Problem 30 - 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 a graphing calculator to estimate how many years it will take for the company's net profit to reach 475 million dollars according to the model.

Step-by-Step Solution

Answer

The solution requires using a graphing calculator to solve the equation \(475 = 6.84 t^{2}-3.76 t+9.29\) for the variable \(t\), representing the number of years since 1993. The answer will vary based on the graphing calculator used, but it should be a positive value that makes sense within this business context.

Step 1: Understand the problem

You are provided with a profit model in the form of a quadratic equation, \(P=6.84 t^{2}-3.76 t+9.29\), where \(P\) represents profit in millions of dollars, and \(t\) is the time in years since 1993. The goal is to determine the number of years it will take for the company's net profit to reach 475 million dollars.

Step 2: Set up the equation

You need to set the net profit \(P\) to be 475 and solve for \(t\). This leads to the equation: \(475 = 6.84 t^{2}-3.76 t+9.29\).

Step 3: Solve the equation

The calculator will now be used to solve for \(t\) in the equation provided in Step 2. Since it is a quadratic equation, it will typically yield two solutions.

Step 4: Interpret the results

The value of \(t\) that we derive from solving the equation is the number of years since 1993 it will take for the company's net profit to reach 475 million dollars. Keep in mind that the solution should be a positive number and make sense in the context of the problem.

Key Concepts

Financial Analysis

Financial analysis is a powerful tool used to evaluate the financial health of a company. In the case of a software company, this involves examining its profit patterns.
In the exercise, we have a quadratic model representing the company's net profit over time. This model helps to predict future profits based on past data, accounting for trends and changes in the company's financial performance.
It is important to understand such models to make informed decisions, forecast outcomes, and strategize accordingly. The quadratic formula is key here as it helps in calculating projections and assessing whether goals like reaching a certain profit are feasible.

Graphing Calculator

Graphing calculators are valuable tools for solving equations, especially quadratic ones. They can graph the equation and find important points like the x-intercepts.
In this problem, you set up the profit model equation, equating it to 475 to find how many years since 1993 it will take to reach that profit.
Using a graphing calculator, you can visually see the intersection points of the curve and the line representing the profit of 475 million dollars. These intersections correspond to the potential values of time, providing visually intuitive solutions.

• Enter the quadratic equation into the graphing calculator.
• Adjust the viewing window to include the point where it intersects the line at 475.
• Identify the point(s) where the curves meet to find the value of "t".

Profit Projection

Profit projection involves estimating future profits based on existing models and data.
The quadratic equation given in the exercise is used as a predictive model to project net profits. This exercise teaches us how to apply mathematical models to anticipate future financial outcomes.
By solving the equation, you can understand how different factors, such as time and company growth, influence profit. It also provides insight into expected profit increases or decreases, helping align strategic planning to meet these financial goals.

Time Calculation

Time calculation plays a crucial role in financial projections and analyses.
In this exercise, you solve for "t," which is the number of years since 1993 when the company's profit reaches a specified value.
Solving the quadratic equation for "t" helps determine specific time frames needed to reach a financial target. This process involves:

• Setting up the equation by substituting 475 for the profit variable.
• Using methods like the quadratic formula or graphing calculators to solve for "t".
• Selecting the appropriate solution that fits the context (usually a positive time in years).

This not only provides clarity on when certain financial goals will be met but also assists in making strategic decisions regarding investments and future ventures.