A manufacturer has cube shaped cardboard boxes with an exact volume of 12000 cubic inches. What is the volum of the largest sphere that can be packed inside the cube shaped box? Give you answer rounded to the nearest whole cubic inch

To solve the initial value problem using the method of Laplace transforms, we'll first take the Laplace transform of both sides of the differential equation.

Taking the Laplace transform of each term, we get:

Ly'' + 2Ly' - 3Ly = 0

Using the properties of Laplace transforms and the initial value theorem, we can write the transformed equation as:

[tex]s^2Y(s) - sy(0) - y'(0) + 2sY(s) - 2y(0) - 3Y(s) = 0[/tex]

Substituting the initial conditions y(0) = 2 and y'(0) = 18, we have:

[tex]s^2Y(s) - 2s - 18 + 2sY(s) - 4 - 3Y(s) = 0[/tex]

Grouping similar terms, we obtain:

[tex](s^2 + 2s - 3)[/tex]Y(s) = 24 + 2s

Now, we can solve for Y(s) by dividing both sides by ([tex]s^2 + 2s - 3)[/tex]

Y(s) = (24 + 2s) /[tex](s^2 + 2s - 3)[/tex]

To find the inverse Laplace transform and obtain the solution y(t), we need to factor the denominator of the expression on the right-hand side:

s^2 + 2s - 3 = (s + 3)(s - 1)

We can rewrite the expression for Y(s) as:

Y(s) = (24 + 2s) / [(s + 3)(s - 1)]

Now, we need to perform partial fraction decomposition to simplify the expression. We write:

Y(s) = A / (s + 3) + B / (s - 1)

Multiplying both sides by (s + 3)(s - 1) to clear the denominators, we get:

24 + 2s = A(s - 1) + B(s + 3)

Expanding and collecting like terms, we have:

24 + 2s = (A + B)s + (3B - A)

To match the coefficients on both sides of the equation, we equate the coefficients of s and the constants:

A + B = 2 (coefficient of s)

3B - A = 24 (constant term)

Solving this system of equations, we find A = 5 and B = -3.

Now, we can rewrite Y(s) as:

Y(s) = 5 / (s + 3) - 3 / (s - 1)

Taking the inverse Laplace transform of Y(s), we can use the table of Laplace transforms or known formulas to find the solution y(t):

y(t) = 5e^(-3t) - 3e^t

Therefore, the solution to the initial value problem is:

[tex]y(t) = 5e^(-3t) - 3e^t[/tex]

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The limits are,

(a) lim(x→0) 4x/(x² + 1) = 0

(b) lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1) = ((1 + √(5))(3 - i))/10

(a) To find the limit of lim(x→0) 4x/(x² + 1), we can directly substitute 0 for x in the expression:

lim(x→0) 4x/(x² + 1) = (4 × 0)/(0² + 1) = 0/1 = 0

Therefore, the limit is 0.

(b) To find the limit of lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1), we can again substitute -1 for z in the expression:

lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1) = (1 + sqrt(2 - 2(-1) + (-1)^2))/(2 - i(-1) - 1)

= (1 + √(2 + 2 + 1))/(2 + i + 1)

= (1 + √(5))/(3 + i)

To simplify this expression further, we need to rationalize the denominator. We can multiply the numerator and denominator by the conjugate of the denominator, which is (3 - i):

lim(z→-1) (1 + √(5))/(3 + i) × (3 - i)/(3 - i)

= ((1 + √(5))(3 - i))/(9 - i²)

= ((1 + √(5))(3 - i))/(9 + 1)

= ((1 + √(5))(3 - i))/10

Therefore, the limit is ((1 + √(5))(3 - i))/10.

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The question is -

Find the following limits:

(a) lim(x->0) 4x/(x^2 + 1)

(b) lim(z->-1) (1 + sqrt(2 - 2z + z^2))/(2 - iz - 1)

Answer: $93649

Step-by-step explanation:

Since this is an exponential growth problem, then we can use the equation 50,000(1.04)^16. Solve it and you get 93649.06228. Round to the nearest dollar, which is probably whole number, so it is 93649.

Answer:

Half of 7 is 3.5

That would be your radius.

3.5^2 x 3.14

12.25 x 3.14 = 38.465 yd2 <--------- area

3.14 x 3.5 x 2 = 21.98yd <------- perimeter

Answer:

8,395

Step-by-step explanation:

21 x 400 = 8,400

is = x

8, 400 - 5 = 8,395

difference = -

Brainlist Pls!

Answer:

I believe the answer is (A)

*Substituting the x and y values from the table into the equation(A) will balance the right side of the equation to the left side of the equation.

