Question

A man rides his motorcycle at the speed of $50km/hr$. He has to spend Rs. $2$ per km on petrol. If he rides it at a faster speed of $80km/hr$ the petrol cost increases to Rs. $3$ km. He has at most Rs. $120$ to spend on petrol and one hour time. Determine the maximum distance that the man can travel.

Answer 1

Let the young man ride with the speed $50km/hr$ for $x$ hours and with the speed $80km/hr$ for $y$ hours respectively.

Money spent at speed $50km/hr=100x$
And Money spent at speed $80km/hr=240y$

Maximum distance :
$Z=50x+80y$ …$\left(1\right)$

Subject to the constraints
$x+y\le 1$ ….$\left(2\right)$
$100x+240y\le 120$ ….$\left(3\right)$
$x,y\ge 0$ ….$\left(4\right)$

Graph these inequalities and identify the corner points.

We get the corner points as $(0,\frac{1}{2}),(\frac{6}{7},\frac{1}{7}),(1,0)$

Substituting these values in $\left(1\right)$ we get maximum value of $z$ is $54.3km$