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Chapter 4: Problem 41

Solve the following exercises on a graphing calculator by graphing an appropriate exponential function (using \(x\) for ease of entry) together with a constant function and using INTERSECT to find where they meet. You will have to choose an appropriate window. A bank account grows at \(6 \%\) compounded quarterly. How many years will it take to: a. double? b. increase by \(50 \%\) ?

Short Answer

Expert verified
a. About 11.9 years to double; b. About 7.3 years to increase by 50%.

Step by step solution

01

Understand the Problem

The problem involves a bank account growing at an interest rate of 6% compounded quarterly. We need to find the number of years it takes for the account balance to double and increase by 50%.
02

Write the Exponential Growth Formula

Use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(A\) is the amount of money accumulated after \(t\) years, including interest, \(P\) is the principal amount (initially 1 for easier calculations), \(r = 0.06\) is the annual interest rate, and \(n = 4\) is the number of times interest is compounded per year.
03

Graph the Exponential Function

For part a, set \(A = 2\) for doubling the money. Rearrange the formula \[1 \left(1 + \frac{0.06}{4}\right)^{4t} = 2\] Graph the function \(y_1 = \left(1 + \frac{0.06}{4}\right)^{4x}\) and the constant function \(y_2 = 2\).
04

Set the Calculator Window

Choose a window to effectively graph the functions. Try setting the x-axis from 0 to 20 (representing years) and the y-axis from 0 to 3 (since you're looking for where the exponentiated value hits 2).
05

Find the Intersection for Doubling

Use the INTERSECT feature of the graphing calculator to find where the graphs \(y_1\) and \(y_2\) intersect. The x-value at this intersection is the number of years it takes for the investment to double.
06

Graph for a 50% Increase

For part b, set \(A = 1.5\) (50% increase). Graph the function \(y_1 = \left(1 + \frac{0.06}{4}\right)^{4x}\) and the constant function \(y_3 = 1.5\).
07

Find the Intersection for a 50% Increase

Use the INTERSECT feature again to find where the graphs \(y_1\) and \(y_3\) intersect. The corresponding x-value tells you how many years it takes for the investment to increase by 50%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compound Interest
Compound interest is an essential concept in financial mathematics that describes how the value of an investment grows over time. It is particularly powerful because the investment earns interest on both the initial principal and the accumulated interest from previous periods. This results in exponential growth.
The compound interest formula is expressed as:\[A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:
  • \(A\) is the ending balance of the investment
  • \(P\) is the principal (initial amount)
  • \(r\) is the annual interest rate (as a decimal)
  • \(n\) is the number of times interest is compounded per year
  • \(t\) is the time the money is invested for in years
When solving problems, you'll often rearrange this formula to find the variable you need, like \(t\) for time. Compound interest is used widely, from savings accounts to loans, making it a valuable tool for everyday finances.
Graphing Calculator
A graphing calculator is a vital tool that extends beyond basic calculations, allowing users to visualize mathematical functions and solve complex equations. It's particularly handy for exponential functions in compound interest problems because it can display growth over time.
When using a graphing calculator to solve for compound interest:
  • Input the exponential function based on the compound interest formula.
  • Create a constant function that represents the target value, such as the amount you want to double.
  • Set an appropriate window for the x-axis (usually time in years) and y-axis (investment value) to fully capture where the lines intersect.
This visual representation helps you understand when the investment reaches a particular value, providing insights that are tougher to gain from numerical analysis alone.
Intersection Method
The intersection method uses graphing techniques to find the point where two functions meet. This is valuable in scenarios involving compound interest, where you might need to know how long it takes for an investment to reach a certain value.
Here鈥檚 how it works:
  • Graph the exponential function of the investment as it grows over time.
  • Simultaneously, graph a constant function representing the target investment amount.
  • Use the graphing calculator's "INTERSECT" function to find where the two graphs meet.
  • The x-coordinate of this intersection point gives you the time in years for the investment to reach the target value.
This approach provides a clear, visual way to solve exponential growth problems, making it a preferred method in both academic and real-world financial contexts.
Financial Mathematics
Financial mathematics involves the study of money and its investment over time, which is vital for making informed economic decisions. Understanding concepts such as compound interest and exponential growth is crucial in this field.
With financial mathematics, you can:
  • Predict how investments will grow with factors like different interest rates and compounding frequencies.
  • Determine future values of assets and how long it will take to reach specific financial goals.
  • Utilize mathematical models to make educated decisions regarding loans, savings, and investments.
By mastering financial mathematics, individuals and businesses can optimize their resources to achieve financial objectives efficiently and effectively. It's a powerful toolset for anyone looking to thrive economically.

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