Problem 81 - College Algebra Essentials

Question

You have \(\$ 10,000\) to invest. One bank pays \(5 \%\) interest compounded quarterly and a second bank pays \(4.5 \%\) interest compounded monthly. a. Use the formula for compound interest to write a function for the balance in each bank at any time \(t\) b. Use a graphing utility to graph both functions in an appropriate viewing rectangle. Based on the graphs, which bank offers the better return on your money?

Step-by-Step Solution

Answer

The compound interest functions for Bank 1 and Bank 2 are \(A = 10000(1 + 0.05/4)^(4t)\) and \(A = 10000(1 + 0.045 / 12)^(12t)\) respectively. After graphing and analyzing these functions over an appropriate time range, one can identify which bank offers better rate of return.

Step 1: Formulation of the compound interest function

The formula for compound interest is \(A = P(1 + r/n)^(nt)\), where: \n\(A\) is the amount of money accumulated after \(n\) years, including interest. \n\(P\) is the principal amount (the initial amount of money). \n\(r\) is the annual interest rate (in decimal). \n\(n\) is the number of times that interest is compounded per unit \(t\). \n\nFor Bank 1, the interest is compounded quarterly, so \(n = 4\) and \(r = 0.05\). The substituted formula for Bank 1 is \(A = 10000(1 + 0.05/4)^(4t)\) \n\nFor Bank 2, the interest is compounded monthly, so \(n = 12\) and \(r = 0.045\). The substituted formula for Bank 2 is \(A = 10000(1 + 0.045/12)^(12t)\)

Step 2: Graph the Functions

Use a graphing calculator or tool to plot these functions. By comparing these functions through a graph, observe the growth over time of your investment from the two banks.

Step 3: Analyze the Graphs

Analyze both graphs and determine which bank offers a better rate of return. If one graph is consistently above the other over the desired duration (time t) then the bank associated with that graph is the better choice.