Problem 108 - Introductory and Intermediate Algebra for College Students 4th

Question

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 8000 & & \$ 12,000 & 2 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The annual interest rate is approximately 0.2 or 20%

Step 1: Identify the known variables

From the table, we know that the invested amount \(P = $8000\), the accumulated amount \(A = $12000\), and the time \(t = 2\) years.

Step 2: Substitute the known values into the formula

Substitute the known values into the continuous compound interest formula. You get \(12000 = 8000e^{2r}\)

Step 3: Solve for \(r\)

Firstly, we can divide both sides of the equation by 8000, which gives us \(e^{2r} = 1.5\). Then we take the natural logarithm (ln) of both sides, which gives us \(2r = ln(1.5)\), and finally we solve for \(r\) by dividing both sides by 2. So, \(r = \frac{ln(1.5)}{2}\)

Step 4: Simplify and round the answer

When you simplify the equation from step 3, using a calculator, you get \(r \approx 0.202733\). Finally, we round the answer to one decimal place, we get \(r \approx 0.2\)

Key Concepts

Compound Interest Formula

Understanding the compound interest formula is crucial when dealing with investment calculations and savings. The continuous compounding formula is a specific type of compound interest, where the interest is compounded infinitely small number of times. This is expressed in the formula \[ A = Pe^{rt} \]where:

• \(A\) is the accumulated amount after time \(t\)
• \(P\) is the principal amount (initial investment)
• \(r\) is the annual interest rate expressed as a decimal
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828
• \(t\) is the time the money is invested for, in years

This formula differs from regular compounding because it assumes continuous compounding, meaning the principal earns interest at every possible instant. This can increase the total interest earned over time.

Natural Logarithm

The natural logarithm, represented as \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. In the context of the exercise, taking the natural logarithm is a step used to solve for the annual interest rate, \(r\). When you have an equation like \[ e^{2r} = 1.5 \],you need to "undo" the exponential function to solve for \(r\). Taking the natural logarithm of both sides allows us to deal directly with the exponent: \[ 2r = \ln(1.5) \]This approach leverages the property of logarithms that states \( \ln(e^x) = x \). Therefore, calculating \(\ln(1.5)\) with a calculator gives us the value needed to find the interest rate when substituting back into the formula.

Annual Interest Rate

An annual interest rate is the percentage increase of an investment or loan, based on the original amount, per year. It is usually denoted by \( r \) in formulas related to compounding interest. In the context of continuous compounding, the formula explains how much growth you can expect annually, considering the compounding effect is constant. When we rearrange the formula \( A = Pe^{rt} \), solving for \( r \) allows us to discover the annual rate needed to grow a certain principal investment \( P \) to an accumulated amount \( A \) over time \( t \). Calculated in step-by-step processes, like in the exercise, you start by isolating \( r \) through algebraic manipulation and use logarithms to solve as shown in \[ r = \frac{\ln(1.5)}{2} \].Remember to convert interest rates to decimal form by dividing percentage values by 100 and vice versa.

Investment Calculations

Investment calculations involving continuous compounding allow investors to grasp how their money grows when subjected to constant interest accumulation. By using the continuous compound interest formula \( A = Pe^{rt} \), you can compute future returns from an initial amount without requiring complex computational tools beyond basic algebra. Key steps include:

• Determining present value or principal \( P \)
• Estimating future value \( A \)
• Assessing the time period \( t \)
• Solving for unknowns such as \( r \), often involving natural logarithms

These calculations serve practical purposes in finance and savings planning by providing insights into how variables interact over time. As seen in the original exercise, knowing some aspects like invested amount, time, and the result, lets you solve for internal components like the interest rate using logical mathematical steps.