Problem 42 - Calculus and its applications

Question

Yukiko opens a savings account to pay for her new baby's college education. She deposits \(\$ 200\) every month into the account at an annual interest rate of \(4.2 \%,\) compounded monthly. a) Find \(A(t),\) the value of Yukiko's account after tyears. b) What is the value of Yukiko's account after 8 yr? c) What is the rate of change in the value of Yukiko's account after 8 yr?

Step-by-Step Solution

Answer

a) The function A(t) represents Yukiko's account value over time. b) After 8 years, the account is valued at approximately \$37,731.68. c) The rate of change at 8 years must be calculated via differentiation or numerical methods.

Step 1: Understand the Formula for Compound Interest with Regular Deposits

The formula to calculate the future value of regular deposits into a compound interest account is given by:\[ A(t) = P \left(\frac{(1 + r/n)^{nt} - 1}{r/n}\right) \]where:- \( A(t) \) is the future value of the investment/loan, including interest,- \( P \) is the monthly deposit (\$200 in this case),- \( r \) is the annual nominal interest rate (as a decimal),- \( n \) is the number of times that interest is compounded per year (12 for monthly), and- \( t \) is the number of years the money is invested or borrowed.

Step 2: Calculate A(t) for 8 Years

We need to find \( A(t) \) after 8 years with monthly deposits. Plug in the values:- \( P = 200 \)- \( r = 0.042 \)- \( n = 12 \)- \( t = 8 \)Substituting these values into the formula:\[ A(8) = 200 \left(\frac{(1 + 0.042/12)^{12 \times 8} - 1}{0.042/12}\right) \].Now, solve inside the parentheses first:\[ A(8) = 200 \left(\frac{(1 + 0.0035)^{96} - 1}{0.0035}\right) \].Calculate further:\[ A(8) = 200 \left(\frac{(1.0035)^{96} - 1}{0.0035}\right) \approx 200 (1.37474/0.0035) \approx 37731.68 \].The value of Yukiko's account after 8 years is approximately \$37,731.68.

Step 3: Compute the Rate of Change After 8 Years

To find the rate of change after 8 years, you can find the derivative of \( A(t) \) and evaluate it at \( t = 8 \). The derivative represents the rate of change or how much the account value increases per year after 8 years. Simplifying it analytically is complex, so we will approximate using numeric methods or specialized software.

Key Concepts

Savings Account

A savings account is a type of bank account specifically designed to hold your money while allowing it to grow with interest over time. It offers a safe way to store cash and often provides added benefits through accumulating interest.
When you deposit money into a savings account, it typically earns interest, which means that your money increases without you having to do much. Over time, this increment helps you build an even larger amount, especially if you keep the money safe and untouched for future needs. This is exactly what Yukiko is doing in the exercise for her baby's college fund.

• Feature: Security - Your cash is securely held by a bank.
• Benefit: Interest - You earn extra money simply by having the savings account.
• Purpose: Planning - Ideal for setting aside funds for future needs and expenses.

In Yukiko's case, the additional benefit is not just the interest but also the regular, planned deposits. This consistent saving routine enhances her financial planning, ensuring she'll have a sizeable account value when it's time for college expenses.

Interest Rate

The interest rate is crucial when it comes to savings accounts because it determines how much your money will grow over time. In Yukiko's savings plan, the account earns at an annual interest rate of 4.2%, which means each year, 4.2% of the total account balance is added to the account.
Interest rates can be categorized into simple and compound interest. Simple interest is calculated only on the principal amount you invest. On the other hand, compound interest - used in Yukiko's case - lets you earn interest on your initial deposit and the interest that accumulates over time.

• Type: Simple and Compound Interest
• Calculation: Annual percentage of the principal and previously earned interest
• Goal: To maximize savings growth

This method of adding interest means the account's value grows faster over time, especially with regular monthly contributions, which compounds the effect even more beneficially.

Future Value Calculation

Calculating the future value of Yukiko's savings account involves using a specific mathematical formula tailored to account for regular deposits into an account earning compound interest. This is important to estimate just how much savings will grow over a certain period, such as eight years in Yukiko's example.
The formula used is: \[ A(t) = P \left(\frac{(1 + r/n)^{nt} - 1}{r/n}\right) \]Here, each parameter plays a role:

• \( P \) is the monthly deposit - \\(200 in Yukiko's case.
• \( r \) is the annual interest rate expressed in decimal (0.042 here).
• \( n \) is the number of times interest compounds per year (12 times, as it's monthly).
• \( t \) is the number of years the money is invested - 8 years here.

The application of this formula gives Yukiko an estimated future value of approximately \( \\)37,731.68 \) at the end of 8 years, reflecting the effect of regular savings and compounded interest.

Rate of Change

The rate of change in the value of a savings account helps understand how quickly or slowly the value is increasing over time. In Yukiko's scenario, calculating the rate of change involves analyzing how swiftly her account grows, particularly after 8 years of regular contributions and compound interest.
To do this, one would typically take the derivative of the future value function, \( A(t) \), with respect to time \( t \). While the calculation of this derivative can be complex analytically, it essentially gives an instantaneous rate showing how much the savings will increase per year after 8 years.

• Concept: Derivative represents the change over time
• Purpose: Decide the increment in the account value
• Importance: Helps in financial planning and decision making

This rate of change can be approximated using numeric methods or specialized financial tools, enabling a better understanding of financial growth and the efficacy of saving and investment strategies.