The balance oif a savings account can be modeled by the function b(t)=5000(1.024)^t, where t is the time in years, To model the monthly balance, a student writes:
b(t)=5000(1.024)^t = 5000(1.024)^(1/12 * 12)^t = 5000(1.024^1/12)^12t = 5000(1.002)^12t
Is the student correct? Explain your reasoning.
To determine if the student's modeling is correct, let's break down the steps they took:
Step 1: The student starts with the original function: b(t) = 5000(1.024)^t.
Step 2: Next, the student tries to model the monthly balance by incorporating the idea of compounding monthly. They rewrite the function as: b(t) = 5000(1.024)^(1/12 * 12)^t.
Step 3: The student then simplifies the exponent, expressing it as 1/12 * 12, which equals 1. Rewrite as: b(t) = 5000(1.024^(1/12))^12t.
Step 4: Lastly, the student further simplifies by evaluating 1.024^(1/12) ≈ 1.002, and then raising it to the power of 12t. They arrive at: b(t) = 5000(1.002)^12t.
To determine if the student's modeling is correct, we compare their model with the original function.
The original function is b(t) = 5000(1.024)^t, which represents the balance after time t in years, compounded annually at an annual interest rate of 2.4% (or 1.024 as a decimal).
The student's modeled function is b(t) = 5000(1.002)^12t, which represents the monthly balance, compounded monthly at a monthly interest rate of approximately 0.2% (or 1.002 as a decimal).
The student's modeling is incorrect because they mistakenly used 1.002 as the decimal equivalent of the monthly interest rate instead of 1.024. To model the monthly balance correctly, they should have used 1.024^(1/12) instead of approximating it as 1.002.
Therefore, the correct function for modeling the monthly balance would be: b(t) = 5000(1.024^(1/12))^12t.