Algebra 2 Problems of the Day

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January 2019, Part IV

The Question in Part IV is worth 6 credits. Partial credit can be earned.


37.Tony is evaluating his retirement savings. He currently has $318,000 in his account, which earns an interest rate of 7% compounded annually. He wants to determine how much he will have in the account in the future, even if he makes no additional contributions to the account.

Write a function, A(t), to represent the amount of money that will be in his account in t years.

Graph A(t) where 0 ≤ t ≤ 20 on the set of axes below.

Tony’s goal is to save $1,000,000. Determine algebraically, to the nearest year, how many years it will take for him to achieve his goal.

Explain how your graph of A(t) confirms your answer.

Answer:
A(t) = 318000(1.07)^t
Put the equation into a graphing calculator and get the table of values. Plot the points.

Tony's goal is $1,000,000. We can see from the Table of Values and the graph that this will happen between years 16 and 17. However, we need to solve this algebraically. But if we don't get an answer between 16 and 17, we know that a mistake was made and can check our work.

Substitute 1000000 for A(t) and solve for t.

1000000 = 318000(1.07)^t
1000000 / 318000 = (1.07)^t
1000 / 318 = (1.07)^t
ln(1000 / 318) = ln(1.07)^t
ln(1000 / 318) = t * ln(1.07)
ln(1000 / 318) / ln(1.07) = t
t = 16.93359...
It will take approximately 17 years.

Explain: On the graph A(17) is approximately $1,000,000. This confirms the answer we found.



Comments and questions welcome.

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