Answer
At $~~y = 17.5~cm,~~$ the net magnetic field is zero.
Work Step by Step
We can write the expression for the magnetic field produced by a current in a straight wire:
$B = \frac{\mu_0~i}{2~\pi~R}$
By the right hand rule, the net magnetic field will be zero at a value of $y$ that is somewhere above the wires where $y \gt 10.0~cm$. At this value of $y$, the magnetic field to to each wire will be equal in magnitude and opposite in direction.
To find $y$, we can equate the magnitude of the magnetic field due to the current in each wire:
$\frac{\mu_0~i_1}{2~\pi~R_1} = \frac{\mu_0~i_2}{2~\pi~R_2}$
$\frac{i_1}{R_1} = \frac{i_2}{R_2}$
$i_1~R_2 = i_2~R_1$
$(i_1)(y-5.00~cm) = (i_2)~(y-10.0~cm)$
$i_1~y-i_2~y = (i_1)(5.00~cm)-(i_2)(10.0~cm)$
$y = \frac{(i_1)(5.00~cm)-(i_2)(10.0~cm)}{i_1-i_2}$
$y = \frac{(6.00~A)(5.00~cm)-(10.0~A)(10.0~cm)}{6.00~A-10.0~A}$
$y = 17.5~cm$
At $~~y = 17.5~cm,~~$ the net magnetic field is zero.
