Problem 114 - College Algebra Essentials

Question

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 17,425 dollar Annual Interest Rate 4.25% Accumulated Amount 25,000 dollar Time \(t\) in Years _______

Step-by-Step Solution

Answer

The time required for the savings account to reach $25,000, given an investment amount of $17,425 and an annual interest rate of 4.25% with continuous compounding, is approximately 7.0 years.

Step 1: Set up the equation

First enter the given values into the continuous compound interest formula \(A=P e^{rt}\). This will give the equation \(25,000=17,425 e^{0.0425t}\).

Step 2: Solving for t

Divide both sides of the equation by 17425. This gives \( e^{0.0425t}= \frac{25000}{17425} \). Then, take the natural logarithm (ln) of both sides and use the property of log that allows the power to come in front when log is taken. This gives \(0.0425t = ln\left(\frac{25000}{17425}\right)\). Finally, solve for \(t\), \(t = \frac{ln\left(\frac{25000}{17425}\right)}{0.0425}\).

Step 3: Calculate t

Use a calculator to compute the value of \(t\), \(t = \frac{ln\left(\frac{25000}{17425}\right)}{0.0425} \approx 7.0 \) years

Key Concepts

Interest Rate

Interest rate is a critical factor when it comes to understanding how savings grow over time, especially with continuous compounding. It is the percentage at which your initial investment (also known as principal) increases over a specified period. For instance, in the formula for continuous compounding:
\[A = Pe^{rt}\]
* \(P\) is the principal amount,* \(r\) is the annual interest rate in decimal form,* \(t\) is the time in years,* \(A\) is the accumulated amount.In our example, the interest rate is 4.25%, which, when expressed as a decimal, becomes 0.0425. This small daily increase compounds over time to significantly grow the original investment. Therefore, even a modest increase or decrease in the interest rate can have a large impact on the final amount accumulated, especially over long periods.

Natural Logarithm

Natural logarithms, often denoted as \(ln\), are essential in solving continuous compounding problems because they simplify exponential equations. In calculus and algebra, logarithms help transform multiplicative relationships into additive ones.
For example, if you have an equation like
\[e^{0.0425t} = \frac{25000}{17425}\]
you take the \(ln\) of both sides to make the equation more solvable:
\[0.0425t = ln\left(\frac{25000}{17425}\right)\]The primary property of logarithms used here is that \(ln(e^x)=x\), allowing powers to "come down" and simplify equations. This manipulation is crucial to isolate variables like \(t\) when solving for unknowns in continuous compounding formulas.

Accumulated Amount

The accumulated amount in context of continuous compounding refers to the total sum in the bank account after a specified amount of time. It includes both the initial principal and the interest earned over that period.
The equation \(A = Pe^{rt}\) shows how the principal grows. In this exercise, we started with an investment of \(17,425 and wanted to know how long it would take to reach \)25,000 at an annual interest rate of 4.25%. By rearranging and solving the continuous compounding formula, it was determined that it would take approximately 7 years.Understanding the accumulated amount helps you set financial goals and track progress over time. This is a powerful demonstration of how small investments or interest rates can significantly grow funds through continuous compounding.