MIXED GROUPING OF BATTERY CELLS (m+n=constant)

m+n=k (keep fixed)
positive number m (vary m)
positive number n 
positive number r   (keep fixed)
n+r
n-r
positive number k  ( > k > , vary k for triangle formation )
m+k
m-k
positive number R (   > R  > , fix R )
Find out    
Area of Δ ABC (nrk)
Area of Δ ADC (mRk)
Total area of ABCD
Area Δ ABC / AreaΔ ADC
nr +mR
mn
(nr +mR) / mn
mn/(nr+mR)
m+n+r+R
Value of m for Minimum Value of mn/ (nr+mR) m=[k/(R-r)]*[r ± √(Rr)] --
Value of n for Minimum Value of mn/ (nr+mR) m=[k/(R-r)]*[r ± √(Rr)] --
Value of Extremum Value of mn/ nr+mR --
   
Whether Δ ADC can be formed with this value of R as above?
   

 

* In electricity, when there is a mixed group of cells of emf E each, internal resistance r each such that there are n cells in series in a row, and there are m equal rows arranged in parallel and the system is linked to an external resistance of R , current in the resistor R is given by I = nE / (R + nr/m ) = mnE / (mR+nr). If mn = k=constant, the arrangement of m and n which results in I being maximum is derived below--

We can get the  result by calculus method by putting m+n=k=constant and putting f(m)= E/ [R/n +r/m] =        E/ [R/(k-m) +r/m]  to be extremum. df(m)/dm = 0 or -r/m2 + R/(k-m)2 = 0  or m2(R-r)+2krm-rk2=0

or m =[k/(R-r)]*[r ± √(Rr)]   and Imax =E / [ r(R-r) /k( rą√(rR)) +R/n ]

* For mn=24, m=4,n=6, r=2, mR + nr will be minimum when R= 6*2/4=3; In other words, when R=3, R/r=3/2. thus n/m =6/4=3/2 will be the combination at which maximum current will flow through the resistance. Similarly, when R=2, R/r=1 and so m/n=1 or m*m=24 or m =4.8 , n=5. And at this combination, maximum current will flow. Rounding up, m=4,n=6 and R=2.
since m,n,r, R i.e. 4 numbers are involved, we have tried to find out whether the numbers can form a quadrilateral . For example, taking mn= 24, the combination can be (m,n)=(24,1) , (1,24) , (12,2), (2,12), (8,3), (3,8), (6,4), (4,6)
taking (24,1) -> and r=2, permissible k=2 only and permissible R=23,24,25. The minimum R which will result in maximum current is not permissible since triangle ACD cannot be constructed with that value of R.