Final answer:
To determine the annual interest rate needed to earn $1200 in interest in 4.4 years from $48,000, use the compound interest formula. The annual interest rate would need to be approximately 2.73%.
Step-by-step explanation:
To determine the annual interest rate needed to earn $1200 in interest in 4.4 years from $48,000, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
- A is the final amount ($48,000 + $1200 = $49,200)
- P is the principal amount ($48,000)
- r is the annual interest rate
- n is the number of times the interest is compounded per year (since it compounds daily, n = 365)
- t is the number of years (4.4)
Substituting the given values, we have:
$49,200 = $48,000(1 + r/365)^(365*4.4)
Solving for r:
Divide both sides by $48,000:
(1 + r/365)^(365*4.4) = $49,200/$48,000 = 1.025
Take the natural logarithm of both sides:
365*4.4 * ln(1 + r/365) = ln(1.025)
Divide both sides by 365*4.4:
ln(1 + r/365) = ln(1.025)/(365*4.4)
Take the antilogarithm of both sides:
1 + r/365 = e^(ln(1.025)/(365*4.4))
Subtract 1 from both sides:
r/365 = e^(ln(1.025)/(365*4.4)) - 1
Multiply both sides by 365:
r = 365 * (e^(ln(1.025)/(365*4.4)) - 1)
Calculating this expression using a calculator, we find that the annual interest rate would need to be approximately 2.73% to earn $1200 in interest in 4.4 years from $48,000.