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The integral representation of the solution to the initial value ordinary differential equation (ODE) x'' - 8x' + 12x = f(t) with f(t) is given by x(t) = x₀ * δ(t) + x₁ * δ'(t) + ∫[a,b] G(t - τ) * f(τ) dτ.

The given ODE is a linear homogeneous second-order ODE with constant coefficients. To find the integral representation of the solution, we introduce the Dirac delta function, δ(t), and its derivative, δ'(t), as the basis for the particular solution.

To set up the integral representation for the solution of the initial value ODE x'' - 8x' + 12x = f(t), we first define the Green's function G(t - τ). The Green's function satisfies the homogeneous equation with the right-hand side equal to zero:

G''(t - τ) - 8G'(t - τ) + 12G(t - τ) = 0.

Next, we set up the integral representation as follows:

x(t) = x₀ * δ(t) + x₁ * δ'(t) + ∫[a,b] G(t - τ) * f(τ) dτ,

The integral represents the convolution of the forcing function f(τ) with the Green's function G(t - τ).

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Answer:

2w+2l.

2(7)+2(5).

14+10.

24.

The length of the radius of the cone is approximately [tex]6.74[/tex] feet.

To find the length of the radius of the cone, we can use the formula for the volume of a cone:

[tex]\[\text{{Volume}} = \frac{1}{3} \pi r^2 h\][/tex]

The concept used to find the length of the radius is the formula for the volume of a cone.

Given that the volume is [tex]640[/tex] ft³ and the height (h) is [tex]12[/tex] ft, we can substitute these values into the formula:

[tex]\[640 = \frac{1}{3} \times 3.14 \times r^2 \times 12\][/tex]

Simplifying the equation:

[tex]\[\frac{640}{12 \times \frac{1}{3} \times 3.14} = r^2\]\[r^2 = \frac{640}{12 \times \frac{1}{3} \times 3.14}\]\[r^2 \approx 45.45\][/tex]

Taking the square root of both sides, we find:

[tex]\[r \approx \sqrt{45.45} \approx 6.74\][/tex]

Rounding the answer to the nearest hundredth, the length of the radius is approximately [tex]6.74[/tex] feet.

In conclusion, the length of the radius of the given cone, with a volume of [tex]640[/tex] ft³ and a height of [tex]12[/tex] feet, is approximately [tex]6.74[/tex] feet (rounded to the nearest hundredth).

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The probability that a randomly dealt 5-card hand contains 2 kings and 3 cards that aren't

kings is approximately 0.0399 or 3.99%.

To find the probability of randomly dealing a 5-card hand containing 2 kings and 3 cards that aren't kings, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

The number of ways to choose 2 kings from the 4 available kings is given by the combination formula:

C(4, 2) = 4! / (2! * (4-2)!) = 6

Similarly, the number of ways to choose 3 non-king cards from the remaining 48 cards (52 cards total - 4 kings) is:

C(48, 3) = 48! / (3! * (48-3)!) = 17,296

Therefore, the number of favorable outcomes (hands with 2 kings and 3 non-king cards) is:

6 * 17,296 = 103,776

The total number of possible 5-card hands that can be dealt from a standard deck of 52 cards is:

C(52, 5) = 52! / (5! * (52-5)!) = 2,598,960

So, the probability of randomly dealing a 5-card hand containing 2 kings and 3 cards that aren't kings is:

P(2 kings and 3 non-kings) = favorable outcomes / total outcomes = 103,776 / 2,598,960 ≈ 0.0399

Therefore, the probability is approximately 0.0399 or 3.99%.

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The sampling distribution of x is N (µx = µ = 81, σx = 2.00).The probability of x > 84.9 is:P(x > 84.9) = P(z > 1.75) = 0.0401.The probability of 78.1 < x < 81 is:P(78.1 < x < 81) = P(-0.95 < z < 0) = 0.3289.

a)Sampling distribution of x

The sampling distribution of x is the probability distribution of all the possible sample means that can be drawn from a population under the same sampling method.

It represents the relative frequency of different values of x (sample mean) that can be obtained when samples of size n are taken from the population.

The sampling distribution of x is approximately normal when the sample size is sufficiently large, i.e. n ≥ 30. In this case, n = 49, which is sufficiently large to assume normality of sampling distribution of x.

The mean of the sampling distribution of x is µx = µ = 81, and the standard deviation is: σx = σ / √n = 14 / √49 = 2.00.

