Problem 33 - 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. $$ P=5,000, r=8 \%, t=20 \text { years } $$

Step-by-Step Solution

Answer

A is approximately 24,765.16.

Step 1: Convert the Interest Rate

The given interest rate is 8%. Convert this percentage into a decimal for calculation by dividing by 100. So, \( r = \frac{8}{100} = 0.08 \).

Step 2: Identify the Parameters

The given formula to find \( A \) is \( A = P e^{rt} \). From the problem, \( P = 5000 \), \( r = 0.08 \) (as calculated), and \( t = 20 \).

Step 3: Substitute the Values

Substitute the identified values into the formula: \[ A = 5000 \times e^{0.08 \times 20} \].

Step 4: Calculate the Exponentiation

Calculate the exponent in the formula: \( rt = 0.08 \times 20 = 1.6 \).So the expression becomes \( A = 5000 \times e^{1.6} \).

Step 5: Calculate \( e^{1.6} \)

Use a scientific calculator to find \( e^{1.6} \).\( e^{1.6} \approx 4.953032424 \).

Step 6: Complete the Multiplication

Multiply the principal \( P \) by \( e^{1.6} \): \[ A = 5000 \times 4.953032424 \approx 24765.16 \].

Step 7: Round the Final Answer

Round \( 24765.16 \) to the nearest hundredth.The calculated value is already at the hundredths place, rounded to \( A \approx 24765.16 \).

Key Concepts

Continuous Compounding

Continuous compounding refers to the process where interest is added to the initial principal continuously, effectively at every possible moment. This is different from more common compounding intervals such as annually, semi-annually, or monthly. The beauty of continuous compounding is that it maximizes the amount of interest earned over a specific period.

• The formula used in continuous compounding is: \[ A = P e^{rt} \]where:
  • \(A\) is the amount of money accumulated after n years, including interest.
  • \(P\) is the principal amount (the initial 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.

Continuous compounding uses the natural exponent \( e \), making it a powerful tool for determining future value, greatly benefitting long-term investments.

Mathematical Exponentiation

Exponentiation is a key mathematical function used in various computations, including the calculation of interest in the continuous compounding formula. In mathematical terms, exponentiation involves raising a number (the base) to the power of an exponent.

• In the formula \( A = P e^{rt} \),
  • \( rt \) is the exponent that multiplies the rate \( r \) and time \( t \).
  • \( e^{rt} \) calls for the base \( e \) to be raised to the power defined by \( rt \).

This operation results in an exponential growth factor, significantly increasing the final amount \( A \). This growth is critical in contexts like financial modeling and population dynamics.

Interest Rate Calculation

Understanding how to calculate and use the interest rate is crucial within the scope of financial formulas like continuous compounding. Interest rates expressed as percentages need to be converted into decimal form before plugging into formulas. This step is vital for proper calculation.

• An interest rate expressed as a percentage (e.g., 8%) should be divided by 100 to convert it to a decimal form (e.g., 0.08).
• The interest rate directly affects how quickly the principal grows over the specified period.
• Correctly understanding and applying the rate can make a major difference in financial projections and outcomes.

Making these precise calculations ensures accurate predictions of financial growth for investments or loans.

Scientific Calculators

Scientific calculators play a pivotal role in solving complex mathematical problems, such as those involving exponentiation with irrational numbers like \( e \). They enable students to manage such calculations with ease, providing reliable results.

• Scientific calculators are equipped to handle exponential, logarithmic, and trigonometric functions with precision.
• In exercises involving continuous compounding, using a scientific calculator becomes essential to determine values like \( e^{1.6} \) accurately.
• Without such tools, manual calculations could lead to errors and inaccuracies, affecting financial decisions and analyses.

Students should familiarize themselves with their calculator's functions to effectively solve which require detailed, accurate computations.