Final answer:
To find the interest rate for an investment that grows with daily compounding, the compound interest formula is used. The rate needed for an $8,000 investment to grow to $12,000 in 4 years is approximately 10.24% annually.
Step-by-step explanation:
To find the interest rate that grows a $8,000 investment to $12,000 in 4 years with daily compounding, we use the formula for compound interest:
A = P(1 + r/n)^(nt).
Here, A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
In this case, the principal (P) is $8,000, the amount (A) is $12,000, and we’re solving for the annual interest rate (r). The compounding is daily so n is 365 and the time (t) is 4 years. Rearranging the formula to solve for r, we get:
r = (n * ((A/P)^(1/(nt)) - 1))
Inserting the given values:
r = 365 * (($12,000/$8,000)^(1/(365*4)) - 1)
After solving this, the desired interest rate is approximately 10.24% annually.
This example demonstrates how powerful compound interest can be over time and the importance of knowing the specifics of how it's calculated, especially the frequency of compounding. With larger amounts and longer periods, the impact of compound interest becomes even more significant.