Working out the current and voltage in a circuit

[Bolter]

The product mn is fixed, so there is only one independent variable.

If product mn is fixed then it can treated as a constant which I'll label as k

Therefore I have total current = kv/mR + nr

What should I differentiate this with respect to?

Edit:

Would this be along the right lines? Where I differentiate with respect to n, set that derivative to zero to find n. Then subsequently get m by 12/n

 

[haruspex]

If product mn is fixed then it can treated as a constant which I'll label as k

Therefore I have total current = kv/mR + nr

What should I differentiate this with respect to?

Edit:

Would this be along the right lines? Where I differentiate with respect to n, set that derivative to zero to find n. Then subsequently get m by 12/n

View attachment 256081

Yes, that looks right. But why use K in some places and 12 in others?

 

[Bolter]

Yes, that looks right. But why use K in some places and 12 in others?

Yes sorry, should've stayed consistent and used k or 12 only throughout.

To differentiate the function I used quotient rule to do this and solve for n and m when letting the derivative be set to 0

And I do get n = 4 and m = 3

 

[haruspex]

Yes sorry, should've stayed consistent and used k or 12 only throughout.

To differentiate the function I used quotient rule to do this and solve for n and m when letting the derivative be set to 0

And I do get n = 4 and m = 3

View attachment 256082
View attachment 256083

Very good.
Note that the m:n ratio turns out always to be the ratio of the resistances, with the higher internal resistance corresponding to the wider, shorter array.