Question

When tracey and Chris’s daughter Emily was born, they set up a trust fund to mature on her 18th birthday. They invested $25,000. When Emily turned 18, the trust fund was worth $100,000. At what continuous rate of interest r was the money invested? (Use A=Pe^rt) enter your answer as a percentage rounded to one decimal point.

Answer 1

To find the continuous rate of interest that increased an investment from $25,000 to $100,000 over 18 years, we use the formula A=Pe^rt. Calculating and applying natural logarithms, the rate is approximately 22.2% when rounded to one decimal point.

Step-by-step explanation:

The student is tasked with finding the continuous rate of interest that grew an investment from $25,000 to $100,000 over the course of 18 years. Utilizing the given formula A = Pert, where A is the future value of the investment, P is the principal amount, e is the base of the natural logarithm, r is the rate of interest, and t is the time in years the money is invested, we can solve for r.

Plugging in the given values, we get:

100,000 = 25,000 × e(r × 18)

To find r, we apply the natural logarithm to both sides:

ln(100,000/25,000) = ln(e(r × 18))

4 = 18r

r = 4/18

r = 0.2222

As a percentage, the continuous compound interest rate is approximately 22.2%, after rounding to one decimal point.

Answer 2

Answer: The continuous interest rate at which the money was invested is approximately 15.4%.

Step-by-step explanation: We can use the formula for continuous compounding to find the value of r:

A = Pe^(rt)

Where A is the final value, P is the initial investment, e is the base of the natural logarithm, r is the continuous interest rate, and t is the time.

Plugging in the given values, we get:

$100,000 = $25,000e^(r*18)

Dividing both sides by $25,000 and taking the natural logarithm of both sides, we get:

ln(4) = 18r

Solving for r, we get:

r = ln(4)/18 ≈ 0.154

Multiplying by 100 to convert to a percentage and rounding to one decimal point, we get:

r ≈ 15.4%

Therefore, the continuous interest rate at which the money was invested is approximately 15.4%.