Monthly sales of a particular personal computer are expected to decline at the following rate of $S^{\prime}(t)$ computers per month, where $t$ is time in months and $S(t)$ is the number of computers sold each month.
\[
S^{\prime}(t)=-10 t^{\frac{2}{3}}
\]
The company plans to stop manufacturing this computer when monthly sales reach 800 computers. If monthly sales now $(t=0)$ are 1,340 computers, find $S(t)$. How long will the company continue to manufacture this computer?

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.