A company estimates that its sales will grow continuously at a rate given by the function
$$S^{\prime }(t)=24e^t$$
where $S^{\prime }(t)$ is the rate at which sales are increasing, in dollars per day, on day $t$.
a) Find the accumulated sales for the first 4 days.
b) Find the sales from the 2 nd day through the 5 th day. (This is the integral from 1 to 5 .)
a) The accumulated sales for the first 4 days is $\mathrm{\$}◻$. (Round to the nearest cent as needed.)

Step 1 ：The problem is asking for the accumulated sales for the first 4 days. This is a calculus problem and can be solved by integrating the given function from 0 to 4. The integral of a function gives the area under the curve, which in this case represents the total sales over a period of time.

Step 2 ：First, we set up the integral: ${\int }_0^4{S}^{\prime }(t)dt={\int }_0^{4}24e^{t}dt$.

Step 3 ：Next, we calculate the integral: $-24+24e^4$.

Step 4 ：This represents the total sales over the first 4 days. However, this is not in a form that is easy to interpret. We need to evaluate this expression to get a numerical value.

Step 5 ：Evaluating the expression gives us approximately 1286.36.

Step 6 ：$\boxed{{\displaystyle \text{The accumulated sales for the first 4 days is approximately \$1286.36.}}}$