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Question
A factory makes tennis rackets and cricket bats. A tennis racte takes 1.5 hour of a machine time and 3 hours of craftman's time in its making white a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman time. Then what number of rackets and lot must be made if the factory is to work at full capacity?
Options
(4s, 12)
(12s, 4)
(5s, 8)
(10s, 18)
Solution
(4s, 12)
Explanation:
Let x tennis rackets and y cricket bats be produced in the factory in one day and have:
Item | Number | Machine hr. |
Craftman's hr. |
Profit |
Tennis racket |
x | 1.5 | 3 | Rs. 30 per item |
Cricket bats |
y | 3 | 1 | Rs. 10 per item |
Total time available |
42 | 24 |
Total Machine hours = 1.5x + 3y
Maximum time available = 42 hours
1.5x + 3y ≤ 42
or 3x + 6y ≤ 84
or x + 2y ≤ 28 ......(i)
Craftinan' s hours = 3x + y
Maximum time available = 24
3x + y ≤ 24
x, y ≥ 0 ......(ii)
(i) Z = x + y
Constraints are x + 2y ≤ 28
3x + y ≤ 24
x, y ≥ 0
(a) The line x + 2y = 28 passes through A(28, 0) and B(0, 14).
Putting x = 0, y = 0 in x + 2y ≤ 28
We get 0 ≤ 28
⇒ x + 2y ≤ 28 reprents the region on and below AB
(b) The line 3x + y = 24 passes through C(8, 0) and D (0, 24).
Putting x = 0, y = 0 in 3x + y ≤ 24 we get 0 ≤ 24 which is true
∴ 3x + y ≤ 24 represents the region and below the line (1)
(c) x ≥ 0 represents the region on and to the right of y-axis.
(d) y ≥ 0 represents the region on and above the of x-axis.
(e) The shaded area B Pco is the feasible region.
The two lines AB and CD are
x + 2y = 28 ......(i)
3x + y = 24 .......(2)
Multiply equation (2) by 2 and subtract (1) from (2)
5x = 48 – 28 = 20
x = 4
From (2), 12 + y = 24, y = 12
Thus these lines meet at P(4, 12)
As a result, 4 tennis rackets and 12 cricket bats are produced.