Final answer:
Kara needs to earn an annually compounded interest rate of approximately 8.447% to increase her investment from $2,000 to $3,000 in five years, using the formula for compound interest and solving for the rate.
Step-by-step explanation:
The question involves finding the annually compounded rate of interest Kara needs to earn to grow her investment from $2,000 to $3,000 in five years. To solve this, we use the formula for compound interest: A = P(1+r)^n, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), and n is the number of years the money is invested. We rearrange the formula to solve for r.
So, substituting the given values into the formula will look like this: $3,000 = $2,000(1+r)^5. Now, we divide both sides by $2,000 to isolate the term with r on one side, resulting in 1.5 = (1+r)^5. To find r, we need to take the fifth root of 1.5 and then subtract 1. Using a scientific calculator, we find that r is approximately 0.08447, or 8.447%.
Thus, Kara needs to earn an annually compounded interest rate of about 8.447% to reach her $3,000 goal in five years.