The quantity, $Q$, of a certain product manufactured depends on the quantity of labor, $L$, and of capital, $K$, used according to the function
$$Q=900L^{\frac{1}{2}}{K}^{\frac{2}{3}}$$
Labor costs $\mathrm{\$}100$ per unit and capital costs $\mathrm{\$}600$ per unit.
What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?
Round your answers to two decimal places.
The minimum cost is dollars.

Step 1 ：We look for the combination of labor $L$ and capital $K$ such that the cost is minimized while producing 36,000 units of goods. The cost of each unit of labor is $100andthecostofeachunitofcapitalis$600. The quantity of goods produced is given by $Q=900L^{\frac{1}{2}}{K}^{\frac{2}{3}}$.

Step 2 ：We set $Q=36000$ and solve for $L$ in terms of $K$: $$36000=900L^{\frac{1}{2}}{K}^{\frac{2}{3}}\Rightarrow L={\left(\frac{36000}{900K^{\frac{2}{3}}}\right)}^2={\left(\frac{40}{K^{\frac{2}{3}}}\right)}^2$$

Step 3 ：The cost $C$ is given by $C=100L+600K=100{\left(\frac{40}{K^{\frac{2}{3}}}\right)}^2+600K$.

Step 4 ：We take the derivative of $C$ with respect to $K$ and set it equal to zero to find the minimum cost: $$\frac{dC}{dK}=-\frac{160000}{K^{\frac{5}{3}}}+600=0\Rightarrow K={\left(\frac{160000}{600}\right)}^{\frac{3}{5}}\approx 56.57$$

Step 5 ：We substitute $K\approx 56.57$ into the equation for $L$ to find $L\approx {\left(\frac{40}{{56.57}^{\frac{2}{3}}}\right)}^2\approx 21.54$.

Step 6 ：We substitute $L\approx 21.54$ and $K\approx 56.57$ into the equation for $C$ to find the minimum cost $C\approx 100(21.54)+600(56.57)\approx \mathrm{\$}35942.20$.

Step 7 ：The combination of labor and capital that should be used to produce 36,000 units of goods at minimum cost is approximately $L\approx 21.54$ and $K\approx 56.57$. The minimum cost is approximately $35942.20.