A financial consultant wishes to invest up to a total of $33,000 in two types of securities, one that yields 8% per year and another that yields 6% per year. Furthermore, she believes that the amount invested in the first security should be at most one-third of the amount invested in the second security. What investment program should the consultant pursue in order to maximize income?
The consultant should invest $___________ in the first security and $_______ in the second security to obtain the maximum income of $___________
See previous post.
To find the investment program that maximizes income, we need to determine the amount to be invested in each security. Let's assume the amount invested in the first security is x dollars.
According to the given condition, the amount invested in the second security should be at least three times the amount invested in the first security, which means the amount invested in the second security is 3x dollars.
The total amount invested is the sum of the amounts invested in each security, which is x + 3x = 4x dollars:
Total amount invested in both securities = x + 3x = 4x
Now, we can determine the income from each investment. The first security earns an 8% annual yield, which is 0.08 times the amount invested in the first security:
Income from the first investment = 0.08 * x = 0.08x
Similarly, the second security earns a 6% annual yield, which is 0.06 times the amount invested in the second security:
Income from the second investment = 0.06 * (3x) = 0.18x
The total income is the sum of the incomes from both investments:
Total income = 0.08x + 0.18x = 0.26x
To maximize income, we want to find the value of x that maximizes the total income.
Since the total income is a linear function of x, the maximum value occurs at the endpoint. The given condition states that the consultant wishes to invest up to a total of $33,000.
Therefore, the maximum amount that can be invested is 33,000 dollars.
We can set up an inequality to find the maximum value of x:
4x <= 33000
Dividing both sides by 4, we get:
x <= 33000/4
x <= 8250
So, the consultant can invest a maximum of $8250 in the first security.
The amount to be invested in the second security is three times that amount:
Amount invested in the second security = 3 * 8250 = $24750
Now, we can calculate the maximum income by substituting these values into the total income formula:
Total income = 0.26x = 0.26 * 8250 = $2145
Therefore, the consultant should invest $8250 in the first security and $24750 in the second security to obtain the maximum income of $2145.