Hence, the sampling distribution of x is N (µx = µ = 81, σx = 2.00).

b)P(x > 84.9)

The z-score is:z = (x - µx) / σx = (84.9 - 81) / 2.00 = 1.75.

Using the standard normal distribution table, the probability of z > 1.75 is 0.0401.

Hence, the probability of x > 84.9 is:P(x > 84.9) = P(z > 1.75) = 0.0401

c)P(x ≤ 76.7)

The z-score is:z = (x - µx) / σx = (76.7 - 81) / 2.00 = -2.15

Using the standard normal distribution table, the probability of z ≤ -2.15 is 0.0150.

Hence, the probability of x ≤ 76.7 is:P(x ≤ 76.7) = P(z ≤ -2.15) = 0.0150d)P(78.1 < x < 81)

The z-score for x = 78.1 is:z1 = (x1 - µx) / σx = (78.1 - 81) / 2.00 = -0.95

The z-score for x = 81 is:z2 = (x2 - µx) / σx = (81 - 81) / 2.00 = 0

Using the standard normal distribution table, the probability of z1 < z < z2 is:P(z1 < z < z2) = P(-0.95 < z < 0) = P(z < 0) - P(z < -0.95) = 0.5000 - 0.1711 = 0.3289.

Hence, the probability of 78.1 < x < 81 is:P(78.1 < x < 81) = P(-0.95 < z < 0) = 0.3289.

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Answer:

0.65

Step-by-step explanation:

Just divide 13 by 20

The volume of a triangular prism with a base 8 cm and height of 4 cm, length of 20 cm = 0.32 liters.

To calculate the volume of a triangular prism, you multiply the area of the base triangle by the length of the prism. Given that the base triangle has a side length of 8 cm and a height of 4 cm, its area can be calculated as (1/2) * base * height = (1/2) * 8 cm * 4 cm = 16 cm².

Multiplying this by the length of the prism, which is 20 cm, we get the volume:

Volume = Base Area * Length = 16 cm² * 20 cm = 320 cm³.

To convert this volume to liters, we know that 1 liter is equal to 1000 cm³. Therefore, we can divide the volume in cm³ by 1000 to obtain the volume in liters:

Volume in liters = 320 cm³ / 1000 = 0.32 liters.

So, the volume of the triangular prism is 0.32 liters.

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The sample variance is 10 and the standard deviation is 3.2.

Given data: 17, 10, 4, 8, 11

The sample size, n = 5

Mean (m) = ∑x / n

m = (17 + 10 + 4 +8 + 11)/5

m = 50/5

m = 10

x           x-m             (x- m)²

17      17-10 = 7         49

10      10-10 = 0        0

4       4-10 = -6         36

8       8 - 10 = -2        4

11       11 - 10 = 1          1

                                90

∑(x- m)² = 90

Sample variance, s² = ∑(x-x)² /(n-1)

Sample variance, s² = 90/(10 - 1)

                                 = 90/9

                                 = 10

Standard deviation (S) = √variance

Standard deviation (S) = √10 = 3.2

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Answer:

Debbie should have included 8 as a possible choice. The correct inequality is p < 9.

Step-by-step explanation:

Given

Passengers = Up to 8

Required

Determine why [tex]p < 8[/tex] is incorrect and make corrections

The inequality [tex]p < 8[/tex] means that the van can hold less than 8 passengers.

To make correction, the digit 8 has to be included in the inequality.

This can be written as:

[tex]p <9[/tex] or[tex]p \le 8[/tex]

Base on the given options, option (c) best answered the question.

Answer:

21

Step-by-step explanation:

Answer:

The slope of the line is 2/3

Answer: 1.5

Step-by-step explanation:

I plugged it in STAT, but you can do rise over run, or use the formula, y2-y2/x2-x1

Answer:

x ≤ 21

Step-by-step explanation:

x/3 ≤ 7

x ≤ 21

Answer:

x ≤ 21

Step-by-step explanation:

With this problem, we have to solve for x, and to do that, we isolate the variable.

To do this, let’s get rid of the /3 from the x by multiplying both sides by 3

now we get x ≤ 7 times 3

finall, we get x≤21

Sherman's total return on his investment in Boca-Cola stock is $5,430.

