Question

Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Rising costs Between 2010 and 2016 , the average rate of inflation was about \(1.6 \% / \mathrm{yr}\). If a cart of groceries cost \(\$ 100\) in 2010 , what will it cost in 2025 , assuming the rate of inflation remains constant at \(1.6 \% ?\)

Short answer

Answer: $126.49

Step-by-step solution

Determine the rate constant k

In order to find the value of k, we need to convert the given inflation rate from a percentage to a decimal. To do that, we simply divide the percentage by 100. The formula to find k is as follows: k = (Inflation Rate %)/100 + 1 Given that the inflation rate is 1.6%, we can plug that value into the formula: k = (1.6)/100 + 1 = 0.016 + 1 = 1.016

Devise the exponential growth function

Now that we have the value of k, we can create the exponential growth function, which has the form: Cost(t) = Initial Cost * (k^t) In this case, the initial cost is $100, and we have found the value of k to be 1.016. So, our exponential growth function is: Cost(t) = 100 * (1.016^t)

Calculate the cost in 2025

We can now use our function to predict the cost of groceries in 2025. First, we need to find the number of years that have passed since 2010: Years since 2010 = 2025 – 2010 = 15 Now, we can plug this value into our function: Cost(15) = 100 * (1.016^15) Cost(15) ≈ 100 * (1.265) Cost(15) ≈ $126.49 So, we can conclude that if the rate of inflation remains constant at 1.6% per year, a cart of groceries that cost \(100 in 2010 will cost about \)126.49 in 2025.

Key concepts

Understanding the Rate Constant in Exponential Growth

The rate constant, often denoted by the symbol 'k', is pivotal in understanding exponential growth and decay processes. In contexts such as finance, population dynamics, and natural phenomena, the rate constant represents the proportionality factor that determines the speed at which a quantity increases or decreases over time. In the case of inflation, the rate constant signifies how quickly the purchasing power of money diminishes, affecting the cost of goods and services.

The calculation for the rate constant is straightforward yet crucial. When you're given an annual percentage rate (APR) of growth or decay, you must first convert this percentage into a decimal form by dividing by 100. This decimal represents the rate of change per time unit. To transform it into the exponential growth constant 'k', you add one to the decimal figure, with the formula: \( k = \frac{{\text{{Inflation Rate %}}}}{100} + 1 \). Once 'k' is determined, it becomes the base for the exponential function representing the growth process, crucial for predicting future values of the quantity in question.

For students and learners, it's important to remember that the rate constant is not the percentage rate itself but a derived value that will be used in the exponential growth equation. It's this subtlety that often needs clarification to properly grasp the concept.

Calculating Inflation Rate and its Impact on Purchasing Power

The process of inflation rate calculation is an application of percentage increase which, when applied to money, reflects the decrease in purchasing power over time. Because inflation reduces the value of currency, it's necessary to understand how it impacts the cost of goods and services in the future. The formula to calculate future costs incorporating inflation uses the rate constant and an exponential function: \( \text{{Cost(t)}} = \text{{Initial Cost}} \times (k^t) \).

In our example, the inflation rate is given as 1.6% per year, meaning every year, the cost of a basket of goods increases by this percentage. By plugging the inflation rate into the exponential growth function, we can predict the increase in cost over a set number of years. Understanding how to calculate inflation rate is not only useful for personal finance budgeting but is also an essential concept for economics students.

When completing such an exercise, you must account for the time period over which inflation applies. Inflation rates are typically given per annum, and when projecting costs into the future, you'll need to raise the rate constant to the power corresponding to the number of years elapsed. This allows you to estimate future costs with reasonable accuracy, assuming a constant rate of inflation.

Applying Exponential Functions to Real-World Scenarios

Exponential functions have widespread applications and are particularly useful in modeling scenarios where growth or decay accelerates over time. The general form of an exponential function is \( y = ab^t \) or, as seen in inflation calculations, \( \text{{Cost(t)}} = \text{{Initial Cost}} \times (k^t) \), where 'a' is the initial quantity, 'b' is the growth factor (similar to 'k' in our problem), and 't' is time. This mathematical model is employed in various fields such as biology (population growth), finance (compound interest), and even physics (radioactive decay).

Understanding the application of exponential functions in these scenarios requires comprehending the base of the function, which represents the rate constant. For students tackling problems like predicting the future cost of groceries factoring in the inflation rate, recognizing the formula's base as the growth factor—1.016 in our case—is critical.

It's worth noting that these functions assume a continuous and steady rate of change. Nevertheless, they provide an excellent approximation for the behavior of the modeled phenomena over time. With this tool, students can analyze and predict outcomes in diverse contexts, making it a powerful component of their mathematical arsenal.