Answer:

-46.677%

Step-by-step explanation:

The computation of the percent error is shown below:

As we know that

Area of the warehouse = length × width

Based on estimated values, the area is

= 400 × 150

= 60,000

And, based on actual values, the area is

= 320 × 100

= 32,000

Now the percent error is

= (32,000 - 60,000) ÷ 60,000 × 100

= -46.677%

Answer:

i got 54ft^2

Step-by-step explanation:

The total number of people surveyed is 75.

The first step is to determine the number of people who had 4 or more rides that preferred a window seat.

= Total number of people that had four or more rides - total number of people who had 4 or more rides that prefer aisle

= 40 - 25 = 15

Total number of people that prefer the window seats= 15 + 20 = 35

Total number of people = total number of people that prefer the window seat + total number of people who prefer the aisle

= 35 + 40 = 75

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Answer: your answer should be 50%

Step-by-step explanation: This is because there are only four marbles in the bag total and only 2 are blue and only 2 are green so your chances of pulling out either is 50%

Answer:

33%

Step-by-step explanation:

2 blue marbles + 2 green marbles = 4 marbles

1st draw for blue:   2/4               (2 blue marbles out of 4 marbles)

2nd draw for green:   2/3          (1 less marble from 4, marble not put back in)

2/4  x  2/3   = 4/12 = 1/3 = 0.33 or 33%

Answer:

5:8

Step-by-step explanation:

yellow:red

10:16

simplified would be 5:8

***important note, when doing ratio, make sure to list the term that is asked for first. example: it's yellow to red skittles and not red to yellow. red to yellow would be 8:5 and that would be a wrong answer, so read carefully:)

Answer:

5:8

Step-by-step explanation: you can divide 10:16 by 2 to make 5:8, and that is the simplest form.

Step-by-step explanation:

C+D=90

2x+3x=90

5x=90

X=90:5=18

Answer:

D is the answer

Step-by-step explanation:

The solution to the system of differential equations with the initial conditions x(0) = 1 and y(0) = 0 is:

x(t) = 10t - 12yt + C₁

y(t) = (1 + C₂exp(-4t)) / 2

To find the solution to the linear system of differential equations x'(t) = 10 - 12y and y'(t) = 4 - 4y, we can solve them separately.

For x'(t) = 10 - 12y:

Integrating both sides with respect to t, we have:

Now, for y'(t) = 4 - 4y:

Rearranging the equation, we have:

This is a first-order linear homogeneous differential equation. To solve it, we use an integrating factor. The integrating factor is given by exp(∫4 dt), which simplifies to exp(4t).

Multiplying both sides of the equation by the integrating factor, we get:

exp(4t) y'(t) + 4exp(4t) y(t) = 4exp(4t)

Now, we can integrate both sides with respect to t:

Integrating, we have:

Simplifying, we get:

Dividing both sides by 2exp(4t), we obtain:

Simplifying further, we have:

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The maximum value of the objective function Z is 120. The optimal values for the decision variables are X1 = 8, X2 = 0, and X3 = 0. The constraints are satisfied, and the optimal solution has been reached using the Simplex Method.

To compute the problem using the Simplex Method, let's convert it into standard form.

Maximize:

Z = 6X1 + 10X2 + 5X3

Subject to the constraints:

X1 + 2X2 + 4X3 <= 8

6X1 + 4X2 <= 24

6X1 + 5X3 <= 30

X1, X2, X3 >= 0

Introducing slack variables S1, S2, and S3 for each constraint, the constraints can be rewritten as equalities:

X1 + 2X2 + 4X3 + S1 = 8

6X1 + 4X2 + S2 = 24

6X1 + 5X3 + S3 = 30

Now, we have the following equations:

Objective function:

Z = 6X1 + 10X2 + 5X3 + 0S1 + 0S2 + 0S3

Constraints:

X1 + 2X2 + 4X3 + S1 = 8

6X1 + 4X2 + S2 = 24

6X1 + 5X3 + S3 = 30

X1, X2, X3, S1, S2, S3 >= 0

Next, we will create the initial simplex tableau:

  | X1 | X2 | X3 | S1 | S2 | S3 | RHS |

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

Z  | 6  | 10 | 5  | 0  | 0  | 0  | 0   |

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

S1 | 1  | 2  | 4  | 1  | 0  | 0  | 8   |

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

S2 | 6  | 4  | 0  | 0  | 1  | 0  | 24  |

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

S3 | 6  | 0  | 5  | 0  | 0  | 1  | 30  |

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

By performing the simplex pivot operations and iterating through the simplex method steps, we find the following tableau:

  | X1 | X2 | X3 | S1 | S2 | S3 | RHS |

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

Z  | 0  | 0  | 5  | -6 | 0  | -60| 120 |

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

X1 | 1  | 2  | 4  | 1  | 0  | 0  | 8   |

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

S2 | 0  | -8 | -24| -6 | 1  | 0  | 0   |

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

S3 | 0  | 0  | -1 | -6 | 0  | 1  | 0   |

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

The optimal solution is Z = 120, X1 = 8, X2 = 0, X3 = 0, S1 = 0, S2 = 0, S3 = 0.

