Problem 55 - Algebra 2

Question

Open-Ended Write a problem that could be modeled with \(y=20(1.1)^{x} .\)

Step-by-Step Solution

Answer

A problem that could be modeled by the given equation \(y = 20(1.1)^x\) is: Assume you deposited $20 in a bank. The bank offers an interest rate of 10% compounded annually. What would be the amount in your bank account after 'x' years?

Step 1: Understand the exponential function

The formula given is of an exponential function \(y = ab^x\), where 'a’ is the initial amount and 'b’ is the growth rate.

Step 2: Identify the initial amount and growth rate

In the function \(y = 20(1.1)^x\), 20 is the initial amount and 1.1 is the growth rate which indicates a 10% increase.

Step 3: Construct realistic problem

From the above data, a problem can be constructed as: Assume you deposited $20 in a bank. The bank offers an interest rate of 10% compounded annually. What would be the amount in your bank account after 'x' years?

Key Concepts

Exponential Function

An exponential function is a mathematical expression that describes the changing behavior of quantities that grow or decay at a constant percentage rate. It is characterized by the general formula \( y = ab^x \), where:

• \( y \) represents the final amount.
• \( a \) is the initial amount, or starting value.
• \( b \) is the base, which represents the growth (if greater than 1) or decay (if between 0 and 1) rate.
• \( x \) is the exponent, often representing time or another influential variable.

The base, \( b \), determines how fast the function's value changes. When it is greater than 1, the function grows exponentially; when it is between 0 and 1, the function decays exponentially. This type of function is beneficial when modeling real-world scenarios, such as population growth, radioactive decay, and finance-related problems like compounded interest. Each aspect of the exponential function provides crucial information that helps predict future outcomes based on current data.

Initial Amount

The initial amount in an exponential function is denoted by the variable \( a \) in the formula \( y = ab^x \). This is the starting value or the quantity at the beginning of the period being considered. For example, in the function \( y = 20(1.1)^x \), the initial amount \( a \) is 20.This initial amount could represent many things:

• A bank deposit or loan balance.
• The initial number of a population in a biological study.
• An initial quantity in any system that is subject to growth or decay over time.

Understanding the initial amount sets the stage for how the function will behave over time. It acts as a baseline from which all future calculations originate and is critical in determining the trajectory of the exponential growth or decay in the system being studied.

Growth Rate

The growth rate in an exponential function is represented by the base \( b \) in the equation \( y = ab^x \). It defines how much the initial amount will increase (if \( b > 1 \)) or decrease (if \( 0 < b < 1 \)) over one unit of time. In \( y = 20(1.1)^x \), the base \( 1.1 \) signifies a growth rate of 10% per time period.It's important to translate the growth base into a percentage:

• If \( b = 1 + r \), where \( r \) is the growth rate, then \( r \) is calculated by: \( r = b - 1 \).
• In this case, \( 1.1 = 1 + 0.1 \), indicating a 10% growth rate.

Identifying the growth rate allows for accurate forecasting and planning in various contexts, such as economics, biology, and environmental science. It helps in understanding and predicting how an entity or a condition will change over time. In financial terms, it also reflects the interest rate in compounded interest calculations.

Compounded Interest

Compounded interest is a financial concept where the interest earned on an initial amount is added back to the principal, so that from that point on, the interest that has been added also starts to earn interest. This is a classic example of utilizing an exponential function, as the growth accumulates at a rate dependent on the current balance.To calculate compounded interest over time, we use the formula:\[A = P(1 + r/n)^{nt}\]Where:

• \( A \) is the future value of the investment or loan, including interest.
• \( P \) is the principal investment amount (initial amount).
• \( r \) is the annual interest rate (as a decimal).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested or borrowed for, in years.

In our context, \( y = 20(1.1)^x \), assuming \( n = 1 \) (compounded annually), matches the form of compounded interest calculations, where each increment in time increases the total by a set rate (10% per year). This formula and concept are widely used in finance to understand the growth of savings, investments, or loans over time.