CAT SCANS $($Refer to Exercise $76.)$ Four X$-$ray beams are aimed at four cells, as shown in the following figure. Source: The Mathematics Teacher.

a$.[$Writing Exercise$]$ Suppose the attenuation values for beams $1,2,3,$ and $4$ are $0\mathrm{.}60,0\mathrm{.}75,0\mathrm{.}65,$ and $0\mathrm{.}70,$ respectively. Do we have enough information to determine the values of $a,b,c,$ and $d?$ Explain.

b$.$ Suppose we have the data from part $($a$),$ as well as the following values for $d.$ Find the values for $a,b,$ and $c,$ and draw conclusions about cells A$,$ B$,$ C$,$ and D in each case.

i$.0\mathrm{.}33$

ii$.0\mathrm{.}43$

c$.$ Two X$-$ray beams are added, as shown in the figure. In addition to the data in part $($a$),$ we now have attenuation values for beams $5$ and $6$ of $0\mathrm{.}85$ and $0\mathrm{.}50.$ Find the values for $a,b,c,$ and $d,$ and make conclusions about cells $A,B,C,$ and $D.$

d$.$ Six X$-$ray beams are not necessary because four appropriately chosen beams are sufficient. Give two examples of four beams $($chosen from beams $1-6$ in part $($c$))$ that will give the solution. $($Note: There are $12$ possible solutions.$)$

e$.[$Writing Exercise$]$ Discuss what properties the four beams selected in part $($d$)$ must have in order to provide a unique solution.