For what positive value of $n$ does $|3+ni|=3\sqrt{10}$?
To solve for $n$, I need to use the definition of the modulus (or absolute value) of a complex number. If $z=a+bi$ is a complex number, then $|z|=\sqrt{a^2+b^2}$. So, in this case, $|3+ni|=\sqrt{3^2+n^2}$. I want this to equal $3\sqrt{10}$, so I can square both sides and get $3^2+n^2=9\cdot 10$. Simplifying, I get $n^2=81$. Taking the square root, I get $n=\pm 9$. But the problem asks for a positive value of $n$, so I can ignore the negative one. Therefore, the answer is $n=9$. 