Problem 14 - 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. $$ A=25, A_{0}=200, r=-50 \%, n=1 $$

Step-by-Step Solution

Answer

The missing variable \( t \) is approximately 3.00.

Step 1: Understand the Problem

We are given the formula \( A = A_{0}\left(1 + \frac{r}{n}\right)^{nt} \). The given values are \( A = 25 \), \( A_{0} = 200 \), \( r = -50\% = -0.50 \), and \( n = 1 \). We need to solve for the variable \( t \).

Step 2: Substitute Known Values

Substitute the known values into the formula: \( 25 = 200\left(1 + \frac{-0.50}{1}\right)^{1\times t} \). This simplifies to: \( 25 = 200\left(0.50\right)^{t} \).

Step 3: Isolate the Exponential Expression

Divide both sides by 200 to solve for the exponential expression: \( \frac{25}{200} = (0.50)^{t} \). Simplifying, this becomes \( 0.125 = (0.50)^{t} \).

Step 4: Take the Logarithm

Take the logarithm of both sides to solve for \( t \): \( \log(0.125) = t \cdot \log(0.50) \).

Step 5: Solve for t

Calculate \( t \) using the logarithm values found in the previous step: \( t = \frac{\log(0.125)}{\log(0.50)} \). Using a calculator, \( \log(0.125) \approx -0.903 \) and \( \log(0.50) \approx -0.301 \). Thus, \( t \approx \frac{-0.903}{-0.301} \approx 3.00 \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. This can model growth or decline over time, making it useful in real-world applications like population studies or bank interest. For instance, when you have a base like 0.50, as in this exercise, we observe exponential decay, which happens when the base is less than 1.

Key characteristics of exponential functions include:

• The function's value changes rapidly, either growing or shrinking based on the exponent's sign.
• They have a constant rate of relative change.
• A horizontal asymptote typically exists, meaning values approach but never reach a certain line.
• Continuous curves without peaks or troughs are notable features.

In exponential decay, the function reduces by a constant percentage over equal time intervals, as seen with the decay from 200 to 25 in the problem. Understanding these functions is crucial, especially when predicting phenomena that rely on consistent rates of change.

Logarithms

Logarithms are the inverse of exponential functions and are essential for solving equations where the unknown variable is an exponent. This is especially useful when you need to solve for time in decay or growth problems, as illustrated in the exercise. The logarithmic relation is expressed as \( \log_b(a) = x \), meaning \( b^x = a \).

Here's why logarithms matter:

• They transform multiplication into addition, simplifying complex calculations, especially for large numbers.
• Logarithms help solve equations where the variable is stuck in the exponent.
• The common logarithm (base 10) and the natural logarithm (base \( e \)) are most frequently used.

In our problem, by applying logarithms, we converted \( (0.50)^t = 0.125 \) into a manageable form, allowing us to find the time \( t \) needed for the quantity to reduce. Therefore, understanding logarithms is crucial for unlocking the solutions to exponential equations.

Compounded Interest Formula

The compounded interest formula calculates the amount of interest earned or paid on an investment or loan where the interest itself is reinvested. The formula \( A = A_{0} \left(1 + \frac{r}{n}\right)^{nt} \) is widely used in finance to model investments where the rate of interest compounds over time. Even though this problem is framed around compounding, notice that the rate \( r \) here is negative, modeling a decay scenario rather than typical interest growth.

Noteworthy points about this formula:

• \( A \) is the final amount after interest.
• \( A_{0} \), the principal amount, is the initial sum of money or principal before interest is applied.
• \( r \) is the annual interest rate, written as a decimal.
• \( n \) denotes the number of times interest is compounded annually, such as quarterly (4), monthly (12), or annually (1).
• \( t \) is the time the money is invested or borrowed, expressed in years.

By plugging in the known values into the formula, one can compute any unknown, which can explain various financial outcomes, from accrued savings to loan payments. Understanding this formula is vital for anyone managing money or investments.