Final answer:
To find out how many years it will take for $30,000 to grow to $210,000 at a 5% annual interest rate, we use the compound interest formula. By rearranging the formula to solve for the time variable t, we calculate it will take approximately 47.75 years to have enough to purchase the Ferrari.
Step-by-step explanation:
To determine how long it will take to save enough money to buy a new $210,000 Ferrari with an initial investment of $30,000 and an annual interest rate of 5%, we'll use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount ($30,000).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested or borrowed for, in years.
Since the bank pays interest annually, n will be 1. We need to solve for t when the amount A is $210,000:
$210,000 = $30,000(1 + 0.05/1)¹ˣ⁺
Rearranging the formula to solve for t, we have:
t = ln($210,000/$30,000) / ln(1 + 0.05)
Calculating this gives us:
t = ln(7) / ln(1.05)
Using a calculator, we find that t is approximately 47.75 years.
It will take almost 48 years for the initial $30,000 to grow to $210,000 at a 5% annual interest rate, assuming the interest is compounded annually and no additional deposits are made.