The formula for the total return on an investment is as follows:

total return = capital gain + dividend yield

Initially, Sherman bought 150 shares of Boca-Cola stock for $15 a share.

Therefore, the initial investment (also known as the initial cost) is:

$15 x 150 = $2,250

Four years later, the stock price of Boca-Cola is $50 per share.

The capital gain is calculated as follows:

capital gain = final share price - initial share price

capital gain = $50 - $15

capital gain = $35

Therefore, the capital gain on Sherman's 150 shares is:

$35 x 150 = $5,250

Next, we need to calculate the total amount of dividends that Sherman received over the 4 years. The dividend per share is $0.30. Therefore, the total amount of dividends received is:

total dividends = dividend per share x number of shares x number of years

Sherman received dividends for 4 years, so:

total dividends = $0.30 x 150 x 4

total dividends = $180

The dividend yield is calculated as follows:

dividend yield = total dividends / initial cost

dividend yield = $180 / $2,250

dividend yield = 0.08 or 8%

Finally, we can calculate the total return:

total return = capital gain + dividend yield

total return = $5,250 + $180

total return = $5,430

Therefore, Sherman's total return on his investment in Boca-Cola stock is $5,430.

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Answer:

An animal that eats dead animals or plants.

Step-by-step explanation:

Its lowkey the defintion lol

Answer:

17

Step-by-step explanation:

This means all students above 20, so 9 + 5 + 3 = 17

Answer:

17

pls mark brainliest

Answer:

3

Step-by-step explanation:

Plug the coordinates into the distance formula to find that they are 3 units apart

Answer:

13 units

Step-by-step explanation:

sqrt (x2 - x1)^2 + (y2 - y1)^2

sqrt (-3 - (-3))^2 + (-8 -5)^2

sqrt (0)^2 + (-13)^2

sqrt 169

13

From the truth table, we can see that p->q and ¬p∨q have the same truth values for all possible combinations of truth values for p and q. Therefore, we can conclude that p->q is logically equivalent to ¬p∨q. In summary, we can see that p->q is logically equivalent to both !q->!p and ¬p∨q.

To prove the logical equivalence of the given statements, we can show that they have the same truth values in all possible cases. We'll use a truth table to demonstrate this.

p | q | p->q | !q | !p | !q->!p | p->q = !q->!p

-------------------------------------------------

T | T |   T  |  F |  F |    T   |      T

T | F |   F  |  T |  F |    F   |      F

F | T |   T  |  F |  T |    T   |      T

F | F |   T  |  T |  T |    T   |      T

From the truth table, we can see that for all possible combinations of truth values for p and q, the statements p->q and !q->!p have the same truth values. Therefore, we can conclude that p->q is logically equivalent to !q->!p.

Now let's consider the second statement, p->q. We can rewrite it as ¬p∨q using the logical equivalence of implication.

The truth table for p->q and ¬p∨q is as follows:

p | q | p->q | ¬p | ¬p∨q

-----------------------------

T | T |   T  |  F |   T

T | F |   F  |  F |   F

F | T |   T  |  T |   T

F | F |   T  |  T |   T

From the truth table, we can see that p->q and ¬p∨q have the same truth values for all possible combinations of truth values for p and q. Therefore, we can conclude that p->q is logically equivalent to ¬p∨q.

In summary, we have shown that p->q is logically equivalent to both !q->!p and ¬p∨q.

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Answer:

y= 3

x= -2

Step-by-step explanation:

..,............

Answer:((2x^3)^4)/1x^2

Step-by-step explanation:

(2x^3)^4)=((2x)^4)(^3 times ^4)=16x^12

16x^12/1x^2=(16x/1x)(x^12-2)

Answer:

Sam's is nonsense and Kim's is sense

Step-by-step explanation:

why? because 1/3 and 1/6 are not equivalent. and 1/3 is greater. so therfore Kim's makes sense

Answer:

[tex]\frac{3}{1}[/tex]

Step-by-step explanation:

Put a 1 under it.

Most likely (3,0)

If it’s reflected across the y axis, then it should be the opposite of (-3,0)

Answer:

yes I guess a rectangle has opposite sides parallel parallelogram has opposite sides parallel a trapezium has opposite sides parallel a rhombus has opposite sides parallel if I'm wrong please connect me right away thank you.

