Question

Use the given information to find the amount \(A\) in the account earning compound interest after 6 years when the principal is \(\$ 3500\).\(r=1.83 \%\), compounded daily

Short answer

To find the amount in the account after 6 years with a 1.83% annual interest rate, compounded daily, we can use the formula for compound interest. This gives us the result after calculating the final amount.

Step-by-step solution

Interpret the given information

The principal here is \$3500, the interest rate \(r\) is 1.83% per annum, it is compounded daily. The time \(t\) is 6 years.

Convert percentages to decimals

The interest rate given is in percent form. Convert it to decimal form by dividing by 100 to get \(r = 0.0183\).

Determine the number of compounding periods per year

With daily compounding, there are approximately 365 days in a year, thus number of compounding periods \(n\) is 365.

Use the formula for compound interest

Now, plug these values into the formula for the compound interest: \(A = P(1+ r/n)^{nt}\). We will have \(A = 3500(1+ 0.0183/365)^(365*6)\).

Calculate the value

By inserting the values into the formula, we could calculate the amount in the account after 6 years.

Key concepts

Understanding Principal Amount

When saving or investing, the initial sum of money put into the account or investment is known as the 'principal amount.' It's the baseline figure on which interest is calculated. In our example, the principal amount is $3500. This is the starting point from which we'll witness the growth of the investment due to compounding interest.

To emphasize, this means that every interest calculation thereafter will be based on the principal amount, as well as the accumulated interest from previous periods if the interest is being compounded. It's like planting a seed – the principal amount is the seed you plant, which will grow over time.

Deciphering Interest Rate

The 'interest rate' is essentially the growth rate at which your money will increase over a certain time period, usually expressed as a percentage. Think of the interest rate like the rate at which a plant grows; the higher the rate, the quicker the growth.

It's important to note that the interest rate given in the problem is an annual rate, even though the compounding occurs more frequently. In this case, the interest rate is 1.83% per year. To work with this in formulas, we convert it into a decimal by dividing by 100, resulting in 0.0183.

Compounding Periods Explained

Compounding periods refer to the frequency with which interest is applied to the principal amount. In our scenario, the compounding is happening daily, which means it occurs 365 times a year. Every single day the interest is calculated on the new amount, which includes the previous day's interest.

With more frequent compounding periods, like daily compounding, the investment grows more rapidly because interest calculations happen more often, leading to interest on top of interest—what's known as compound interest.

Experiencing Exponential Growth

Interest that compounds leads to exponential growth of the principal amount. It's not linear, where you'd get the same amount of interest every compounding period; it's exponential, meaning every compounding period, you're earning interest on the new total, which includes the last period's interest.

The formula we use reflects this growth:
\(A = P(1+ \frac{r}{n})^{nt}\).
Here, \(A\) is the amount in the account after time \(t\), \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of compounding periods per year, and \(nt\) is the total number of times the interest is compounded over the entire period. This formula gives us the final amount, showing the power of exponential growth at work.