Problem 20 - 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 tenth. $$ P=50,000, r=-0.12, \text { and } t=50 $$ $$ A=50,000 e^{(\quad)(50)} $$ $$ =50,000 e $$ $$ \approx $$

Step-by-Step Solution

Answer

The final amount \(A\) is approximately 123.9.

Step 1: Understanding the Formula

The formula given is the compound interest formula: \(A = Pe^{rt}\), where \(A\) is the amount of money accumulated after \(t\) years, including interest. \(P\) is the principal amount, \(r\) is the rate of interest, and \(t\) is the time in years.

Step 2: Substituting Values

Substitute the values \(P = 50,000\), \(r = -0.12\), and \(t = 50\) into the formula. This gives us:\[ A = 50,000 e^{-0.12 \times 50} \]

Step 3: Calculating the Exponent

Calculate the exponent by multiplying \(-0.12\) and \(50\):\[ -0.12 \times 50 = -6 \]

Step 4: Simplifying the Expression

Replace the exponent in the expression:\[ A = 50,000 e^{-6} \]

Step 5: Evaluating the Exponential Expression

Calculate \(e^{-6}\) using a calculator:\[ e^{-6} \approx 0.00247875 \]

Step 6: Calculating Final Amount

Multiply \(50,000\) by the calculated exponential value:\[ A = 50,000 \times 0.00247875 \approx 123.9375 \]

Step 7: Rounding to the Nearest Tenth

Round the result \(123.9375\) to the nearest tenth:\[ A \approx 123.9 \]

Key Concepts

Compound Interest Formula

The compound interest formula is a powerful tool to understand the growth or decay of an investment over time. It is expressed as:\[ A = Pe^{rt} \] Here,

• \( A \) represents the total amount of money you accumulate, including the initial principal and any interest earned.
• \( P \) is the principal amount or the initial sum of money.
• \( e \) is the mathematical constant, approximately equal to 2.718, used in exponential growth or decay calculations.
• \( r \) is the rate of interest per time period, expressed as a decimal.
• \( t \) is the time period the money is invested for.

The formula can describe how quantities grow or diminish rapidly over time. In this exercise, the interest rate is negative, indicating that the principal will decrease over time—essentially modeling an exponential decay.

Exponential Function Calculation

Now, let's dive into calculating the exponential part of the formula. For exponential growth and decay, the term involved is \( e^{rt} \). Here, you first need to find the exponent by multiplying the rate \( r \) by the time \( t \). In this example:

• The rate \( r = -0.12 \) and the time \( t = 50 \).
• The calculation for the exponent is: \[ -0.12 \times 50 = -6 \]

Once you have the exponent \(-6\), substitute it back into the expression \( e^{-6} \) and evaluate this using a calculator. For practical purposes, \( e^{-6} \) is approximately equal to 0.00247875. This value represents the factor by which the principal will be multiplied to account for the decay over 50 years.

Rounding Decimals

After calculating the expression, you often end up with a decimal that needs rounding. Rounding helps to present the answer in a simpler form. In situations like these, you round to the nearest tenth when asked to present the results with a specific precision. To round any number, observe the digit in the place immediately after the place you want to round to. For instance, when rounding 123.9375 to the nearest tenth:

• Focus on the digits after the tenths place: 123.9 375. The digit 3, in this case, is less than 5.
• Because it is less than 5, you round down, and the tenths digit (9) remains the same.

Thus, rounding 123.9375 to the nearest tenth gives you 123.9, making it easier to read and use in further calculations or reporting.