Example
The cookie factory has to produce 10,000 cookies tomorrow using 100 ovens (K). The cookie production function is $$ q \left( K, L \right) = K L^2 $$
How many grandmas (L) does the factory need? $$ \begin{align*} q \left( K, L \right) &= K L^2 \\ 10000 &= 100 L^2 \\ 100 &= L^2 \\ L &= 10 \end{align*} $$
The cookie factory needs 10 grandmas to produces 10,000 cookies tomorrow.
Question
The production function is `q \left( K, L \right) = 9 K^{7} L^{}`. The factory has to produce 1032192 cookies in a short period of time with K = 4 ovens.
How many workers does the factory need?
Plugging the values we know gives $$ \begin{align*} q \left( K, L \right) &= 9 K^{7} L^{} \\ 1032192 &= 9 \times 4^{7} L^{} \end{align*} $$
Solving for L: $$ \begin{align*} L^{} &= \frac{1032192}{9 \times 4^{7} } \\ L^{} &= 7 \end{align*} $$
The factory needs `7` grandmas to bake cookies tomorrow.