## 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) = 4 K^{} L^{}`. The factory has to produce 1072 cookies in a short period of time with K = 4 ovens.

How many workers does the factory need?

Plugging the values we know gives

$$ \begin{align*} q \left( K, L \right) &= 4 K^{} L^{} \\ 1072 &= 4 \times 4^{} L^{} \end{align*} $$Solving for L:

$$ \begin{align*} L^{} &= \frac{1072}{4 \times 4^{} } \\ L^{} &= 67 \end{align*} $$The factory needs `67` grandmas to bake cookies tomorrow.