. Margaret is planning to invest up to $22,000 in certificates of deposit at City Bank and People’s Bank. She wants to invest at least $2,000 but no more than $14,000 at City Bank. People’s Bank does not insure more than a $15,000 investment, so she will invest no more than that in People’s Bank. The interest is 6% at City Bank and 6½% at Peoples Bank. This is simple interest for one year. How much should she invest in each bank in order to maximize her income? What is the maximum income?
Best Investment:
1. Peoples Bank, 15000 @ 6.5 %.
City Bank, 7000 @ 6%.
2. Peoples Bank:
Pt = Po + r*t*Po,
Pt = 15000 + 0.065 / yr *1yr * 15000,
Pt = 15000 + 975 = 15975.
City Bank:
Pt = 7000 + 0.06 / yr * 1yr * 7000,
Pt = 7000 + 420 = 7420.
Total Income = 15975 + 7420 = 23395.
To solve this problem, we can use a two-variable linear inequalities approach. Let's define the variables:
- Let x represent the amount invested at City Bank, in dollars.
- Let y represent the amount invested at People's Bank, in dollars.
According to the given information, we have the following constraints:
1. 2000 ≤ x ≤ 14000 (Invest at least $2,000 but no more than $14,000 at City Bank).
2. y ≤ 15000 (Invest no more than $15,000 at People's Bank).
3. x + y ≤ 22000 (The total investment should not exceed $22,000).
We aim to maximize Margaret's income, which is determined by the interest earned. The interest from City Bank is calculated as 6% of x, and the interest from People's Bank is calculated as 6.5% of y.
Thus, the objective function (income) is:
Income = 0.06x + 0.065y
To find the maximum income, we need to solve this optimization problem by finding the feasible region and then evaluating the objective function at each corner point.
First, let's graph the feasible region:
1. Plot the line x = 2000 (vertical line at x = 2000).
2. Plot the line x = 14000 (vertical line at x = 14000).
3. Plot the line y = 15000 (horizontal line at y = 15000).
4. Plot the line x + y = 22000 (diagonal line from point (2000, 20000) to point (14000, 8000)).
5. Shade the region that satisfies all the given constraints.
Now, let's find the corner points of the feasible region by solving the corresponding equations:
1. (2000, 15000) [intersection of x = 2000 and y = 15000]
2. (14000, 15000) [intersection of x = 14000 and y = 15000]
3. (14000, 8000) [intersection of x = 14000 and x + y = 22000]
Finally, evaluate the objective function (income) at each corner point:
1. For (2000, 15000): Income = 0.06(2000) + 0.065(15000) = 1800 + 975 = $2775
2. For (14000, 15000): Income = 0.06(14000) + 0.065(15000) = 840 + 975 = $1815
3. For (14000, 8000): Income = 0.06(14000) + 0.065(8000) = 840 + 520 = $1360
From the three possibilities, the maximum income of $2775 is obtained when Margaret invests $2,000 at City Bank and $15,000 at People's Bank.