As a consultant in a manufacturing company, you are required to propose

solutions for the following scenarios that the company is experiencing:

$$ There is a glitch in one of the computer programs where the problem has

been narrowed down to three areas: either $($i$)$ a decimal number $19,($ii$)$ a

value of $371410,$ or $($iii$)$a binary number $110101112.$

$$ The dimensions to manufacture a shipping box are expressed in complex

number form. All dimensions must be positive for the expression to be

inputted into the system.

$(2+$ j$)(3$ j$2)=$ a $+$ jb$,$ and $($x $$ j$2$y$)($y $$ jx$)=2+$ j$,$ are the two

expressions recommended to manufacture the shipping box, where the

combination of a and b or x and y can represent the dimensions to be

inputted into the system.

$$ The phase shift between two systems has been expressed as:

$2<120+5\mathrm{.}2<581\mathrm{.}6<40.$

The correction to this phase shift is expressed as $(15$j$).$

$$ The current flowing through an electric motor is expressed as $(3$ j$4)$A and

the voltage is expressed as $(1$ j$2)$V$.$ This motor is also connected to a

pump where the vibration experienced by the pump is expressed as cos$3\backslash$theta $.$

FORMAL REPORT

Your report must include using De Moivre$$s theorem to test the correctness of the

vibration experienced by the pump, where cos$3\backslash$theta $=4$cos$3\backslash$theta $3$cos$\backslash$theta $.$

After testing the correctness above, you must deduce the solution for the current

flowing through and the voltage across the electric motor.

Once the previous two sections are completed, you need to first perform arithmetic

operations, using polar form of complex numbers for the phase shift, and

exponential form of complex numbers for the correction to the phase shift. Then,

using complex number theory to manufacture the shipping box, solve for a and b

in the first expression, then solve for x and y in the second expression, followed by

recommending which expression should be inputted into the system and give a

reason for your selection. Lastly, by applying addition and multiplication methods

for the three problematic areas in the computer glitch, convert the decimal number

$19$ to a binary number, the number $371410$ to an octal number, and the binary

number $110101112$ to a hexadecimal number.