Therefore, the maximum value of Z is 120, and the values of X1, X2, and X3 that maximize Z are 8, 0, and 0, respectively.

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The area of the circle with a diameter of 20 inches is 100π square inches, and the circumference of the circle is 20π inches.

To determine the area and circumference of a circle with a diameter of 20 inches, you need to use the formulas for these measures.

A circle is a set of points that are equidistant from the center point, and the diameter of a circle is the longest line that can be drawn from one point on the circle to another while passing through the center point. The formulas for the area and circumference of a circle are as follows:

A = πr²C = πd

where A is the area of the circle, C is the circumference of the circle, r is the radius of the circle, d is the diameter of the circle, and π (pi) is a mathematical constant that approximates to 3.14.

To find the area of a circle with a diameter of 20 inches, you need to find the radius of the circle first. The radius is half of the diameter, so r = d/2 = 20/2 = 10 inches. Therefore, the area of the circle is:A = πr² = π(10)² = 100π square inches (rounded to two decimal places).

To find the circumference of a circle with a diameter of 20 inches, you can either use the formula C = πd or you can use the formula C = 2πr. Since you already know the diameter, let's use the first formula. C = πd = π(20) = 20π inches (rounded to two decimal places).

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Answer:

5..

Step-by-step explanation:

Answer: One answer

Step-by-step explanation:

The statement " for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1" is proved.

If η is the Euler totient function defined by η(n)=n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk) then for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.

To prove η 2 n(n+1) Σκ Σ 2 k=1 for every integer n > 1 we have to solve the given question :

1) We know that η(n) = n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk).and

let S = Σκ Σ 2 k=1

2) For n = 2 we have η(2) = 2 * (1 - 1/2) = 1

Hence, S = Σκ Σ 2 k=1 = 1*2=2

Now, η(4) = 4 * (1 - 1/2)(1 - 1/2) = 2 and η(6) = 6 * (1 - 1/2)(1 - 1/3) = 2

Therefore, η 2 n(n+1) Σκ Σ 2 k=1

Hence, S = Σκ Σ 2 k=1 = 2* (2 + 1) * 2 = 12.

3) For n=3, we haveη(3) = 3 * (1 - 1/3) = 2S = Σκ Σ 2 k=1 = 1 * 2 + 2 * 3 = 8

Also, η(6) = 6 * (1-1/2)(1-1/3) = 2

Hence, η 2 n(n+1) Σκ Σ 2 k=1

Thus, S = Σκ Σ 2 k=1 = 2* (3 + 1) * 2 = 16

Therefore, for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.

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The inverse Laplace transform of (-5s - 36) / ((s + 2)(s²+ 9)) is [tex]-4e^{-2t}[/tex]+ (-cos(3t) + 8sin(3t))/3.

To find the inverse Laplace transforms of the given expressions, we can use partial fraction decomposition and known Laplace transform pairs. Let's solve each one step by step:

To find the inverse Laplace transform of (-2s² - 3s - 2) / (s(s + 1)²):

Step 1: Factorize the denominator:

s(s + 1)² = s(s + 1)(s + 1)

Step 2: Perform partial fraction decomposition:

(-2s² - 3s - 2) / (s(s + 1)²) = A/s + (B/(s + 1)) + (C/(s + 1)²)

Multiplying through by the common denominator, we get:

-2s² - 3s - 2 = A(s + 1)² + B(s)(s + 1) + C(s)

Expanding and equating coefficients, we find:

-2 = A

-3 = A + B

-2 = A + B + C

Solving these equations, we find: A = -2, B = 1, C = 0.

Step 3: Express the inverse Laplace transform in terms of known Laplace transform pairs:

[tex]L^{-1(-2s^{2} - 3s - 2) }[/tex]/ (s(s + 1)²) = [tex]L^{-1(-2/s)}[/tex] + [tex]L^{-1(1/(s + 1)) }[/tex]+ [tex]L^{-1(0/(s+1)^{2} }[/tex]

= -2 + [tex]e^{-t}[/tex]+ 0t[tex]e^{-t}[/tex]

Therefore, the inverse Laplace transform of (-2s² - 3s - 2) / (s(s + 1)²) is -2 + [tex]e^{-t}[/tex].

