A company manufactures bicycles and tricycles each of which must be processed through machines $A$ and $B$. Machine $A$ has maximum of $120$ hours available and machine $B$ has maximum of $180$ hours available. Manufacturing a bicycle requires $6$ hours on machine $A$ and $3$ hours on machine $B$. Manufacturing a tricycle requires $4$ hours on machine $A$ and $10$ hours on machine $B$. If profits are Rs. $180$ for a bicycle and Rs. $220$ for a tricycle, formulate and solve the L.P.P. to determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.

A factory produces two types of chemicals $A$ and $B$ . The following table gives the units of ingredients $P$ and $Q$ (per kg) of chemicals $A$ and $B$ as well as minimum requirements of $P$ and $Q$ and also cost per kg of chemicals $A$ and $B$.

$$\begin{gathered} \begin{array}{|cccc|}\hline & \begin{array}{c}\text{ Chemical }\\ \text{ (in units) }\end{array}& \begin{array}{c}\text{ Chemical} \\ \left(\mathrm{in} \mathrm{units}\right)\end{array}& \\ & \mathit{\text{A}}& \mathit{\text{B}}& \mathrm{Minimum} \\ \mathrm{requirements}\\ \begin{array}{l}\text{ Ingradient }\mathit{\text{P}}\\ \text{ (per kg) }\end{array}& 1& 2& 80\\ \begin{array}{l}\text{ Ingradient}\mathit{\text{Q}}\\ \text{ (per kg) }\end{array}& 3& 1& 75\\ \text{ cost (in ₹) }& 4& 6& \\ \hline\end{array} \end{gathered}$$

Find the number of units of chemicals $A$ and $B$ should be produced so as to minimise the cost.

A company produces mixers and food processors. Profit on selling one mixer and one food processor is $₹2000$ and$₹3000$ respectively. Both the products are processed through three machines $A, B, C.$ The time required in hours by each product and total time available in hours per week on each machine are as follows.

$\begin{array}{|cccc|}\hline \text{ Machine }& \text{ Mixer }& \begin{array}{c}\text{ Food }\\ \text{ processor }\end{array}& \begin{array}{c}\text{ Available }\\ \text{ time }\end{array}\\ \mathit{A}& 3& 3& 36\\ \mathit{ }\mathit{B}& 5& 2& 50\\ \mathit{C}& 2& 6& 60\\ \hline\end{array}$

How many mixers and food processors should be produced in order to maximize the profit?