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in Linear Programming by (47.4k points)
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Anil wants to invest at most Rs 12000 in Saving Certificates and National Saving Bonds. According to rules, he has to invest at least Rs 2000 in Saving Certificates and at least 4000 in National Saving Bonds. If the rate of interest on saving certificate is 8% per annum and the rate of interest on National Saving Bonds is 10% per annum, how much money should he invest to earn maximum yearly income? Find also his maximum yearly income.

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Let Anil invests Rs x and Rs y in saving certificate (SC) and National saving bond (NSB) respectively.

Since, the rate of interest on SC is 8% annual and on NSB is 10% annual. So, interest on Rs x of SC is \(\frac{8x}{100}\) and Rs y of NSB is \(\frac{10x}{100}\) per annum.

Let Z be total interest earned so,

Given he wants to invest Rs 12000 is total

x + y ≤ 12000

According to the rules he has to invest at least Rs 2000 in SC and at least Rs 4000 in NSB.

x ≥ 2000

y ≥ 4000

Hence the mathematical formulation of LPP is to find x and y which

Maximizes Z

The region represented by x ≥ 2000: line x = 2000 is parallel to the y - axis and passes through (2000, 0).

The region not containing the origin represents x ≥ 2000

As (0, 0) doesn’t satisfy the inequation x ≥ 2000

The region represented by y ≥ 4000: line y = 4000 is parallel to the x - axis and passes through (0, 4000).

The region not containing the origin represents y ≥ 4000

As (0, 0) doesn’t satisfy the inequation y ≥ 4000

Region represented by x + y ≤ 12000: line x + y = 12000 meets axes at A(12000, 0) and B(0, 12000) respectively. The region which contains the origin represents the solution set of x + y ≤ 12000

as (0, 0) satisfies the inequality x + y ≤ 12000.

Region x, y ≥ 0 is represented by the first quadrant.

The corner points are E(2000, 10000), C(2000, 4000), D(8000, 4000).

The values of Z at these corner points are as follows:

The maximum value of Z is Rs 1160 which is attained at E(2000, 10000).

Thus the maximum earning is Rs1160 obtained when Rs 2000 were invested in SC and Rs 10000 in NSB.

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