A company that produces ribbon has found that the marginal cost of producing $x$ yards of fancy ribbon is given by $C^{\prime }(x)=-0.00002x^2-0.04x+44$ for $x\le 700$, where $C^{\prime }(x)$ is in cents. Approximate the total cost of manufacturing 700 yards of ribbon, using 5 subintervals over $[0,700]$ and the left endpoint of each subinterval.
The total cost of manufacturing 700 yards of ribbon is approximately $\mathrm{\$}$
(Do not round until the final answer. Then round to the nearest cent as needed.)

Step 1 ：The total cost of manufacturing 700 yards of ribbon can be approximated by integrating the marginal cost function over the interval [0, 700]. Since we are asked to use 5 subintervals and the left endpoint of each subinterval, we can use the Left Riemann Sum to approximate the integral.

Step 2 ：The Left Riemann Sum is given by the formula: $$\sum _{i=0}^{n-1}f(a+i\mathrm{\Delta }x)\mathrm{\Delta }x$$ where $\mathrm{\Delta }x=\frac{b-a}{n}$ is the width of each subinterval, $a$ is the left endpoint of the interval, $b$ is the right endpoint of the interval, and $n$ is the number of subintervals. In this case, $a=0$, $b=700$, and $n=5$.

Step 3 ：Substituting the given values into the formula, we get $\mathrm{\Delta }x=\frac{700-0}{5}=140.0$.

Step 4 ：Then, we calculate the total cost by summing up the product of the marginal cost at the left endpoint of each subinterval and the width of the subinterval. The marginal cost function is given by $C^{\prime }(x)=-0.00002x^2-0.04x+44$.

Step 5 ：By performing the calculations, we find that the total cost of manufacturing 700 yards of ribbon is approximately 213.136000000000 dollars.

Step 6 ：Rounding to the nearest cent, the total cost is approximately $213.14.

Step 7 ：Final Answer: The total cost of manufacturing 700 yards of ribbon is approximately $\boxed{{\displaystyle 213.14}}$ dollars.