question_answer

David wants to invest at most Rs. 12000 in bonds A and B. According to the rule, he has to invest at least Rs. 2000 in bond A and at least Rs. 4000 in bond B. If the rates of interest on bonds A and B respectively are 8% and 10% per annum. Formulate the problem as linear programming problem and solve it graphically for maximum interest. Also, determine the maximum interest received in a year. Why investment is important for future life?

Answer:

Let David invests Rs. x in bond A and Rs. y in bond B. The given information can be written in tabular form as

Bonds	Interest rates	Restrictions
A(x)	8%	At least Rs. 2000
B(y)	10%	At least Rs. 4000

Then, required linear programming problem is Maximize interest \[Z=8%\]of \[x+10%\]of y             \[=0.08x+0.10y\] Subject to the constraints             \[x+y\le 12000,\] \[x\ge 2000,\] \[y\ge 4000\] and       \[x\ge 0,\,\,y\ge 0\] Consider the constraints as equations, we get \[x+y=12000\]                            ...(i) \[x=2000\]                                   ...(ii) \[y=4000\]                                   ...(iii) and         \[x,\text{ }y=0\] Table for \[x+y=12000\] is         

x	0	12000
y	12000	0

So, line passes through the points (0, 12000) and (12000, 0). Putting (0, 0) in the inequality \[x+y\le 12000,\] we get             \[0+0\le 12000\] \[\Rightarrow \]   \[0\le 12000\]                             [true] \[\therefore \] The shaded region is towards the origin. \[\because \] Line x = 2000 is parallel to Y-axis. Putting (1000, 0) in the inequality \[x\ge 2000,\] we get \[1000\ge 2000\]                                [false] \[\therefore \] The shaded region is at the right side of the line. \[\because \] Line y = 4000 is parallel to X-axis. Putting (0, 6000) in the inequality \[y\ge 4000,\] we get             \[6000\ge 4000\]                         [true] \[\therefore \] The shaded region is above the line.          The intersection point of lines (ii) and (iii), (i) and (iii), (i) and (ii) are respectively, A (2000, 4000), B (8000, 4000) and C(2000,10000) Now, plot the graph of the system of inequalities. The shaded portion ABC represents the feasible region which is bounded. And the coordinates of the corner points are A(2000, 4000), 6(8000, 4000) and C(2000,10000), respectively Now, the values of Z at each comer point are given below

Corner points	\[\mathbf{Z=0}\mathbf{.08x+0}\mathbf{.10y}\]
A(2000, 4000)	\[Z=0.08(2000)+0.10(4000)\] \[=160+400=560\]
B(8000, 4000)	\[Z=0.08(8000)+0.10(4000)\]\[=640+400=1040\]
C(2000, 10000)	\[Z=0.08(2000)+0.10(10000)\]\[=160+1000=1160\](maximum)

\[\therefore \] Maximum value of Z is 1160 at (2000, 10000). Hence, maximum profit is Rs.1160 when Rs. 2000 are invested in bond A and Rs. 10000 are invested in bond B. Value The growth of money is also important to fulfil basic needs of life and investing can help a person achieve long life goal easily