Problem 15 - Intermediate Algebra

Question

For Problems \(15-23\), use the formula \(A=P e^{r t}\) to find the total amount of money accumulated at the end of the indicated time period by compounding continuously. $$\$ 400$$ for 5 years at \(7 \%\)

Step-by-Step Solution

Answer

The total amount is approximately $567.63.

Step 1: Identify Known Values

We need to determine the total amount using the formula \(A=Pe^{rt}\). The given values are:- Principal \(P = 400\)- Rate \(r = 7\% = 0.07\)- Time \(t = 5\) years.

Step 2: Substitute Values Into Formula

Substitute the known values into the formula: \(A = 400 \, e^{0.07 \times 5}\).

Step 3: Calculate the Exponent

Calculate the exponent part first: \(0.07 \times 5 = 0.35\). Hence, \(A = 400 \, e^{0.35}\).

Step 4: Use the Exponential Function

Find \(e^{0.35}\). The approximate value can be calculated using a calculator: \(e^{0.35} \approx 1.419067\).

Step 5: Compute the Final Amount

Multiply the principal by the exponential: \(A = 400 \times 1.419067 \approx 567.63\).

Key Concepts

Continuous Compounding

When we talk about continuous compounding, we’re thinking about a beautiful concept where interest doesn't just accumulate at the end of the year, month, or even daily. Instead, it is being added constantly, almost like it's being compounded every second, of every day.
Continuous compounding involves using a mathematical constant known as Euler's number, denoted as \(e\), which is approximately equal to 2.71828.
This fascinating number is crucial in calculating interest through continuous compounding. It allows us to consider situations where compounding happens an infinite number of times within a fixed period, giving us the maximum possible interest from a given rate and principal.
This method of compounding results in slightly more interest earned than regular compounding methods over the same period at the same rate, which is why it's so appealing!

Compound Interest Formula

The compound interest formula for continuous compounding is given by: \[ A = Pe^{rt} \] Here,

• \(A\) represents the total amount after time \(t\).
• \(P\) is the principal amount or the initial sum of money.
• \(r\) is the annual interest rate (as a decimal).
• \(t\) is the time the money is invested or borrowed for, in years.

This formula may look a bit different from the compound interest formulas you are used to because it uses the number \(e\), which reflects constant growth. Instead of calculating compound interest periodically, like annually or monthly, this formula applies the concept of compounding continuously.
The power of continuous growth ensures that the interest is always being added, maximizing the effect of compounding and providing fully realized growth over time.

Exponential Functions

Exponential functions are all about growth. When you see an exponential function in mathematics, you're typically looking at something growing or decaying; in our context, it's usually associated with growth, such as money growing over time with compound interest.
Exponential functions take the form \(f(x) = a \, e^{bx}\), where \(a\) and \(b\) are constants, and \(e\) is the base of the exponential function. In continuous compounding, the formula \(A = Pe^{rt}\) is an example of an exponential function where \(A\) grows exponentially based on the principal \(P\) and the rate \(r\) applied over time \(t\).
This behavior is what makes exponential functions so powerful and pervasive—the higher the rate, the faster the growth. In practical applications, this means the more frequent the compounding, the greater the accumulation of wealth or quantity, which makes understanding exponential functions central to mastering continuous compounding and similar real-world scenarios.