Step 1 :Given the rate of decline of sales $S^{\prime}(t)=-10 t^{\frac{2}{3}}$, we can find the function $S(t)$ by integrating $S^{\prime}(t)$ with respect to $t$.
Step 2 :The integral of $-10 t^{\frac{2}{3}}$ with respect to $t$ is $-30 t^{\frac{5}{3}} + C$, where $C$ is the constant of integration.
Step 3 :We know that at $t=0$, $S(0)=1340$ computers. We can use this information to find the value of $C$.
Step 4 :Substituting $t=0$ and $S(0)=1340$ into the equation $S(t)=-30 t^{\frac{5}{3}} + C$, we get $C=1340$.
Step 5 :So, the function $S(t)$ is $S(t)=-30 t^{\frac{5}{3}} + 1340$.
Step 6 :The company plans to stop manufacturing this computer when monthly sales reach 800 computers. So, we need to solve the equation $S(t)=800$ for $t$.
Step 7 :Substituting $S(t)=800$ into the equation $S(t)=-30 t^{\frac{5}{3}} + 1340$, we get $-30 t^{\frac{5}{3}} + 1340 = 800$.
Step 8 :Solving this equation for $t$, we get $t=\left(\frac{540}{30}\right)^{\frac{3}{5}}$.
Step 9 :Calculating the value of $t$, we get $t=\boxed{3.42}$ months.