The production $q($in units$)$ of a certain good in terms of the amount $q_L$ of labour invested and the amount $q_C$ of capital invested is given by the production function $q=5*q_L^{\frac{2}{3}}*q_C^{\frac{1}{3}}.$ At this moment, $512$ units of labour and $1000$ units of capital are invested daily. This means that at this moment, $3200$ units are produced daily. The graph below gives the contour line of level $3200$ of the production function.

Figure $8$: Contour line of level $3200$ of the production function $q.$

$($a$)$ Give an implicit and explicit equation of this contour line.

Assume for this and later parts that a unit, of labour costs $1\mathrm{.}89$ units of money and that a unit of capital costs $1\mathrm{.}25$ units of money. Moreover, there is a fixed cost of $14\mathrm{.}3$ units of money a day.

$($b$)$ Give the equation of the cost function $c(q_L,q_C).$

$($c$)$ Graph, in Figure $8$ above, a line that allows you to decide that there exists another combination $(q_L,q_C)$ to generate $3200$ units a day with exactly the same cost as the combination $(q_L,q_C)=(512,1000).$

$($d$)$ Explain how you can approximately read this second combination from the graph above.

$($e$)$ Calculate this second combination exactly. Make use of your graphing calculator.

$($f$)$ Explain how you can graphically find the combinations to produce $3200$ units a day in a cheaper way than by investing $(q_L,q_C)=(512,1000).$

$($g$)$ Describe the method to find the minimal budget to reach the daily production level $q=3200$ geometrically.

$($h$)$ Calculate the minimal cost to realize production level $q=3200$ and the number of units of capital and labour that have to be invested.