Answer
The tension should be multiplied by four.
Work Step by Step
The formula for the velocity of a wave on a string is equal to $$v=\sqrt{\frac{F_T}{\mu}}$$ The ratio of the new speed to the old speed is $\frac{32m/s}{16m/s}=2.0$, so the new speed must be twice the original speed. This means that $$2v=\sqrt{\frac{nF_T}{\mu}}$$ where $n$ is the multiplication factor of the tension. This means that $$2v=\sqrt{n} \times \sqrt{\frac{F_T}{\mu}}=v\sqrt{n}$$ This means that $\sqrt{n}=2$, or $n=4$. This means that the tension should be multiplied by four to double the speed to 32m/s.
