Calculus word problem help?

Question

The impedance Z in Ohms in an electric circuit is given by
Z=Squareroot[ R^2+(X-Y)^2 ]. If R=2500 Ohms and X=1500 Ohms, what value of Y makes the impedance a minimum?

Answer 1

minimizing z is equivalent to minimizing z^2, since z is always positive
z^2 = r^2 + (x-y)^2
0 = d(z^2) /dy = -2(x-y)
so x=y=1500 ohms minimizes z^2 and also z
you could do the same process
and equate dz/dy = 0 but it is more difficult to differentiate and will give you the same answer

note: the solution makes logical sense without calculus, if you look at the equation for z you can see that it will be smallest when the (x-y)^2 term disappears ie when x = y

to check if it is minimum/maximum find
d^2(z^2) /dy^2 = 2 >0 so it is a minimum

the end
.

Answer 2

We readily see that Z is minimum when X - Y = 0, that is when X = Y. So, since X =1500, Y = 1500. We don't need calculus here. The minimum value for z is sqrt(2500) = 50.