Question

A library has to accommodate two different types of books on a shelf. The books are 6cm and 4cm thick and weigh 1kg and 1.5kg respectively. The shelf is 96cm long and at most can support a weight of 21kg. How should the shelf be filled with the books of two types in order to include the greatest number of books? Make it as an LLP and solve it graphically.

Answer 1

Consider the problem

Let two type of books be $x$ and $y$,

The required LLP is maximize $Z=x+y$ Subject to constraints

$\begin{array}{c}6x+4y\le 96Or3x+2y\le 48\\ x+\frac{3}{2}y\le 21Or2x+3y\le 42\end{array}$

and $x,y\ge 0$

On considering the inequalities as equations, We get

$\begin{array}{c}3x+2y=48...\left(i\right)\\ 2x+3y=42....\left(ii\right)\end{array}$

Now tablefor line $3x+2y=48$ is

x	0	16
y	24	0

So, it passes through $(0,0)$ and $(16,0)$

On putting $(0,0)$ in $3x+2y\le 48$ we get

$\begin{array}{c}0+0\le 48\\ Or0\le 48\left[whichistrue\right]\end{array}$

so, the half plane is towards the origin.

And Table for $2x+3y=42$ is

x	0	21
y	14	0

So, it passes through $(0,14)$ and $(21,0)$.

On putting $(0,0)$ in $2x+3y\le 42$ We get

$\begin{array}{c}0+0\le 42\\ Or0\le 42\left[whichistrue\right]\end{array}$

On solving equation $\left(i\right)$ and $\left(ii\right)$ we get

$x=12$ and $y=6$

Thus, the point of intersection is $B(12,6)$

And from the graph $OABCD$ is the feasible region which is bounded. The corner points are $O(0,0),A(0,14),B(12,16),C(16,0)$.

And the value of $Z$ at corner points are

Corner points	Value of $Z=x+y$
$O(0,0)$	$Z=0+0=0$
$A(0,14)$	$Z=0+14=14$
$B(12,16)$	$Z=12+6=18\left(maximum\right)$
$C(16,0)$	$Z=16+0=16$

From the table the maximum value of $Z$ is $18$ at $B(12,6)$.

Hence, The maximum number of books of $I$ type is $12$ and books of $II$ type is $6$.