Answer:

1.Yes, 2 pairs

2.Yes, 2 pairs

3.Yes, 1 pair

4.Yes, 2 pairs

Step-by-step explanation:

1.Each pair of co-interior angles are supplementary, because two right angles add to a straight angle, so the opposite sides of a rectangle are parallel. This means that a rectangle is a parallelogram, so: Its opposite sides are equal and parallel. Its diagonals bisect each other.

2.A parallelogram is a four sided figure where the opposite sides are parallel.

3.A trapezoid is a quadrilateral with one pair of opposite sides parallel. It can have right angles (a right trapezoid), and it can have congruent sides (isosceles), but those are not required.

4.Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides.

The given partial differential equation is given by; Uz(x, t) + 2xut(x, t) = 2x

The Laplace transform of Uz(x, t) + 2xut(x, t) is given as follows; L[Uz(x,t)] + 2x L[ut(x,t)] = L[2x]sU(x,s) - u(x,0) + 2x[sU(x,s)-u(x,0)] = 2x/sU(x,s) + 2x/s^2 - 1(2x/s)U(x,s) = 2x/s^2 - 1 + sU(x, s)U(x, s) = [2x/s^2 - 1]/[2x/s - s]U(x, s) = s(2x/s^2 - 1)/(2x - s^2) = s/(2x - s^2) - 1/(2(s^2 - 2x))

By using the inverse Laplace transform, we have; u(x, t) = [1/s] * e^(s^2t/2x) - (1/2)sinh(t sqrt(2x)) / sqrt(2x)

Thus, the solution to the given partial differential equation is given as follows; u(x, t) = [1/s] * e^(s^2t/2x) - (1/2)sinh(t sqrt(2x)) / sqrt(2x)Where, u(x,0) = 1 and u(0,t) = 1.

The integral transform known as the Laplace transform is particularly useful for solving ordinary differential equations that are linear. It finds extremely wide applications in var-ious areas of physical science, electrical designing, control engi-neering, optics, math and sign handling.

The mathematician and astronomer Pierre-Simon, marquis de Laplace, gave the Laplace transform its name because he used a similar transform in his work on probability theory.

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Answer:

To determine the solutions x⁵ + 2x³ = 3, you can use numerical methods or approximation techniques to estimate the values of x that satisfy the equation.

Let's solve the equation step by step to find the solutions.

Starting with the given equation:

log₃(x) + log₃(x² + 2) = 1 - 2log₃(x)

Now, let's simplify the equation using logarithmic properties. The sum of logarithms is equal to the logarithm of the product, and the difference of logarithms is equal to the logarithm of the quotient:

log₃(x(x² + 2)) = 1 - log₃(x²)

Next, we can simplify further by using the properties of exponents. The logarithmic equation can be rewritten in exponential form as:

([tex]3^{(log3(x(x^{2} +2)))} = 3^{(1-log3((x^{2}))}[/tex]

The base of the logarithm and the exponent cancel each other out, resulting in:

x(x² + 2) = [tex]3^{(1-log3(x^{2}))}[/tex]

Now, let's simplify the right-hand side by applying the power rule of logarithms:

x(x² + 2) = 3 / [tex]3^{(log3(x^{2} ))}[/tex]

Since  [tex]3^{(log3(x^{2} ))}[/tex]       is equal to x² by the definition of logarithms, the equation becomes:

x(x² + 2) = 3 / x²

Expanding the left-hand side:

x³ + 2x = 3 / x²

Multiplying through by x² to eliminate the fraction:

x⁵ + 2x³ = 3

This is a quadratic equation, which does not have a general algebraic solution that can be expressed in terms of radicals. Therefore, it is challenging to find the exact solutions analytically.

To determine the solutions, you can use numerical methods or approximation techniques to estimate the values of x that satisfy the equation.

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Answer:

its 64 because 4x16= 64

Step-by-step explanation:

Answer:

4.6

Step-by-step explanation:

3.. litteraly anything less than that

Answer:

0

Step-by-step explanation:

|4.7| = 4.7

Any number less than a positive 4.7 is less than |4.7|.

Answer:

Answer is B

Step-by-step explanation:

Answer:

The answer is B

Step-by-step explanation:

The answer is B

Answer:

it is 14

Step-by-step explanation:

Answer:

C) 14

Step-by-step explanation:

Volume of a Sphere:

V = 4/3πr³

V = 4/3(3.14)1.5³

V = 4/3(3.14)3.375

V = 14.13

14.13 ≈ 14