Question

MODELING WITH MATHEMATICS For the 12 years that a grocery store has been open, its annual revenue \(R\) (in millions of dollars) can be modeled by the function $$ R=0.0001\left(-t^4+12 t^3-77 t^2+600 t+13,650\right) $$ where \(t\) is the number of years since the store opened. In which year(s) was the revenue \(\$ 1.5\) million?

Short answer

The solution to the problem will be the year(s) after the store opened when the annual revenue was $1.5 million. This can be found by solving the polynomial equation in Step 3. Note that a specific answer cannot be provided without solving the equation.

Step-by-step solution

Set up the equation

First, the equation provided for the revenue has to be set equal to 1.5. The problem is asking for the year(s) when the revenue was $1.5 million. So, \( R = 1.5 \) and the equation becomes: \( 1.5 = 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,650) \)

Simplify the equation

Simplify the equation to isolate terms containing \( t \): \( -t^4 + 12t^3 - 77t^2 + 600t + 13,650 = 15000 \)

Solve for \( t \)

The equation is a polynomial equation of degree 4, which may have 4 real or complex roots. The roots of the equation should be determined using a suitable method, such as the rational root theorem, synthetic division, or using a polynomial solver tool.

Interpret the results

The solution for \( t \) will give the year(s) after the store opened when the revenue was $1.5 million. Since \( t \) represents years since the store opened, only positive, real values of \( t \) are meaningful in the context of this problem. Ignore any negative or complex solutions.

Key concepts

Polynomial Equations

Understanding polynomial equations is essential when dealing with complex algebraic expressions. A polynomial is an expression consisting of variables, coefficients, and non-negative integer exponents of variables. The degree of a polynomial is the highest exponent on its variable. In our exercise, we have a fourth-degree polynomial equation, which can be written as:

1.5 = 0.0001(-t^4 + 12t^3 - 77t^2 + 600t + 13,650).

Solving such equations involves isolating the variable and finding the roots, or solutions, for the variable. The given polynomial represents the revenue in millions of dollars and includes both positive and negative powers of our variable, time (t), indicating the dynamic nature of the revenue over time.

To simplify and solve the equation, we first clear the coefficient by multiplying both sides of the equation, obtaining:

-t^4 + 12t^3 - 77t^2 + 600t + 13,650 = 15000.

The roots of this equation, corresponding to the time in years, are the solutions we seek. Note that since a fourth-degree polynomial can have up to four real or complex roots, not all solutions may be meaningful in the context of the problem – we will be looking for the positive, real solutions that make sense within the given scenario.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world situation. In the case of our exercise, the polynomial equation models the annual revenue of a grocery store over 12 years.

Mathematical models are used to predict and understand the behavior of systems in a reliable and quantifiable way. They can be simple linear functions or, as in our example, more complex polynomial functions. The coefficients in the model (-t^4, 12t^3, -77t^2, 600t, and 13,650) reflect various factors that impact the store's revenue.

These coefficients may account for the initial investment (13,650), growth over time (terms with t^3 and t^2), and other factors influencing the revenue trajectory. When using such a model, it's important to understand its limitations and assumptions. For example, polynomial models may overfit past data and might not accurately predict future trends beyond the observed interval. Nevertheless, they are incredibly valuable for examining trends and making data-driven decisions within their applicable range.

Algebraic Problem-Solving

Algebraic problem-solving is a systematic approach to solving equations or systems of equations. It involves understanding the problem, identifying what is known and what needs to be found, and then applying algebraic methods to find the solution. In our grocery store example, the problem-solving steps are clearly demonstrated.

Step 1 was to understand the question and set up the equation accordingly. The known was the revenue (1.5 million), and we needed to find the year(s) when this revenue was achieved.

Step 2 required simplifying the equation to isolate the terms containing the variable t. This step is crucial for decreasing the complexity of the problem and preparing the equation for solving.

Step 3, the actual problem-solving step, involves finding the roots of the polynomial, which may require methods such as synthetic division, the use of polynomial solvers, or numerical approaches like graphing. Solving higher-degree polynomials analytically can be challenging, and sometimes approximation methods are preferred.

Step 4 is interpreting the results in the context of the problem, ensuring we consider only the realistic and meaningful solutions. This step closes the loop of problem-solving by connecting our algebraic solution back to the real-world scenario.