Problem 29 - Algebra 1

Question

FINANCIAL ANALYSIS In Exercises 29 and \(30,\) use a graphing calculator and the following information. You are a financial analyst for a software company. You have been asked to project the net profit of your company. The net profit of the company from 1993 to 1998 can be modeled by \(P=6.84 t^{2}-3.76 t+9.29\) where \(P\) is the profit in millions of dollars and \(t\) represents the number of years since \(1993 .\) Use the model to predict whether the net profit will reach 650 million dollars.

Step-by-Step Solution

Answer

Solving the quadratic equation will suggest the actual years when the estimated profit will reach 650 million dollars. The precise years will depend on the exact solutions of the equation.

Step 1: Understand the given model

The given model is expressed as a quadratic equation \(P=6.84 t^{2}-3.76 t+9.29\), where \(P\) stands for the profit in millions of dollars, and \(t\) represents the years past since 1993.

Step 2: Plug in the desired profit into the model

Plugging \(P = 650\) into the equation to determine the year when the profit reaches this amount, we end up solving the following quadratic equation: \(650 = 6.84t^{2} - 3.76t + 9.29\).

Step 3: Solve the quadratic equation

Subtract 650 from both sides and then rearrange the resulting equation to have it equal to zero. This will result in \(6.84t^{2} - 3.76t - 640.71 = 0\). Solve this equation for \(t\) using the quadratic formula \(t = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\).

Step 4: Interpret the result

The solutions of the equation correspond to the years when the profit will be exactly 650 million dollars. Note that we are only interested in positive real solutions, since \(t\) cannot be negative (it represents time past since 1993) and it must be a real number (we can't have a fraction of a year). Since we are working with years since 1993, add 1993 to the solution to find the actual year when the profit will be 650 million dollars.

Key Concepts

Financial Analysis

Financial analysis in the realm of a software company involves the examination of a company's financial data to understand its financial health and predict future profits. One such method of analysis is modeling profit over time using mathematical equations. In this case, a quadratic equation is used to model the company's net profit. This equation can help analysts visualize past trends and make future predictions.The equation given, \(P=6.84 t^{2}-3.76 t+9.29\), expresses profit \(P\) in millions of dollars, with \(t\) representing years since 1993. By analyzing the equation, one can determine how changes in time affect profits. Quadratic equations, like this one, typically describe trends where the rate of growth can accelerate over time, suggesting increasing returns as time progresses. Financial analysts use such models to provide valuable insight into future financial performance, guiding decision-making within the company.

Profit Prediction

Profit prediction is an essential task for financial analysts, aiming to anticipate the future profitability of a business. Using the quadratic model \(P=6.84 t^{2}-3.76 t+9.29\), analysts seek to determine when the company's profit reaches a specified level, such as 650 million. Plugging the desired profit into the quadratic equation helps analysts solve for the year \(t\) when this profit occurs.To predict when the net profit will reach 650 million dollars, we substitute \(P = 650\) into the equation and solve for \(t\). This forms a new equation: \(650 = 6.84 t^{2} - 3.76 t + 9.29\). By rearranging and solving the resulting quadratic equation \(6.84 t^{2} - 3.76 t - 640.71 = 0\), analysts can find the value of \(t\) that represents the number of years after 1993 when this profit level is expected. This predicted timeline aids in formulating strategic objectives and future planning.

Graphing Calculator Usage

Graphing calculators are valuable tools in solving complex mathematical equations, such as the quadratic equation used for profit prediction. These devices not only compute solutions but also graphically represent the equation's behavior over time. Visualizing the graph can offer additional insights into the profit trend and other potential profit levels.When solving the equation \(6.84 t^{2} - 3.76 t - 640.71 = 0\), a graphing calculator assists by quickly calculating potential solutions for \(t\). Moreover, it helps visualize the polynomial curve generated by the equation. By examining the graph, analysts can determine when the curve intersects the line \(y = 650\), indicating the years when the profit will hit the targeted 650 million.Aside from merely providing numerical results, the graphical representation can highlight additional points of interest, such as maximum profit or inflection points, contributing to broader financial insights. This makes graphing calculators an integral element of modern financial analysis.