P$1)$ You are the engineer on a construction project that requires $4800m^3$ of sand, $5800m^3$

of fine gravel and $5700m^3$ of coarse gravel. These materials can be obtained from $4$ pits

whose compositions are:

You, as the project engineer, need to determine the volume of material that needs to be

acquired from each of the four pits so that the requirements of sand, fine gravel, and coarse

gravel are met.

$($a$)$ Define the unknown variables, model the problem as a linear system of equations

and write it in terms of $[A]\{x\}=\{B\}$

$($b$)$ Can you determine the volume of material that needs to be acquired from each of

the four pits by using $\{x\}=[A{]}^{-1}\{B\}?$ If not, how could you determine the solution?

$($c$)$ Determine the volume of material that needs to be acquired from each of the four

pits to meet the project requirements using pseudo$-$inverses. You can use MATLAB

as a calculator to perform matrix multiplication, transpose and inverses $($note that you

can use $A^{\text{'}}$ to get $[A{]}^T$ in MATLAB$).$ Submit screenshots of command window for all

MATLAB calculations.

$($d$)$ Has the solution truly met the requirements of the project? If it has, is this solution

unique? If it has not, why not?

$($e$)$ We know that $\frac{A}{\frac{?}{?}}B$ in MATLAB is equivalent to $[A{]}^{-1}\{B\}.$ But the backslash operator

$\frac{\frac{?}{?}}{?}$ is programmed in MATLAB such that when it detects $[$A$]$ not being a square matrix

$($implying an underdetermined or overdetermined system$),$ it automatically solves

the system using something akin to pseudo$-$inverses. What is the solution you get by

using $\frac{A}{\frac{?}{?}}B$ in MATLAB? Does this solution truly meet the requirements of the

project? If it is different from the solution obtained in $($c$),$ explain how this possible?

Submit screenshots of command window for all MATLAB calculations.