Problem 25 - Algebra 2

Question

Find the amount in a continuously compounded account for the given conditions. principal: \(\$ 400\) annual interest 7.6\(\%\) time: 1.5 yr

Step-by-Step Solution

Answer

The total amount in the account after 1.5 years with continuous compounding will be approximately $447.78.

Step 1: Convert the Annual Interest Rate to a Decimal

The annual interest rate is given as 7.6\%. This needs to be converted to a decimal by dividing by 100, so \(r = 7.6 / 100 = 0.076\).

Step 2: Substitute values into the Formula

Using the formula for continuously compounded interest \(A = P \times e^{rt}\), substitute \(P = 400\), \(r = 0.076\), and \(t = 1.5\) into the formula to obtain \(A = 400 \times e^{0.076 \times 1.5}\).

Step 3: Calculate the Final Amount

Perform the exponent operation first, then the multiplication to get the final amount. Using a calculator, find the value of \(e^{0.076 \times 1.5}\) to get approximately \(1.119463211\). Multiply this by 400 to get the final amount \(A = 400 \times 1.119463211 = 447.78\). Therefore, the total amount in the account after 1.5 years with continuous compounding will be approximately $447.78.

Key Concepts

Interest Rate Conversion

Interest rate conversion is essential when dealing with percentages in financial mathematics. Often, interest rates are expressed as percentages, which need to be converted into decimals before using them in mathematical formulas. This transformation is straightforward: divide the percentage by 100. For example, converting a 7.6% annual interest rate into a decimal involves dividing 7.6 by 100, resulting in 0.076.

Understanding this conversion helps in simplifying calculations, especially when plugging the rate into formulas for continuously compounded interest or other financial calculations. Always remember to convert percentage rates to decimals to ensure accurate computation.

Exponential Growth Formula

The exponential growth formula is a handy tool in calculating how money (or other quantities) grow over time continuously. The formula is expressed as:\[ A = P \times e^{rt} \]

• \(A\) stands for the final amount.
• \(P\) refers to the principal, or initial amount.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(r\) is the interest rate (as a decimal).
• \(t\) is the time period under consideration.

This formula is particularly used for continuously compounded interest, which implies that interest calculations are made an infinite number of times per period. Exponential growth naturally suits this kind of calculation because of its continuous nature.

Compound Interest

Compound interest refers to the concept where interest earned is added to the principal balance, which then earns interest in the next period. Essentially, you're earning interest on both your initial investment and the interest accrued over time.

There are two main types:

• Discrete Compound Interest: Interest is compounded at regular intervals (e.g., yearly, quarterly).
• Continuous Compound Interest: Here, interest is compounded an unlimited number of times, theoretically every moment, using the formula \(A = P \times e^{rt}\).

Continuously compounded interest results in slightly higher amounts compared to discrete compounding over the same time period and interest rate. It's particularly useful in scenarios involving continuous growth or decay, like population growth models or radioactive decay calculations.

Financial Mathematics

Financial Mathematics is a crucial field that incorporates mathematical formulas and principles to solve various financial problems. It involves concepts like interest rates, annuities, loans, investments, and risk assessment. One key concept is understanding how different interest computations affect financial projections and decisions.

In the realm of continuously compounded interest, Financial Mathematics makes use of calculus and advanced algebra to determine future values of investments, helping investors estimate growth over time with great precision. This branch of mathematics ensures a comprehensive understanding of how money can work effectively, considering both time and interest rates.
Financial Mathematics provides tools to evaluate different financial scenarios systematically and apply mathematical reasoning to optimize results.