Problem 22 - Algebra 2

Question

OPEN ENDED Give an example of a quantity that grows or decays at a fixed rate. Write a real-world problem involving the rate and solve by using logarithms.

Step-by-Step Solution

Answer

A bank balance with 5% annual interest takes about 14.21 years to double.

Step 1: Identify a Real-World Quantity

Consider a bank account where the balance grows due to interest. Specifically, let's use an account with compound interest. Suppose the account balance grows by 5% per year.

Step 2: Create a Mathematical Model

The balance of the account each year can be modeled by the formula: \[ A = P(1 + r)^t \]where:- \( A \) is the amount of money accumulated after \( t \) years, including interest.- \( P \) is the principal amount (initial balance).- \( r \) is the annual interest rate (in decimal).- \( t \) is the time in years.Let's assume an initial balance of $1000 (i.e., \( P = 1000 \)).

Step 3: Formulate a Problem Using the Model

Suppose you want to find out how long it will take for the balance to reach $2000. Using the model, you'll solve for \( t \) in the equation:\[ 2000 = 1000(1 + 0.05)^t \]

Step 4: Simplify the Equation

First, divide both sides by 1000:\[ \frac{2000}{1000} = (1.05)^t \]This simplifies to:\[ 2 = (1.05)^t \]

Step 5: Solve Using Logarithms

To solve for \( t \), take the logarithm of both sides:\[ \log(2) = \log((1.05)^t) \]Using the logarithm power rule \( \log(a^b) = b \cdot \log(a) \), we have:\[ \log(2) = t \cdot \log(1.05) \]Now, solve for \( t \):\[ t = \frac{\log(2)}{\log(1.05)} \]

Step 6: Calculate the Result

Use a calculator to find the values of the logarithms:\[ t = \frac{\log(2)}{\log(1.05)} \approx \frac{0.3010}{0.0212} \approx 14.21 \]So, it will take approximately 14.21 years for the balance to grow to $2000.

Key Concepts

Exponential Growth

Exponential growth happens when the rate of growth of a quantity is proportional to its current value, leading it to increase at an accelerating pace.
This concept is widely applicable in contexts where growth compounds over time, such as populations, finances, or even viral videos on social media.
When applied to compound interest, exponential growth helps model how money in a bank account increases over time.
By using the formula \( A = P(1 + r)^t \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the starting principal, and \( r \) is the interest rate as a decimal, we can predict future balances.
A vital part of understanding this is recognizing how effortless, yet powerful, the doubling effect can be.
For example, if an account doubles its balance in a set amount of time due to a steady interest rate, these compounding increments significantly contribute to financial growth or debts over the long term.
This principle of exponential growth underscores the importance of savvy financial planning.

Logarithms

Logarithms are mathematical tools used to handle exponential functions and are essential for solving problems involving exponential growth.
They effectively reverse exponential processes, allowing us to solve for time or rates when other aspects of exponential formulas are known.
Key properties of logarithms include:

• Inverse Relationship: Logarithms are the inverse operation to exponentiation, helping "undo" exponential equations.
• Power Rule: The rule \( \log(a^b) = b \cdot \log(a) \) shows how exponents can be moved as coefficients, making complex problems solvable.

In the context of compound interest, if you need to calculate how long it'll take for your money to double, logarithms simplify the process.
For example, to find \( t \) when \( 2 = (1.05)^t \), taking the logarithm of both sides transforms the equation into \( \log(2) = t \cdot \log(1.05) \), making it straightforward to isolate \( t \).
Thus, logarithms make dealing with exponential equations accessible, allowing us to apply these concepts to real-world scenarios effectively.

Real-World Applications

Real-world applications of exponential growth and logarithms are everywhere and provide significant practical value.
Whether it's predicting how quickly a population can grow, how investments can multiply, or even estimating how a disease spreads, these concepts are indispensable.
In finance, compound interest is an excellent illustration.
It influences decisions in savings, loans, and mortgages, teaching us not only how to manage our money but also how to build wealth strategically.
Understanding the nature of exponential growth allows for better planning in these realms. Exponential growth can also be seen in technology and communication, where user bases on social networks can explode overnight.
Companies use these principles to analyze how quickly they need to scale resources or roll out products in response to a rapidly increasing customer base.
By grasping exponential growth and leveraging logarithms to make calculations easier, we empower ourselves to make informed decisions based on prediction and analysis.
This builds an understanding of both the immediate and projected impacts of compound growth in these varied domains.