To find the inverse Laplace transform of (-5s - 36) / ((s + 2)(s² + 9)):

Step 1: Factorize the denominator:

(s + 2)(s² + 9) = (s + 2)(s + 3i)(s - 3i)

Step 2: Perform partial fraction decomposition:

(-5s - 36) / ((s + 2)(s² + 9)) = A/(s + 2) + (Bs + C)/(s² + 9)

Multiplying through by the common denominator, we get:

-5s - 36 = A(s² + 9) + (Bs + C)(s + 2)

Expanding and equating coefficients, we find:

-5 = A + B

0 = 2A + C

-36 = 9A + 2B

Solving these equations, we find: A = -4, B = -1, C = 8.

Step 3: Express the inverse Laplace transform in terms of known Laplace transform pairs:

[tex]L^{-1(-5s - 36)}[/tex] / ((s + 2)(s² + 9)) = [tex]L^{-1(-4/(s + 2))}[/tex] + [tex]L^{-1((-s + 8)/(s^2 + 9)}[/tex])

= [tex]-4e^{-2t}[/tex] + (-cos(3t) + 8sin(3t))/3

Therefore, the inverse Laplace transform of (-5s - 36) / ((s + 2)(s²+ 9)) is [tex]-4e^{-2t}[/tex]+ (-cos(3t) + 8sin(3t))/3.

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Among the statements provided, the only true statement is that 235 is congruent to 23 modulo 13.

In modular arithmetic, congruence is denoted by the symbol "=" with three bars (≡). It indicates that two numbers have the same remainder when divided by a given modulus.

Let's evaluate each statement:

1. 57 ≡ -13 (mod 7): This statement is false. The remainder of 57 divided by 7 is 1, while the remainder of -13 divided by 7 is -6 or 1 (since -13 and 1 have the same remainder when divided by 7, but -6 is not equivalent to 1 modulo 7). Therefore, 57 is not congruent to -13 modulo 7.

2. 235 ≡ 23 (mod 13): This statement is true. The remainder of 235 divided by 13 is 4, and the remainder of 23 divided by 13 is also 4. Hence, 235 is congruent to 23 modulo 13.

3. 57 ≡ 13 (mod 7): This statement is false. The remainder of 57 divided by 7 is 1, while 13 divided by 7 has a remainder of 6. Thus, 57 is not congruent to 13 modulo 7.

4. 2 - 14 ≡ -28 (mod 7): This statement is false. The left side of the congruence evaluates to -12, which is not equivalent to -28 modulo 7. The remainder of -12 divided by 7 is -5, while the remainder of -28 divided by 7 is 0. Hence, -12 is not congruent to -28 modulo 7.

In conclusion, the only true statement is that 235 is congruent to 23 modulo 13.

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Answer:

A standard form of a linear equation is Ax + By = C

Step-by-step explanation:

For example, 3x + 4y = 7 is a linear equation in standard form. When an equation is given the form it ia pretty easy to find the both intercepts of (x and y). It can be useful when solving a two linear equation.

Answer:

thx for the points

Step-by-step explanation:

Answer:

Where is the question tho whaaAAaaaa

Step-by-step explanation:

2) a= -3/8 and b= -5/3

a×b= b×a

-3 × -5 = -5 × -3

8. 3. 3. 8

15 = 15

24. 24

3)a=8/11 and b= -6/11

a×b=b×a

8 × -6 = -6 × 8

11. 11. 11. 11

-48 = -48

121. 121

4) a= -9/15 and b= -7/2

a×b=b×a

-9 × -7 = -7 × -9

15. 2. 2. 15

63 = 63 , let's divide both by 3

30. 30

21 = 21

10. 10

The correlation coefficient for the data-set in this problem is given as follows:

r = 0.9553.

The coefficient is obtained inserting the points in a data-set in a correlation coefficient calculator.

The input and the output of the data set are given as follows:

From the table, the points are given as follows:

(100, 25), (150, 35), (200, 32), (250, 37), (300, 48), (350, 49), (400, 52).

Inserting these points into the calculator, the correlation coefficient is given as follows:

r = 0.9553.

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Answer:

I believe the answer is 1,800 :)

Step-by-step explanation:

1,500x0.20=300+1,500=1,800

Hope this helped!

Answer:

100 i think

Step-by-step explanation:

Answer:

Here is the answer

Step-by-step explanation:

That will show you.

Answer:

2.25 or 2(1/4)

Step-by-step explanation:

Type into a calc :)