Problem 12 - Algebra 2 and Trigonometry

Question

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A=400, r=5 \%, n=4, t=3 $$

Step-by-Step Solution

Answer

The initial amount, \(A_0\), is approximately 344.24.

Step 1: Identifying Known and Unknown Variables

We start by identifying the given values in the formula. Here, we have:- Final amount, \(A = 400\) - Interest rate, \(r = 5\% = 0.05\)- Number of times interest is compounded per year, \(n = 4\)- Number of years, \(t = 3\)We need to find the initial amount, \(A_0\).

Step 2: Substitute the Known Values into the Formula

Substitute the given values into the formula:\[A = A_0 \left(1 + \frac{r}{n}\right)^{nt}\]\[400 = A_0 \left(1 + \frac{0.05}{4}\right)^{4 \times 3}\]

Step 3: Simplify Inside the Parentheses

Calculate the expression inside the parentheses:\[1 + \frac{0.05}{4} = 1 + 0.0125 = 1.0125\]

Step 4: Calculate the Exponent

Raise the result from the previous step to the power of \(nt\), which is \(4 \times 3 = 12\):\[1.0125^{12} \approx 1.1616\]

Step 5: Divide Both Sides to Solve for the Initial Amount

Now that we have the expression simplified, solve for \(A_0\):\[400 = A_0 \times 1.1616\]Divide both sides by 1.1616:\[A_0 = \frac{400}{1.1616} \approx 344.24\]

Step 6: Round to the Nearest Hundredth

Confirm that the calculated result is already rounded to the nearest hundredth. Here, the value we found is \(344.24\), which is already rounded correctly.

Key Concepts

Exponential Growth

Exponential growth refers to the increase in quantity by a consistent percentage over consistent time intervals. In the context of compound interest, this concept means that the initial amount of money not only grows by the interest rate, but each new interest calculation is then applied to the total amount accumulated so far. This creates a snowball effect, where the amount of growth continuously accelerates as the base becomes larger.

• When calculating compound interest, this exponential growth is represented within the formula by raising the base (1 plus the interest rate divided by the frequency of compounding) to the power of the total number of compounding periods.
• In simple terms, with each compounding period, we see increasingly larger increases as the periods progress. Even small interest rates can make a large impact over a long period of time due to this exponential effect.

Understanding exponential growth helps to appreciate why long-term savings can grow considerably even with modest annual interest rates. Knowledge of this concept is crucial in planning investments or savings that take advantage of compounding.

Interest Rate

The interest rate is a percentage that determines how much interest will be paid on the initial amount over a set period. A higher interest rate will result in faster growth of your total amount due to the effect of compounding.
In the compound interest formula, the interest rate is denoted by the symbol "r." However, it's important to note the interest rate must be converted into a decimal for calculations. For instance, if an interest rate is stated as 5%, you would convert this to a decimal by dividing the percentage by 100, which results in 0.05.
The formula considers how often the interest is applied, which can be annually, semi-annually, quarterly, monthly, or even daily. If the interest rate is adjusted for these different compounding periods, it must be divided by the number of times interest is compounded per year—denoted as "n." Adjustments like this ensure accurate modeling of growth over time.

• For instance, when interest is compounded quarterly in a year, you would divide the annual interest rate by the number of quarterly periods, which is 4.
• This division is critical for ensuring that each compounding period uses the correct smaller rate to calculate growth accurately.

Initial Amount Calculation

The initial amount, or present value, is what you start with before any interest accrues. Calculating this amount correctly is often key to solving problems involving compound interest. In the given formula, the initial amount is symbolized as "\(A_0\)," representing the principal before interest is added.
To find the initial amount when other variables are known, the final formula rearranges to solving for \(A_0\):

• First, substitute all known values into the compound interest formula.
• Calculate the interest accrued and apply it by dividing the future value \(A\) by the compounded interest factor.

Simplifying inside parentheses and exponents ensures that all growth via compounding interest is correctly reflected in the answer.
Accurate calculation, as highlighted in the example provided, ensures the right principal amount is identified, demonstrating how the starting point grows to the known future value \(A\). Understanding and applying the method of calculating \(A_0\) allows individuals to determine how much they need to invest or save initially to reach their financial targets.
This concept helps guide financial decisions, investments, and understand growth over time related to specific financial goals.