Problem 48 - Algebra 1

Question

A TV station's local news program has \(50,000\) viewers. The managers of the station plan to increase the number of viewers by \(2 \%\) per month. Write an exponential growth model to represent the number of viewers in \(t\) months.

Step-by-Step Solution

Answer

The exponential growth model that represents the number of viewers in t months is \(N = 50,000 * 1.02^t\).

Step 1: Identify the Knowns

The initial number of viewers is \(N_0 = 50,000\). The rate of growth is r = 2%. To use in the formula, convert the percent to a decimal: \(r = 0.02\). The time in duration will be t months.

Step 2: Apply the Exponential Growth Formula

Substitute the knowns into the formula. Therefore, the exponential growth model for the TV station will be \(N = 50,000 * (1 + 0.02)^t\).

Step 3: Simplify the Exponential Growth Model

To make the equation easier to use, simplify the growth factor in the equation. Hence, the exponential growth model is \(N = 50,000 * 1.02^t\) which can be used to calculate the number of viewers in t months.

Key Concepts

Exponential Growth Formula

When trying to understand how something like an audience can grow exponentially over time, it's essential to start with the exponential growth formula. The formula is used widely to predict the increase in numbers for various applications, especially where growth is not linear but accelerates over time dynamically.
The standard equation for exponential growth is: \[ N = N_0 \times (1 + r)^t \] Here is what these symbols mean:

• **\( N_0 \):** This is the initial quantity before growth begins—in our TV station example, it is 50,000 viewers.
• **\( r \):** The growth rate expressed as a decimal. In the case of the TV station, the rate is 2%, which converts to 0.02.
• **\( t \):** Time. This is the period over which growth is measured and is typically in months or years, depending on the scenario.

Essentially, you start with your initial value (\( N_0 \)), and then every period (\( t \)), it grows by a factor of \( (1 + r) \). This formula gives you a clear mathematical model to predict how your numbers will change over time.

Growth Rate

Understanding the growth rate is crucial because it tells us how quickly something is expanding.
The growth rate is the percentage by which the original amount increases, and it is a key component of the exponential growth formula. In the example of the TV station, the growth rate is 2%. This means that each month, the number of viewers is expected to increase by 2% of the current number of viewers.
To use the growth rate in calculations, it must be converted from a percentage into a decimal. This involves simply dividing the percentage by 100. So, from 2% to 0.02.

• A high growth rate means faster increases, leading to a much larger number after a series of time periods.
• A low growth rate means slower increases, resulting in a smaller number after the same period.

The transformation of a percentage to a decimal is critical as it integrates seamlessly into the exponential growth formula, driving the calculations necessary to forecast future growth.

Mathematical Model

A mathematical model is a valuable tool to represent real-world situations with the help of mathematical concepts and languages. In our exercise, the exponential growth model forms part of such a mathematical model, allowing us to predict future trends effectively.
Mathematical models are essential because they:

• Provide a simplified way of understanding complex systems or predictions.
• Allow for simulations to predict outcomes based on different growth rates and time periods.
• Can be adjusted for accuracy as more data becomes available, refining predictions.

In this scenario, by using the exponential growth model \( N = 50,000 \times 1.02^t \), it becomes straightforward to predict the number of viewers in any future month \( t \), providing critical insights for decision makers. Mathematical models like this enable media managers to plan marketing strategies, resource allocation, and set goals by anticipating future viewer growth based on current trends.