A school principal wants to buy some book cases for the school library. She can choose between two types of book case. Type A costs $10 and it requires 0.6 m of floor space and holds 0.8 m of books. Type B costs $20 and it requires 0.8 m of floor space and holds 1.2 m of books. The maximum floor space available is 7.2 m and the budget is $140 (but the school would prefer to spend less). What number and type of book cases should the principal buy to get the largest possible storage space for books ?
Answers
Answer:
Consider the problem
Let two type of books be x and y,
The required LLP is maximize Z=x+y Subject to constraints
6x+4y≤96Or3x+2y≤48
x+
2
3
y≤21Or2x+3y≤42
and x,y≥0
On considering the inequalities as equations, We get
3x+2y=48...(i)
2x+3y=42....(ii)
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≤48 we get
0+0≤48
Or0≤48[whichistrue]
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≤42 We get
0+0≤42
Or0≤42[whichistrue]
On solving equation (i) and (ii) 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(maximum)
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.
Answer:
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