Problem 40 - Elementary and 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. $$ P=110, r=-0.25 \%, t=9 \text { years } $$

Step-by-Step Solution

Answer

A is approximately 107.55.

Step 1: Understand the Formula

The formula given is \( A = P e^{r t} \), where \( A \) is the amount you want to find, \( P \) is the principal amount, \( r \) is the rate of interest as a decimal, and \( t \) is the time in years. The variable \( e \) is the base of natural logarithms, approximately equal to 2.71828.

Step 2: Convert the Rate to Decimal

The rate \( r \) is given as \(-0.25\%\). To convert this percentage to a decimal, divide by 100: \( r = \frac{-0.25}{100} = -0.0025 \).

Step 3: Substitute the Values into the Formula

Substitute \( P = 110 \), \( r = -0.0025 \), and \( t = 9 \) into the formula: \[ A = 110 e^{-0.0025 \times 9} \]

Step 4: Calculate the Exponent

Calculate the product of the rate and time: \[ -0.0025 \times 9 = -0.0225 \]This means you now have: \( A = 110 e^{-0.0225} \).

Step 5: Evaluate the Exponential Term

Use a calculator to find \( e^{-0.0225} \). \( e^{-0.0225} \approx 0.977766 \) (rounded to six decimal places).

Step 6: Compute the Final Value for A

Now multiply the result from the previous step by \( P \):\[ A = 110 \times 0.977766 \]\( A \approx 107.55426 \).

Step 7: Round the Result

Round \( 107.55426 \) to the nearest hundredth.The rounded value is \( 107.55 \).

Key Concepts

Natural Exponents

In mathematics, natural exponents refer to the exponential function with the base of the natural logarithm, denoted as "e." The constant \( e \) is approximately 2.71828 and is unique in its role in various mathematical and financial calculations. This special number appears frequently when dealing with exponential growth or decay situations because of its natural properties.

In exponential functions, \( e \) often replaces other numerical bases, leading to smoother and more continuous growth curves. The formula \( A = P e^{rt} \) represents a function of natural exponents, where \( A \) stands for the final amount, \( P \) is the principal amount, \( r \) is the rate of interest or decay as a decimal, and \( t \) is time.

A core property of \( e \) is its derivative. The rate of change of \( e^x \) is \( e^x \), meaning its slope at every point equals the value of the function, providing a natural mode for growth and decay calculations, especially in continuous compounding scenarios.

Interest Rate Conversion

Interest rate conversion involves changing a percentage interest rate into a decimal form. This is a small but crucial step within the process of calculating financial figures like compound interest or continuously compounding earnings. For instance, if given an interest rate of \(-0.25\%\), convert it by dividing by 100 to obtain \(-0.0025\).

• Move the decimal point two places to the left.
• This change in format allows the rate to be directly used in calculations.
• In negative rates, as shown here, your wealth decreases over time instead of increasing.

For daily or periodic interest calculations, always ensure the rate conversions reflect the actual compounding periods involved.

Rounding Decimals

Rounding decimals helps in making precise yet manageable numerical results. In financial contexts, this may mean rounding to the nearest cent, as monetary values usually represent two decimal places (hundredths).To round a decimal number to the nearest hundredth:

• Identify the hundredths place (second decimal point).
• Observe the number in the thousandths place (third decimal point).
• If this number is 5 or greater, add 1 to the hundredths place.
• If it's less than 5, leave the hundredths place unchanged.

In the exercise, rounding the computed \(A\), which was \(107.55426\), to the nearest hundredth results in \(107.55\). Accurate rounding ensures precision is maintained without overstating or neglecting minor effects, especially crucial in financial reports.