Question

Janice would like to have $40,000 to help pay for college in 8 years. Currently, she has $1000. What interest rate, when compounded yearly, would help her reach her goal?

c. Janice's friend Sarah starts with $7800 and wants to have $18, 400 twenty years from now. What interest rate does she need (compounded yearly)?

b. If y represents the amount of money and x represents the number of years after today, find an equation that models Janice's financial situation. What interest rate does she need to earn?

a. What type of function would best model this situation? Explain how you know and write the general form of this function.

d. Is Janice's goal or Sarah's goal more realistic? Justify your response.

Answer 1

a. The situation can be modeled using exponential growth function since the amount of money is increasing at a constant rate over time. The general form of the function is: A = P(1 + r)^t, where A is the future value, P is the present value, r is the interest rate, and t is the time in years.

b. Using the given information, we can write the equation as:

40,000 = 1,000(1 + r)^8

Dividing both sides by 1,000, we get:

40 = (1 + r)^8

Taking the eighth root of both sides, we get:

1.047 ≈ 1 + r

r ≈ 0.047 or 4.7%

Therefore, Janice needs an interest rate of approximately 4.7% compounded yearly to reach her goal.

c. Using the same formula as above, we can write:

18,400 = 7,800(1 + r)^20

Dividing both sides by 7,800, we get:

2.359 ≈ (1 + r)^20

Taking the twentieth root of both sides, we get:

1.044 ≈ 1 + r

r ≈ 0.044 or 4.4%

Therefore, Sarah needs an interest rate of approximately 4.4% compounded yearly to reach her goal.

d. Janice's goal is less realistic since she would need to earn a higher interest rate in order to reach her goal. Additionally, starting with only $1,000 and expecting to have $40,000 in 8 years is a very ambitious goal, whereas Sarah's goal seems more achievable as she has a longer time horizon and a larger initial investment.