Question

Sung Lee invests $3,000 at age 18. He hopes the investment will be worth $6,000 when he turns 25. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Round to the nearest tenth of a percent.

Answer 1

Sung Lee needs an approximate annual growth rate of 9.9% for his $3,000 investment to grow to $6,000 in 7 years with continuous compounding. This rate was found by rearranging the formula for continuous compound interest and solving for the rate.

Step-by-step explanation:

When Sung Lee invests $3,000 at age 18 and wants it to grow to $6,000 by age 25, he is looking to double his investment over 7 years. If the interest compounds continuously, we can use the formula for continuous compound interest which is A = Pert, 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 t is the time in years.

To find the interest rate needed, we'll rearrange the formula to solve for r:

1. Set A to $6,000, P to $3,000, and t to 7 years.
2. Substitute the values into the formula: $6,000 = $3,000 * e7r.
3. Divide both sides by $3,000 to get 2 = e7r.
4. Take the natural logarithm (ln) of both sides: ln(2) = 7r.
5. Divide by 7 to isolate r: r = ln(2) / 7.
6. The approximate rate of growth r needed is 0.099 or 9.9% when rounded to the nearest tenth of a percent.

This shows that Sung Lee will need an approximate annual growth rate of 9.9% for his investment to grow to $6,000 in 7 years through continuous compounding.

Answer 2

10.4%

Step-by-step explanation:

Sung Lee invests $3,000 at age 18.

He hopes the investment will be worth $6,000 when he turns 25, that is, after 7 years.

The interest compounds (Compound Interest). The formula for the final amount in a compound interest is:

`A = P(1 + R)^T`

where P = Principal (Amount invested) = $3000

A = final amount = $6000

R = rate

T = number of years = 7 years

This implies that:

`6000 = 3000(1 + R)^7\\\\(6000)/(3000) = (1 +R)^7\\\\2 = (1 + R)^7`

Find the 7th root of 2:

`\sqrt[7]{2} = (1 + R)\\\\1.104 = (1 + R)\\\\=> R = 1.104 - 1\\\\R = 0.104`

=> R = 10.4%

Hence, the rate would need to be 10.4% for him to achieve his goal.