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) = 7 K^{8} L^{}`. The factory has to produce 266 cookies in a short period of time with K = 1 ovens.
How many workers does the factory need?
Plugging the values we know gives $$ \begin{align*} q \left( K, L \right) &= 7 K^{8} L^{} \\ 266 &= 7 \times 1^{8} L^{} \end{align*} $$
Solving for L: $$ \begin{align*} L^{} &= \frac{266}{7 \times 1^{8} } \\ L^{} &= 38 \end{align*} $$
The factory needs `38` grandmas to bake cookies tomorrow.