# A man rides his motorcycle at the speed of 50km/hour.He has to speed Rs 2 per km on petrol.If he rides it at at a faster speed of 80km/hour,the petrol cost increases to Rs 3 per km.He has at most Rs120 to spend on petrol and one hour's time.He wishes to find the maximum distance that he can travel.Express this problem as a linear programming problem.

$\begin{array}{1 1}(A)\;Z=x+y\; subjected \;to \;constraints 8x+5y\leq 120,2x+3y\leq 120 \\ (B)\;Z=x+y\; subjected \;to \;constraints \;5x+8y\leq 400,2x+3y\leq 120 \\(C)\;Z=x+y\; subjected\; to \;constraints 8x+5y\leq 400,2x+3y\leq 120 \\ (D)Z=x+y\; subjected \;to\; constraints \;8x+5y\leq 100,2x+3y\leq 120\end{array}$

Toolbox:
• Let $R$ be the feasible region for a linear programming problem and let $Z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region.
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R
Step 1:
It is given that the normal speed of the motorcycle is $50km/hr$
His increased speed is 80km/hr
he has at most of 1hour's time.
Let $x$ and $y$ be the time,which we takes,when the speed is normal
$\therefore 80x+50y\leq 4000$
$\Rightarrow 8x+5y\leq 400$
The cost for the petrol when goes in the normal speed is Rs.2 per kms.
The cost for the petrol when goes in the increased speed is Rs.3 per kms.
He has at most of Rs 120 to spend.
Step 2:
Therefore $2x+3y\leq 120$
$x\geq 0,y\geq 0$
The objective function is $Z=x+y$ subjected to the constraints $8x+5y\leq 400,2x+3y\leq 120,x,y\geq 0$
$Z=x+y$ subjected to constraints $8x+5y\leq 400,2x+3y\leq 120$