Carla wants to start a college fund for her daughter Lila. She puts $63,000 into an account that grows at a rate of 2.55% per year, compounded monthly. Write a function, C (t), that represents the amount of money in the account t years after the account is opened, given that no more money is deposited into or withdrawn from the account. Calculate algebraically the number of years it will take for the account to reach $100,000, to the nearest hundredth of a year.

Question

Ashanti Arnold

Mathematics

14 July, 06:29

Carla wants to start a college fund for her daughter Lila. She puts $63,000 into an account that grows at a rate of 2.55% per year, compounded monthly. Write a function, C (t), that represents the amount of money in the account t years after the account is opened, given that no more money is deposited into or withdrawn from the account. Calculate algebraically the number of years it will take for the account to reach $100,000, to the nearest hundredth of a year.

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Hanna Lowe · Answer 1

Step-by-step explanation:

We would apply the formula for determining compound interest which is expressed as

A = P (1+r/n) ^nt

Where

A = total amount in the account at the end of t years

r represents the interest rate.

n represents the periodic interval at which it was compounded.

P represents the principal or initial amount deposited

From the information given,

P = 63000

r = 2.55% = 2.55/100 = 0.0255

n = 12 because it was compounded 12 times in a year.

Therefore, function, C (t), that represents the amount of money in the account t years after the account is opened is

C (t) = 63000 (1 + 0.0255/12) ^12t

C (t) = 63000 (1.002125) ^12t

For C (t) = 100000,

100000 = 63000 (1.002125) ^12t

100000/63000 = (1.002125) ^12t

1.587 = 1.002125) ^12t

Taking log of both sides

Log 1.587 = log 1.002125) ^12t

Log 1.587 = 12tlog 1.002125) ^

0.2005 = 12t * 0.00092

0.2005 = t * 0.01104

t = 0.2005/0.01104

t = 18.16 years