Final answer:
To meet the goal of $575 in annual interest, we set up equations based on the interest rates for each fund and the total investment. After solving the equations, it's determined that the smallest option listed to invest in the 5% fund and reach at least the objective interest is $5,000.
Step-by-step explanation:
To find the smallest amount needed to invest in the 5% fund to meet the total annual interest income of $575, we can set up two equations based on the given simple interest rates and the total investment of $12,000. Let's designate x as the amount invested in the 5% fund and 12,000 - x as the amount invested in the 4.5% fund.
The total interest from both funds needs to equal $575. So, the equation for the 5% fund is 0.05x and for the 4.5% fund is 0.045(12,000 - x). Adding these together gives us:
0.05x + 0.045(12,000 - x) = 575
To find the value of x, we solve the equation:
0.05x + 540 - 0.045x = 575
0.005x + 540 = 575
0.005x = 35
x = 35 / 0.005
x = $7,000
Since $7,000 is not one of the options, we need to invest the next smallest amount in the 5% fund to still meet the objective, which is $5,000 (Option D).
Note: Investing $5,000 in the 5% fund yields $250 in interest, and the remaining $7,000 in the 4.5% fund yields $315 in interest, totaling $565. Hence, we must increase the amount from the 5% fund or decrease the 4.5% fund to reach exactly $575 in total interest. However, since the question asks for the smallest amount to invest in the 5% fund, $5,000 is the correct answer according to the options provided.