The First Street branch of Capitol Bank has a sum of $70 million earmarked for home loans and commercial-development loans. The bank expects to realize and 8% annual rate of return on the home loans and a 6% annual rate of return on the commercial-development loans. Management has decided that the total amount of home loans is greater than or equal to 3 times the total amount of commercial-development loans. Owing to prior commitment, at least $11 million of the funds has been designated for commercial-development loans. Determine the amount of each type of loan the bank should extend to maximize it returns. What is the maximum return?
A) Bank should extend $48 million in home loans, $48 million of commercial-development loans to attain a maximum return of $5.38 million.
B) Bank should extend $59 million in home loans, $11 million of commercial-development loans to attain a maximum return of $5.38 million.
C) Bank should extend $11 million in home loans, $59 million of commercial-development loans to attain a maximum return of $5.38 million.
D) Bank should extend $11 million in home loans, $48 million of commercial-development loans to attain a maximum return of $5.38 million.
E) Bank should extend $48 million in home loans, $11 million of commercial-development loans to attain a maximum return of $5.38 million.

Question

Quizplus · Accepted Answer

Let x be the total amount of commercial-development loans. Then, the total amount of home loans should be at least 3x. Also, at least $11 million should be designated for commercial-development loans. Therefore, we have:

x ≥ 11 million
Total amount of home loans ≥ 3x
Total amount of home loans + Total amount of commercial-development loans = $70 million

To maximize the returns, we need to maximize the total amount of money invested in both types of loans while satisfying the above conditions. Let's express the total return as a function of x:

Return = 0.08(Total amount of home loans) + 0.06(Total amount of commercial-development loans)
Return = 0.08(3x) + 0.06(x) = 0.3x

Now, we want to maximize this function subject to the conditions above. Using the third condition, we can express the total amount of home loans in terms of x:

Total amount of home loans = 70 million - Total amount of commercial-development loans
Total amount of home loans = 70 million - x
Total amount of home loans ≥ 3x
70 million - x ≥ 3x
4x ≤ 70 million
x ≤ 17.5 million

Now, we need to check the other two conditions:

- x ≥ 11 million: This condition is already satisfied since we set the minimum value of x to be 11 million.
- Total amount of home loans ≤ $70 million: We can substitute x = 17.5 million to find the maximum value of the total amount of home loans:

Total amount of home loans = 70 million - x
Total amount of home loans = 70 million - 17.5 million
Total amount of home loans = 52.5 million

Therefore, the bank should extend $11 million of commercial-development loans and $52.5 million of home loans to maximize its returns, resulting in a total return of:

Return = 0.3x = 0.3(17.5 million) = $5.25 million

The closest option to this answer is B: Bank should extend $59 million in home loans, $11 million of commercial-development loans to attain a maximum return of $5.38 million. However, this option is incorrect because it violates the condition that the total amount of home loans should be greater than or equal to 3 times the total amount of commercial-development loans.

The First Street Branch of Capitol Bank Has a Sum