Problem 37 - Intermediate Algebra

Question

Find A using the formula \(A=P e^{r t}\) given the following values of \(P, r,\) and \(t .\) Round to the nearest hundredth. $$ P=15,895, r=-2 \%, t=16 \text { years } $$

Step-by-Step Solution

Answer

The value of \(A\) is approximately \(11,537.18\).

Step 1: Understand the Given Values

We are given the values of principal \(P\), rate \(r\), and time \(t\). Specifically, \(P = 15,895\), \(r = -2\%\), and \(t = 16\) years. We will use these values in the formula \(A = P e^{rt}\).

Step 2: Convert the Rate to Decimal

The rate \(r\) is given in percentage form, \(-2\%\). To use it in our formula, we need to convert it to a decimal. Divide by 100: \(-2\% = -0.02\).

Step 3: Substitute Values in the Formula

Substitute the given values into the formula: \(A = 15,895 \, e^{-0.02 \times 16}\). This represents our expression for \(A\).

Step 4: Calculate the Exponent

First calculate the exponent: \(-0.02 \times 16 = -0.32\). This is the power to which \(e\) will be raised.

Step 5: Evaluate the Exponential Expression

Calculate \(e^{-0.32}\) using a calculator. This evaluates to approximately \(0.7261\).

Step 6: Multiply by the Principal

Multiply the result of the exponential expression by the principal \(P\): \(A = 15,895 \times 0.7261\).

Step 7: Calculate and Round

Carry out the multiplication to get \(A \approx 11,537.1795\). Round this to the nearest hundredth to find \(A \approx 11,537.18\).

Key Concepts

Compound Interest Formula

The Compound Interest Formula is a fundamental tool in understanding how investments grow over time. While it may seem complex at first glance, breaking it down into simple terms can make it quite approachable. In this formula, we calculate the amount of money that will be accumulated after a certain period of time, which is known as the accumulated amount or future value.Let's take a closer look at the components of the formula:

• \(A\) is the future value of the investment/loan, including interest.
• \(P\) is the principal investment amount (the initial deposit or loan amount).
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(r\) is the annual interest rate (expressed as a decimal).
• \(t\) is the time the money is invested or borrowed for, in years.

In the equation \(A = P e^{rt}\), the term \(e^{rt}\) represents the compound growth factor. This fascinating concept relies on the use of exponential functions to grow the principal amount over the designated time period. When dealing with negative growth rates, as we have here with -2%, \(e^{rt}\) accounts for depreciation over time. This is perfect for modeling situations like population decay and diminishing investments.

Exponential Functions

Exponential functions are a crucial mathematical concept, representing situations where a quantity grows or decays at a rate proportional to its current value. This property makes them an essential part of understanding both exponential growth and decay.Key aspects of exponential functions include:

• The formula, \(y = a \, \cdot \, e^{bx}\), where \(a\) is the initial amount and \(b\) is the rate of growth or decay.
• The constant \(e\), which is the base of the natural logarithm, allowing the function to smoothly and continuously increase or decrease.
• The power of exponential functions lies in their ability to model real-world scenarios, such as population growth, interest accumulation, or radioactive decay.

In our exercise, the use of \(e^{-0.32}\) demonstrates an exponential function at work. The negative exponent indicates a decrease, as seen with a -2% rate, resulting in a decay over time. This function describes how the investment's value decays, producing a smaller accumulated value at the end of the 16 years compared to the original principal.

Rounding Decimals

Rounding decimals is a simple yet important skill, especially in financial calculations where precision is key. When dealing with money or measurements, rounding ensures the figures are practical without being overly precise.Here’s how to round a number to the nearest hundredth:

• Identify the digit in the hundredths place – this is the second digit to the right of the decimal point.
• Look at the next digit (the thousandths place). If it is 5 or greater, round the hundredth digit up by one. If it’s less than 5, keep the hundredth digit the same.
• Drop all digits following the hundredths place after rounding.

In the given exercise, the final amount calculated was approximately \(11,537.1795\). To round this to the nearest hundredth, observe that the third decimal number, which is 9, necessitates an increase in the second decimal portion. Therefore, the value becomes \(11,537.18\), presenting a clear and concise result suitable for reporting.