Math Problem Statement

You deposit $200 in an investment account that earns 6% annual interest compounded monthly. You also have a savings account with a constant balance of $800. Identify the function B(t) that represents the total balance of the two accounts after t years.

Solution

This problem involves calculating the total balance B(t)B(t) in two accounts:

  1. An investment account that compounds monthly with a 6% annual interest rate.
  2. A savings account with a constant balance of $800.

Breaking Down the Problem:

  • The formula for compound interest is:
    A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}
    where:
    PP = principal amount ($200),
    rr = annual interest rate (6% = 0.06),
    nn = number of compounding periods per year (12 for monthly),
    tt = number of years.

Thus, the balance of the investment account after tt years is:
200(1+0.0612)12t=200(1.005)12t.200 \left(1 + \frac{0.06}{12}\right)^{12t} = 200(1.005)^{12t}.

  • Adding the savings account balance of $800, the total balance B(t)B(t) becomes:
    B(t)=200(1.005)12t+800.B(t) = 200(1.005)^{12t} + 800.

Correct Answer:

The correct choice is: (B)B(t)=200(1.005)12t+800.\text{(B)} \quad B(t) = 200(1.005)^{12t} + 800.

Let me know if you want further clarification!


Here are 5 related questions to expand your understanding:

  1. What happens to the total balance if the savings account increases by $100 annually?
  2. How does the compounding frequency affect the total balance in the investment account?
  3. If the interest rate is increased to 8%, how would that change the formula?
  4. How can you graph B(t)B(t) to visualize the growth over time?
  5. What is the balance after 5 years?

Tip: Always confirm the compounding frequency when working with compound interest problems—it directly impacts the formula and the results.

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Exponential Growth
Arithmetic Addition

Formulas

Compound Interest Formula: A = P(1 + r/n)^(nt)
Total Balance: B(t) = Investment Balance + Savings Balance

Theorems

Properties of Exponential Functions

Suitable Grade Level

Grades 9-11