Minimize Z = 6x + 2y

English0Maharashtra State BoardHSC Science (General) 12th Standard Board ExamQuestion PapersQuestion Papers328Textbook Solutions13169MCQ Online Mock Tests73Important Solutions9041Concept Notes & Videos444Time Tables32SyllabusSolve the Following Linear Programming L. P. P. Graphically Minimize Z = 6x + 2y Subject to 5x + 9y ≤ 90 - Mathematics and Statistics

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Question

Solve the following L. P. P. graphically:Linear Programming

Minimize Z = 6x + 2y

Subject to

5x + 9y ≤ 90

x + y ≥ 4

y ≤ 8

x ≥ 0, y ≥ 0

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SolutionShow Solution

To draw the feasible region, construct table as follows:

Inequality	5x + 9y ≤ 90	x + y ≥ 4	y ≤ 8
Corresponding equation (of line)	5x + 9y = 90	x + y = 4	y = 8
Intersection of line with X-axis	(18, 0)	(4, 0)	−
Intersection of line with Y-axis	(0, 10)	(0, 4)	(0, 8)
Region	Origin side	Non-origin side	Origin side

Shaded portion ABCDE is the feasible region, whose vertices are A(4, 0), B(18, 0), C,D(0, 8) and E(0, 4).

C is the point of intersection of the lines y = 8 and 5x + 9y = 90.

Putting y = 8 in 5x + 9y = 90, we get

5x + 72 = 90

∴ x = 18/5

∴ C = `(18/5, 8)`

Here, the objective function is Z = 6x + 2y,

Z at A(4, 0) = 6(4) + 2(0) = 24

Z at B(18, 0) = 6(18) + 2(0) = 108

Z at `C(18/5,8) = 6(18/5)` + 2(8)

= 188/5 = 37.6

Z at D(0, 8) = 6(0) + 2(8) = 16

Z at E(0, 4) = 6(0) + 2(4) = 8

∴ Z has minimum value 8 at E(0, 4).

∴ Z is minimum, when x = 0 and y = 4.

shaalaa.comGraphical Method of Solving Linear Programming Problems Report Error Is there an error in this question or solution?Q 3.2.2Q 3.2.1Q 3.2.32012-2013 (October)

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2012-2013 (October) (with solutions)Q 3.2.2 | 4 marks

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