Suppose a company wants to introduce a new machine that will produce a rate of annual savings (in dollars) given by the function $S^{\prime }(x)$, where $x$ is the number of years of operation of the machine, while producing a rate of annual costs (in dollars) given by the function $C^{\prime }(x)$
$$S^{\prime }(x)=100-x^2,C^{\prime }(x)=x^2+\frac{14}{3}x$$
a. For how many years will it be profitable to use this new machine?
The number of profitable years is 6
b. What are the net total savings during the first year of use of the machine?
The net total savings during the first year of use of the machine is $\mathrm{\$}97$. (Round to the nearest dollar as needed.)
c. What are the net total savings over the entire period of use of the machine?
The net total savings over the entire period of use of the machine is $\mathrm{\$}$ (Round to the nearest dollar as needed.)

Step 1 ：Define the savings rate function as $S^{\prime }(x)=100-x^2$ and the cost rate function as $C^{\prime }(x)=x^2+\frac{14}{3}x$.

Step 2 ：Find the number of profitable years by solving for $x$ when $S^{\prime }(x)>C^{\prime }(x)$. The solution is $x=6$ years.

Step 3 ：Find the net total savings during the first year by subtracting the cost from the savings for the first year, i.e., $S(1)-C(1)$, where $S(x)$ and $C(x)$ are the antiderivatives of $S^{\prime }(x)$ and $C^{\prime }(x)$ respectively. The solution is $97.

Step 4 ：Find the net total savings over the entire period of use by subtracting the total cost from the total savings over the entire period of use, i.e., $S(n)-C(n)$, where $n$ is the number of profitable years. The solution is $372.

Step 5 ：Final Answer: The number of profitable years is $\boxed{{\displaystyle 6}}$ years. The net total savings during the first year of use of the machine is $\boxed{{\displaystyle \mathrm{\$}97}}$. The net total savings over the entire period of use of the machine is $\boxed{{\displaystyle \mathrm{\$}372}}$.