Problem 15 - 5 Steps to A 5: AP Calculus AB 2021

Question

The sales of an item in a company follow an exponential growth/decay model, where \(t\) is measured in months. If the sales drop . from 5000 units in the first month to 4000 units in the third month, how many units should the company expect to sell during the seventh month?

Step-by-Step Solution

Answer

Answer: The company should expect to sell approximately 3207 units during the seventh month.

Step 1: Understand the problem

To solve this problem, we will use the exponential decay formula \(N(t) = N_0 * e^{-kt}\), where \(N(t)\) is the number of units at time \(t\), \(N_0\) is the initial number of units, \(k\) is the decay constant, and \(t\) is the time in months.

Step 2: Find the decay constant, k

We have two data points to find the decay constant, k: 1. At t=1, N(1) = 5000 units 2. At t=3, N(3) = 4000 units Using the first data point, \(N_0\) = 5000. Now, to find the value of k, we will use the second data point and plug in the values into the exponential decay formula: \(4000 = 5000 * e^{-k * 3}\) \(\Rightarrow \frac{4}{5} = e^{-3k}\) Now, take the natural logarithm of both sides: \(log_e(\frac{4}{5}) = -3k\) \(\Rightarrow k \approx 0.04575729\) So, the decay constant, k, is approximately 0.04575729.

Step 3: Determine the expected number of units to sell in the seventh month

Now, we have the exponential decay equation: \(N(t) = 5000 * e^{-0.04575729t}\). To find the number of units to be sold in the seventh month, plug in the value of t = 7: \(N(7) = 5000 * e^{-0.04575729*7}\) \(\Rightarrow N(7) \approx 3206.526\) Therefore, the company should expect to sell approximately 3207 units during the seventh month (rounding to the nearest whole number).

Key Concepts

Exponential Growth

Exponential growth is a powerful concept often seen in nature, economics, and many other fields. It describes situations where quantities increase at a constant rate over equal intervals of time. This means that as time progresses, the quantity grows larger and larger, much like how compound interest increases the balance of a bank account.

Exponential growth is mathematically represented by the formula:

• \(N(t) = N_0 * e^{kt}\)

where:

• \(N(t)\) is the amount at time \(t\)
• \(N_0\) is the initial amount
• \(k\) is the growth constant
• \(e\) is the base of the natural logarithms

Understanding exponential growth helps us comprehend how small, consistent changes can lead to large impacts over time. It is the opposite of exponential decay, which we will discuss further.

Mathematical Modeling

Mathematical modeling is a method used by scientists and engineers to represent real-world scenarios with mathematical formulas. This process helps predict and analyze the behavior of different systems by simplifying them into equations.

In the context of the problem, mathematical modeling allows us to predict the future sales of the item by using the exponential decay model. This model provides a way to represent the decrease in sales over time based on observed data.

By using mathematical modeling, one can not only analyze current data but also make informed decisions about the future. This is particularly useful in business environments for forecasting sales and adjusting strategies.

Decay Constant

The decay constant, often represented by \(k\), is a crucial component in modeling exponential decay situations. It measures the rate at which a quantity decreases over time. A higher decay constant indicates a faster decline of the quantity.

In exponential decay, the formula is given by:

• \(N(t) = N_0 * e^{-kt}\)

where:

• \(N(t)\) is the quantity at time \(t\)
• \(N_0\) is the initial quantity

To find the decay constant \(k\), we use initial data points and solve the resulting equation. Accurately determining \(k\) allows for precise predictions about how a quantity will diminish over time. Knowing the decay constant is key for strategy adjustments in scenarios like stock depletion and resource management.

Natural Logarithm

The natural logarithm, denoted as \(\log_e\) or simply \(\ln\), is the logarithm to the base \(e\). The number \(e\) is approximately 2.71828 and is an irrational constant that appears frequently in mathematics, especially in calculus and exponential functions.

In exponential decay problems, like the one in the exercise, the natural logarithm is used to simplify logarithmic expressions. When dealing with equations such as \(e^{-kt}\), we often take the natural logarithm to find \(k\).

For example, if you have the equation \( \frac{4}{5} = e^{-3k} \), taking the \(\ln\) of both sides allows for solving \(k\):

• \(\ln(\frac{4}{5}) = -3k\)

The natural logarithm helps break down exponential relationships, making them more manageable for solving and understanding complex mathematical problems.