Methods of structuring organizations to enable them produce large amounts of goods is known as mass production.
<h3>What is mass production?</h3>
Mass production is also referred to as continuous production and it can be defined as a method of production that focuses on manufacturing large quantities of standardized goods.
This ultimately implies that, mass production is a method of structuring organizations that is mainly focused on making a large number of a few goods.
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It looks like this is a system of linear ODEs given in matrix form,
![x' = \begin{bmatrix}10&-1\\5&8\end{bmatrix} x](https://tex.z-dn.net/?f=x%27%20%3D%20%5Cbegin%7Bbmatrix%7D10%26-1%5C%5C5%268%5Cend%7Bbmatrix%7D%20x)
with initial condition x(0) = (-6, 8)ᵀ.
Compute the eigenvalues and -vectors of the coefficient matrix:
![\det\begin{bmatrix}10-\lambda&-1\\5&8-\lambda\end{bmatrix} = (10-\lambda)(8-\lambda) + 5 = 0 \implies \lambda^2-18\lambda+85=0 \implies \lambda = 9\pm2i](https://tex.z-dn.net/?f=%5Cdet%5Cbegin%7Bbmatrix%7D10-%5Clambda%26-1%5C%5C5%268-%5Clambda%5Cend%7Bbmatrix%7D%20%3D%20%2810-%5Clambda%29%288-%5Clambda%29%20%2B%205%20%3D%200%20%5Cimplies%20%5Clambda%5E2-18%5Clambda%2B85%3D0%20%5Cimplies%20%5Clambda%20%3D%209%5Cpm2i)
Let v be the eigenvector corresponding to λ = 9 + 2i. Then
![\begin{bmatrix}10-\lambda&-1\\5&8-\lambda\end{bmatrix}v = 0 \implies \begin{bmatrix}1-2i&-1\\5&-1-2i\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D10-%5Clambda%26-1%5C%5C5%268-%5Clambda%5Cend%7Bbmatrix%7Dv%20%3D%200%20%5Cimplies%20%5Cbegin%7Bbmatrix%7D1-2i%26-1%5C%5C5%26-1-2i%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7Dv_1%5C%5Cv_2%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D0%5C%5C0%5Cend%7Bbmatrix%7D)
or equivalently,
![\begin{cases}(1-2i)v_1-v_2=0 \\ 5v_1-(1+2i)v_2=0\end{cases} \implies 5v_1 - (1+2i)v_2 = 0](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%281-2i%29v_1-v_2%3D0%20%5C%5C%205v_1-%281%2B2i%29v_2%3D0%5Cend%7Bcases%7D%20%5Cimplies%205v_1%20-%20%281%2B2i%29v_2%20%3D%200)
Let
; then
, so that
![\begin{bmatrix}10&-1\\5&8\end{bmatrix}\begin{bmatrix}1\\1-2i\end{bmatrix} = (9+2i)\begin{bmatrix}1\\1-2i\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D10%26-1%5C%5C5%268%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20%3D%20%289%2B2i%29%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D)
and we get the other eigenvalue/-vector pair by taking the complex conjugate,
![\begin{bmatrix}10&-1\\5&8\end{bmatrix}\begin{bmatrix}1\\1+2i\end{bmatrix} = (9-2i)\begin{bmatrix}1\\1+2i\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D10%26-1%5C%5C5%268%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D%20%3D%20%289-2i%29%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D)
Then the characteristic solution to the system is
![x = C_1 e^{(9+2i)t} \begin{bmatrix}1\\1-2i\end{bmatrix} + C_2 e^{(9-2i)t} \begin{bmatrix}1\\1+2i\end{bmatrix}](https://tex.z-dn.net/?f=x%20%3D%20C_1%20e%5E%7B%289%2B2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20%2B%20C_2%20e%5E%7B%289-2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D)
From the given condition, we have
![\displaystyle \begin{bmatrix}-6\\8\end{bmatrix} = C_1 \begin{bmatrix}1\\1-2i\end{bmatrix} + C_2 \begin{bmatrix}1\\1+2i\end{bmatrix} \implies C_1 = -3-\frac i2, C_2=-3+\frac i2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Bbmatrix%7D-6%5C%5C8%5Cend%7Bbmatrix%7D%20%3D%20C_1%20%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20%2B%20C_2%20%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D%20%5Cimplies%20C_1%20%3D%20-3-%5Cfrac%20i2%2C%20C_2%3D-3%2B%5Cfrac%20i2)
and so the particular solution to the IVP is
![\displaystyle \boxed{x = -\left(3+\frac i2\right) e^{(9+2i)t} \begin{bmatrix}1\\1-2i\end{bmatrix} - \left(3-\frac i2\right) e^{(9-2i)t} \begin{bmatrix}1\\1+2i\end{bmatrix}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cboxed%7Bx%20%3D%20-%5Cleft%283%2B%5Cfrac%20i2%5Cright%29%20e%5E%7B%289%2B2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1-2i%5Cend%7Bbmatrix%7D%20-%20%5Cleft%283-%5Cfrac%20i2%5Cright%29%20e%5E%7B%289-2i%29t%7D%20%5Cbegin%7Bbmatrix%7D1%5C%5C1%2B2i%5Cend%7Bbmatrix%7D%7D)
which you could go on to rewrite using Euler's formula,
![e^{(a+bi)t} = e^{at} (\cos(bt) + i \sin(bt))](https://tex.z-dn.net/?f=e%5E%7B%28a%2Bbi%29t%7D%20%3D%20e%5E%7Bat%7D%20%28%5Ccos%28bt%29%20%2B%20i%20%5Csin%28bt%29%29)
<span>the income and size of a family is used to determine: C. Poverty Level.
The threshold for the poverty level will be depended on the total income compared to the amount of children that the family has.
This data will be later used to determine which families are eligible to get government Aid.</span>
The magnitude of the gravitational force between the two spheres is approximately; F = 1.67 x 10⁻⁹ N
<h3>Gravitational Force</h3>
We are given;
Mass 1; M = 15 kg
Mass 2; m = 15 kg
distance of separation; r = 3 m
Formula for gravitational force using newtons law of gravitation is;
F = GMm/r²
where;
M is mass 1
m is mass 2
G is gravitational constant = 6.67 × 10⁻¹¹ N.kg²/m²
r is distance of separation
Plugging in the relevant values, we have;
F = (6.67 × 10⁻¹¹ × 15 × 15)/3²
F = 1.67 x 10⁻⁹ N
Read more about Gravitational Force at; brainly.com/question/1602310