Consider a scenario where a manufacturer creates widgets using a production function as represented by $\backslash ($ Y$=8\mathrm{.}00$ L$^\{0\mathrm{.}90\}$ K$^\{0\mathrm{.}50\}\backslash ),$ where $\backslash ($ Y $\backslash )$ is widget output in thousands, $\backslash ($ L $\backslash )$ is the number of hours of labor employed, and $\backslash ($ K $\backslash )$ is the number of hours that the production machinery is in use. Additionally, let's assume that the firm has contracted $40\mathrm{.}00$ hours of machine time rented at a cost of $\backslash (\backslash$$ $18\mathrm{.}77\backslash )$ per hour. The widgets can be sold at a rate of $\backslash (\backslash$$ $60\mathrm{.}00\backslash )$ per thousand widgets, and the manufacturer pays its workers a wage of $\backslash (\backslash$$ $14\mathrm{.}06\backslash )$ per hour.

$3$rd attempt

Part $1(1\mathrm{.}7$ points$)$

How many hours of labor should the manufacturer hire to maximize profit? $($Do not round until your final answer, then round to two decimal places.$)$

hours

Part $2(1\mathrm{.}7$ points$)$

See Hint

How many thousands of widgets will be produced? $($Round to two decimal places.$)$ thousands

Part $3(1\mathrm{.}7$ points$)$

See Hint

What is the level of profits earned? $($Round to two decimal places.$)$

$\backslash$$