A financial manager wants to invest up to $400,000 of his client’s money in two investment plans. Plan A pays an annual interest rate of 6% and Plan B pays an annual interest rate of 8%. Although Plan B pays a higher interest rate, there is some risk involved. The manager and the client have agreed to invest at most 40% of the total in Plan B. The manager feels that in order to get the best return, the client should invest no less than $180,000. How much should be invested in each plan to maximize the return?
a. Let x represent the amount invested in Plan A. Write an expression for the interest received from this investment.
b. Let y represent the amount invested in Plan B. Write an expression for the interest received from this investment.
c. Use parts a and b to write the objective function for the investment return.
d. Write an expression that represents the total amount that can be invested.
e. Write the constraint that represents the limit on the total amount that can be invested. This will be one of the constraints that form the feasible region.
f. Write an expression that represents 40% of the total amount found in part d.
g. The manager is recommending that the amount invested in Plan B be no more than the amount found in part f. Write this inequality.
h. Use the distributive property to simplify the inequality found in g.
i. Combine like terms. Simplify the inequality. This will be one of the constraints that form the feasible region.
j. Write the inequality representing the manager's recommendation for the total amount that should be invested. This will be one of the constraints that form the feasible region.
k. Graph the feasible region.
l. Determine the vertices of the feasible region.
m. Which vertex of the feasible region will maximize the return on the investment? What is that maximum return?