Answer
$7.9\;\rm cm$
Work Step by Step
Since the two identical speakers are placed on the same line then the amplitude of the net wave is given by
$$A=2a\cos\left(\dfrac{\Delta \phi}{2}\right)=2a\cos\left(\frac{1}{2}\left[\dfrac{2\pi \Delta x}{ \lambda}+\Delta \phi_0\right]\right)$$
where $\Delta \phi_0=0$ since the speakers are emitting identical waves so they have identical phase constants.
$$A= 2a\cos\left(\frac{1}{ \color{red}{\bf\not} 2}\left[\dfrac{ \color{red}{\bf\not} 2\pi \Delta x}{ \lambda}+0\right]\right)=2a\cos\left( \dfrac{ \pi \Delta x}{ \lambda}\right)$$
Recalling that $v=\lambda f$, so $\lambda=v/f$
$$A= 2a\cos\left( \dfrac{ \pi f\Delta x}{ v}\right)$$
We know that the net amplitude at his point is 1.5 times that of each speaker alone, so $A=1.5 a$;
$$1.5 \color{red}{\bf\not} a= 2 \color{red}{\bf\not} a\cos\left( \dfrac{ \pi f\Delta x}{ v}\right)$$
Hence,
$$\dfrac{3}{4}=\cos\left( \dfrac{ \pi f\Delta x}{ v}\right)$$
$$ \dfrac{ \pi f\Delta x}{ v}=\cos^{-1}\left( \dfrac{3}{4}\right)$$
$$ \Delta x =\dfrac{v}{\pi f}\;\cos^{-1}\left( \dfrac{3}{4}\right)$$
Plugging the known;
$$ \Delta x =\dfrac{343}{\pi (1000)}\;\cos^{-1}\left( \dfrac{3}{4}\right)=0.0789\;\rm m$$
$$\Delta x=\color{red}{\bf 7.90}\;\rm cm$$
