Problem 111 - 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 8000 dollar Annual Interest Rate 8% Accumulated Amount Double the amount invested Time \(t\) in Years _______

Step-by-Step Solution

Answer

Based on the given values and the calculated steps, the time required in years to double the initial investment amount, rounded to one decimal place, is approximately \(t = 8.7\) years.

Step 1: Understand the Given Problem

First, it's important to identify the values given. The amount invested, \(P\), is \$8000, the annual interest rate, \(r\), is 8%, and the accumulated amount, \(A\), is double the amount invested, therefore \$16000.

Step 2: Convert Annual Interest Rate

The annual interest rate should be expressed as a decimal in the compounding formula, divide the percentage by 100: \(r = 0.08\)

Step 3: Implement Given Values Into the Formula

Next, implement the given values into the formula \(A = P e^{rt}\), so it becomes: \(16000 = 8000e^{0.08t}\)

Step 4: Manipulate the Formula to Solve for Time

In this step, the aim is to isolate \(t\) in the equation. First, divide both sides by 8000 to eliminate the coefficient of \(e^{0.08t}\): \[2 = e^{0.08t}\]Then, take the natural logarithm of both sides to solve for \(0.08t\): \[\ln(2) = \ln(e^{0.08t})\]The exponential and the natural logarithm cancel each other out, which leads to: \[0.693 = 0.08t\]

Step 5: Final Calculation of Time

Finally, divide both sides by 0.08 to solve for \(t\): \[t = \frac{0.693}{0.08}\]After calculating this expression, round the answer to one decimal place to get the final value for \(t\).

Key Concepts

Compound Interest

Compound interest represents the concept of earning interest on interest. It's a powerful financial mechanism that allows an initial amount of money, known as the principal, to grow at an accelerating rate over time. Unlike simple interest, where interest is only calculated on the principal, compound interest is calculated on the principal plus the accumulated interest from previous periods.

In the case of our exercise, the formula for continuous compounding is leveraged, which takes the relationship even further by compounding interest at an infinite number of times, essentially continuously. This results in the formula: \(A = Pe^{rt}\), where \(A\) is the accumulated amount, \(P\) is the principal, \(e\) is the base of the natural logarithm, \(r\) is the annual interest rate, and \(t\) is the time in years. This formula reflects the concept of exponential growth, which is essential to understanding compound interest and its potential for increasing wealth over time.

Exponential Functions

Exponential functions are mathematical expressions that describe situations where a quantity grows at a rate proportional to its current value. Such functions are commonly represented in the form \(f(x) = a\text{e}^{bx}\), where \(a\) and \(b\) are constants, and \(e\) is the mathematical constant approximately equal to 2.71828. This constant is the base of the natural logarithm and is a key player in continuous compounding interest formulas.

Exponential functions have the distinctive feature of increasing very rapidly, which is why they're ideally suited for modeling compound interest. The higher the rate \(b\), or in our financial context, the interest rate \(r\), the more quickly the function grows. It's crucial for problem solving in finance to grasp how exponential functions operate, as they can illustrate how investments evolve over time under continuous compounding.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the inverse operation of exponentiating with base \(e\). This means that if \(y = \ln(x)\), then \(e^{y} = x\). The natural logarithm is especially handy in algebra when dealing with equations involving \(e\), as it can help to isolate variables compounded at a continuous rate.

In our exercise, taking the natural logarithm of both sides of the equation was the key step that allowed us to simplify and ultimately solve for the variable \(t\), the time required for the investment to double. By understanding the relationship of \(e\) and natural logarithms, we can manipulate exponential equations effectively, making them simpler to solve. The natural logarithm and its properties are essential tools in algebraic problem solving for continuous compounding scenarios.

Algebraic Problem Solving

Algebraic problem solving involves finding unknown variables through operations and manipulations of equations. It requires a clear understanding of algebraic principles and the ability to apply appropriate techniques to isolate and solve for variables.

In the context of our exercise involving continuous compounding, the step-by-step solution demonstrates algebraic problem solving in action. Firstly, we simplify the equation by normalizing the percentage rate and applying it to the continuous compounding formula. Then, to isolate \(t\), we use division to simplify the exponential equation, followed by the application of the natural logarithm to both sides. This eliminates the exponential part of the equation, making it linear and easily solvable for \(t\). The step-by-step approach allows students to follow the logic and understand how each action brings them closer to the solution.