Problem 61 - College Algebra and Calculus: An Applied Approach

Question

Profit The net profits \(P\) (in billions of dollars per year) for The Hershey Company from 2002 through 2005 can be modeled by \(P=\sqrt{0.00645 t^{2}+0.1673}, \quad 2 \leq t \leq 5\) where \(t\) is time in years, with \(t=2\) corresponding to 2002 . Find the average net profit over that time period. (Source: The Hershey Co.)

Step-by-Step Solution

Answer

The average net profit for The Hershey Company from 2002 to 2005 is given by \(\frac{1}{3} \int_{2}^{5} \sqrt{0.00645t^2 + 0.1673} dt\). Note that the integral needs to be evaluated to obtain a numerical answer, which may require the use of a calculator or computer algebra system.

Step 1: Identify the endpoints of the interval

The time period under consideration is from \(t = 2\) (2002) to \(t = 5\) (2005). So, \(a = 2\) and \(b = 5\).

Step 2: Write down the formula for the average value of a function

The formula for the average value of a function over an interval \([a, b]\) is \(\frac{1}{b - a} \int_{a}^{b} f(x) dx\).

Step 3: Substitute the values into the formula

From Step 1, \(a = 2\) and \(b = 5\). The function is \(f(t) = \sqrt{0.00645t^2 + 0.1673}\). Substituting these into the formula gives us: \(\frac{1}{5 - 2} \int_{2}^{5} \sqrt{0.00645t^2 + 0.1673} dt\).

Step 4: Evaluate the integral

The integral can be evaluated using tabular integration or numerical methods, which may require the use of a calculator or computer algebra system.

Step 5: Compute the final answer

Once the integral is evaluated, divide the result by \(5 - 2 = 3\) to get the average net profit.

Key Concepts

Integration

Understanding integration is crucial for students delving into calculus. Let's illustrate its importance by considering a practical situation. Imagine we're interested in calculating the total amount of something that accumulates over time—this could be distance traveled, money earned, or, as in our example, net profit.

Integration is the mathematical process that allows us to add up tiny pieces of a quantity to find the whole. It's the counterpart to differentiation, which deals with rates of change. Think of integration as piecing together a puzzle where each piece is an infinitesimally small part of the total picture. In calculus terms, integration computes the area under a curve, denoted as the integral of the function. The symbol for integration is the elongated 'S', which stands for 'sum'—a nod to the idea of adding up all those little pieces.

When it comes to net profit, as in the exercise with The Hershey Company, integration enables us to sum up all the little bits of profit over time to get the total profit. It's the combined area under the profit curve from one time point to another. By calculating this area, we're essentially adding up all the profits between 2002 and 2005, giving us a much clearer picture of the company's financial performance over those years.

Modeling Net Profit

Modeling net profit using calculus is an excellent example of applying mathematics to real-world situations. In our exercise, the net profit of The Hershey Company is represented by a function of time, specifically a square root function.

The function has been derived from data or assumptions about the company's earnings and expresses how profit changes over time. The variable 't' represents time in years, and the model is only valid for the years between 2002 and 2005. This ability to model economic outcomes is incredibly powerful for businesses, as it helps predict future profits, evaluate performance, and make informed decisions.

Understanding the context of modeling is important as well. For instance, the time frame we’re looking at can impact the accuracy of our model. In the case of The Hershey Company, administrators would be interested not just in profits for a single year, but how the average net profit shapes up over a period, which represents a more stable financial metric than profits of a single year which can be volatile.

Evaluating Integrals

Evaluating integrals is akin to solving the puzzle of integration. It's about finding the exact numerical value of the area under the curve. In our exercise, we face the task of integrating a square root function. This can seem daunting, but with the right techniques and tools, it becomes manageable.

There are several methods to evaluate an integral. In calculus classes, you might encounter techniques such as substitution, integration by parts, or special integration rules for trigonometric functions. However, for more complex functions, numerical methods or computer algebra systems may be used to approximate the integral.

These computations can yield the total net profit over a certain period, or in more advanced applications, they can help engineers determine the amount of material needed for a construction project or how much medicine a patient receives over time through an IV drip. In all cases, the integral gives us valuable information about the total quantity of something when that quantity is changing and needs to be accumulated.