Question

Mai invests $20,000 at age 20. She hopes the investment will be worth $500,000 when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal? Round to the nearest tenth of a percent.

Answer 1

Mai needs an annual growth rate of approximately 17.9%, rounded to the nearest tenth, for her investment of $20,000 to reach $500,000 in 20 years with continuous compounding, using the formula A = Pe^(rt).

Step-by-step explanation:

The question involves finding the necessary rate of growth for an investment to grow from $20,000 to $500,000 over 20 years, with continuous compounding. The formula for continuous compounding is A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, and t is the time in years.

Let's try to find the rate 'r' that Mai needs for her investment:

• $500,000 = $20,000e20r
• Divide both sides by $20,000 to get e20r = 25
• Take the natural logarithm of both sides to find ln(e20r) = ln(25)
• Apply the power rule of logarithms to get 20r = ln(25)
• Divide by 20 to find the rate r = ln(25)/20
• Use a calculator to find r ≈ 0.1791, or about 17.9%

Therefore, to reach her goal of $500,000 in 20 years with continuous compounding, Mai would need an annual growth rate of 17.9%, rounded to the nearest tenth of a percent.