multiplication.

An isometry is a linear transformation that preserves distances, meaning the norm of the transformed vector is equal to the norm of the original vector. To show that an isometry must be regular (i.e., invertible), we can assume there exists a non-invertible isometry matrix A. In this case, there exists a nonzero vector x such that Ax = 0. However, this contradicts the property of an isometry since ||Ax|| = ||0|| = 0, but ||x|| ≠ 0. Thus, an isometry must be regular.

The set of isometries forms a group under matrix multiplication because it satisfies the group axioms: closure (the product of two isometries is an isometry), associativity (matrix multiplication is associative), identity (identity matrix is an isometry), and inverses (the inverse of an isometry is also an isometry).

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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]

Answer:

The triangle is isosceles.

Step-by-step explanation:

This means that one angle's sine is the same as the other's cosine.

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.

We first must use the z-score formula, as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

In which:

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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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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Answer:

y=2x+5

Step-by-step explanation:

u find the slope and since the y intercept is (0,5), that's the b value

hope this helps

Answer: 2x+5

Step-by-step explanation: calculate the slope of the points with the formula m= (y2-y1)/(x2-x1) then use the y-int to complete the equation

Answer:

Quotient: p + 7

Remainder: 10

Step-by-step explanation:

To find the quotient of the expression (2p² + 7p - 39) ÷ (2p - 7), we can use long division or synthetic division. Let's use long division:

           ____________________

2p - 7  |   2p² + 7p - 39

We start by dividing the first term of the dividend by the first term of the divisor, which gives us 2p² ÷ 2p = p. We then multiply p by the divisor (2p - 7) and subtract it from the dividend:

         p

    ____________________

2p - 7  |   2p² + 7p - 39

        - (2p² - 7p)

       14p - 39

    ____________________

2p - 7  |   2p² + 7p - 39

        - (2p² - 7p)

        ___________

               14p - 39

We repeat the process by dividing the first term of the new dividend (14p - 39) by the first term of the divisor (2p - 7). This gives us (14p - 39) ÷ (2p - 7) = 7. We then multiply 7 by the divisor (2p - 7) and subtract it from the new dividend:

         p + 7

    ____________________

2p - 7  |   2p² + 7p - 39

        - (2p² - 7p)

        ___________

               14p - 39

        - (14p - 49)

        ___________

                10

We are left with a remainder of 10. Therefore, the quotient is p + 7 with a remainder of 10.

Quotient: p + 7

Remainder: 10

Hope this helps!

p + 7 should be it.

I am not 100% sure?

Answer:

20.00

Step-by-step explanation: