Example
The cookie factory has to produce 10000 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 10000 cookies tomorrow.
Question
The production function is `q \left( K, L \right) = 7 K^{3} L^{}`. The factory has to produce 423549 cookies in a short period of time with K = 9 ovens.
How many workers does the factory need?
Plugging the values we know gives
$$ \begin{align*} q \left( K, L \right) &= 7 K^{3} L^{} \\ 423549 &= 7 \times 9^{3} L^{} \end{align*} $$Solving for L:
$$ \begin{align*} L^{} &= \frac{423549}{7 \times 9^{3} } \\ L^{} &= 83 \end{align*} $$The factory needs `83` grandmas to bake cookies tomorrow.