Problem 34 - Elementary and Intermediate Algebra

Question

Find A using the formula \(A=P e^{r t}\) given the following values of \(P, r,\) and \(t .\) Round to the nearest hundredth. $$ P=15,000, r=6 \%, t=40 \text { years } $$

Step-by-Step Solution

Answer

A = 165,348.00

Step 1: Convert the Interest Rate

The formula uses the interest rate \( r \) in decimal form, not percentage. Convert 6% to a decimal by dividing by 100: \( r = \frac{6}{100} = 0.06 \).

Step 2: Plug Values into the Formula

The formula is \( A = Pe^{rt} \). Substitute the given values: \( P = 15,000 \), \( r = 0.06 \), and \( t = 40 \). The equation becomes \( A = 15,000 \times e^{0.06 \times 40} \).

Step 3: Simplify the Exponent

Calculate the exponent: \( 0.06 \times 40 = 2.4 \). The equation now is \( A = 15,000 \times e^{2.4} \).

Step 4: Calculate \( e^{2.4} \)

Use a calculator to find \( e^{2.4} \). It gives approximately 11.0232.

Step 5: Find A by Multiplying

Multiply the principal by \( e^{2.4} \): \( A = 15,000 \times 11.0232 \). Calculating this gives \( 165,348 \).

Step 6: Round to the Nearest Hundredth

Round the result \( 165,348 \) to the nearest hundredth, but since it is already an integer, the rounded value remains \( 165,348.00 \).

Key Concepts

Interest Rate Conversion

Converting interest rates from a percentage to a decimal is crucial in many mathematical formulas, including those for calculating exponential growth. When you come across an interest rate like 6%, it essentially suggests that for every hundred units of currency, six additional units would be added over the specified period. However, in mathematical formulas, particularly the one used in exponential growth, the interest rate needs to be in its decimal form instead. This ensures that the calculations accurately reflect compound growth over time.
To convert a percentage to a decimal, you simply divide by 100. For example, converting 6% involves the straightforward calculation:

• 6 divided by 100 equals 0.06.

This decimal representation is used consistently in the formula to predict the future value of the principal amount after a specified time under continuous growth conditions.

Principal Amount

The principal amount, often denoted as \( P \), is the original sum of money invested or borrowed before any interest is applied or accrued. In the context of exponential growth, this base value represents the starting point from which the compound growth calculations are derived.
In our given exercise, the principal amount is \( 15,000 \). This initial value forms the foundation for calculating the entire growth over time using the exponential formula \( A = P e^{rt} \). Here the computation of the future amount \( A \) depends heavily on the initial principal:

• An increase in \( P \) would proportionally increase the final amount \( A \).
• Conversely, a decrease in \( P \) would reduce \( A \).

Therefore, understanding the principal amount's role helps grasp how initial investments grow over time when an interest rate is applied continuously.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify figures, making them easier to work with. This process involves adjusting the digits of a number so that it is closer to its actual value, but expressed in a less precise form. It's often done for convenience or to fit a result into a certain format, like currency.
In this problem, we consider rounding to the nearest hundredth. The 'hundredth' in a decimal number is the second digit to the right of the decimal point. Thus, we should check the third digit to determine whether to round up the second digit:

• If the third digit is 5 or greater, you round up.
• If it is 4 or less, you keep the hundredth digit as is.

For example, for a number like 165,348.XXXX, if XXXX involves digits beyond the integer, decisions on rounding are required. However, since our result was an exact integer, expressing it as 165,348.00 ensures the proper format without changing its value.