Problem 14 - Applied Calculus

Question

When a deposit of $$\$ 1000$$ is made into an account paying \(8 \%\) interest, compounded annually, the balance, $$\$ B$$, in the account after \(t\) years is given by \(B=1000(1.08)^{t}\). Find the average rate of change in the balance over the interval \(t=0\) to \(t=5\). Give units and interpret your answer in terms of the balance in the account.

Step-by-Step Solution

Answer

The average rate of change is approximately \$93.87 per year, indicating the balance increases by this amount each year.

Step 1: Calculate Balance at Start

To find the balance when \(t = 0\), substitute \(t = 0\) into the formula \(B = 1000(1.08)^{t}\). This results in \(B = 1000\).

Step 2: Calculate Balance at End

Next, find the balance at \(t = 5\) by substituting \(t = 5\) into the formula. So, \(B = 1000(1.08)^{5} = 1000 imes 1.46933 \approx 1469.33\).

Step 3: Find Change in Balance

To find the change in balance over the interval, subtract the balance at \(t = 0\) from the balance at \(t = 5\). This gives \(1469.33 - 1000 = 469.33\).

Step 4: Calculate Average Rate of Change

The average rate of change is the change in balance divided by the change in time, \(t=5 - t=0 = 5\). So, the average rate of change is \(\frac{469.33}{5} = 93.866\approx 93.87\).

Step 5: Interpret the Result

The average rate of change is approximately \\(93.87 per year. This means that, on average, the balance in the account increases by about \\)93.87 each year over the interval from \(t=0\) to \(t=5\).

Key Concepts

Compound Interest

Compound interest is a fascinating concept that describes how your money grows when the interest earned itself earns interest. Unlike simple interest, where only the principal earns interest, compound interest accelerates growth by reinvesting earned interest back into the principal.
This cycle of earning interest on interest creates exponential growth in your balance.

• Initially, your deposit might seem to grow slowly, but as time progresses, the compounding effect becomes more pronounced.
• In this exercise, the formula \(B = 1000(1.08)^{t}\) represents compound interest where \(1000 is the initial deposit, or principal, and 1.08 accounts for both the initial amount and the 8% interest rate compounded annually.

With each passing year, the interest not only builds on the original \)1000 but also on the previously accumulated interest, explaining the increasing rate of growth over time.

Interest Rate

The interest rate is an incredibly pivotal factor in determining how money grows over time. An interest rate, quoted as a percentage, indicates how much of the principal you'll earn back as interest over a given period, typically one year.

• In the exercise, you encounter an 8% annual interest rate. This rate is compounded, meaning the interest earned also generates interest in subsequent periods.
• The formula \(B = 1000(1.08)^{t}\) encapsulates this rate by multiplying the principal $1000 by 1.08 for each year \(t\), mimicking the compounding process.

Higher interest rates generally imply faster growth of your investment, but they also come with potential risks or conditions—so always read the fine print.

Exponential Growth

Exponential growth stands in contrast to linear growth, as it involves rapid increases over time due to the compounding effect of interest. This type of growth is best illustrated through situations where the rate of change itself is growing, like in compound interest.

• In the formula \(B = 1000(1.08)^{t}\), the variable \(t\) in the exponent signals an exponential function, which means the balance grows faster and faster each year.
• The exercise showcases how over a five-year interval, beginning with \(1000, the balance reaches approximately \)1469.33 due to exponential growth—a result that would not occur with simple, linear interest.

Exponential growth can drastically increase the final amount in your account over a number of years, highlighting the importance of understanding time and interest rates in personal finance.