Problem 77 - Introductory Algebra for College Students

Question

The alligator, at one time an endangered species, is the subject of a protection program. The formula $$P=-10 x^{2}+475 x+3500$$ models the alligator population, \(P,\) after \(x\) years of the protection program, where \(0 \leq x \leq 12 .\) Use the formula to solve Exercises \(76-77\) After how long is the population up to \(7250 ?\)

Step-by-Step Solution

Answer

Given the values, after solving the equation, the solutions for \(x\) will be obtained. Check the solution(s) to make sure they fall within the valid range. If they do, that's your answer.

Step 1: Understand the Problem

The formula \(P=-10 x^{2}+475 x+3500\) represents the model for the alligator population where \(P\) is the population and \(x\) is the time in years. We need to find the number of years required for the population to reach 7250.

Step 2: Substitute the Population Value

Substitute \(P=7250\) into the equation. This results in the equation \(7250=-10 x^{2}+475 x+3500\).

Step 3: Solve the Equation

Simplify the equation, it will be a quadratic equation in the form \(a x^{2} + bx + c = 0\). Solve this quadratic equation by using the quadratic formula \(x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\) where \(a=-10\), \(b=475\), and \(c=3500-7250=-3750\).

Step 4: Check the Conditions

Make sure the solutions fall within the valid range, that is, \(0\leq x\leq 12\)

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is fundamental in algebra. These are second-degree equations, generally in the form of \( ax^2 + bx + c = 0 \) where \( a \) is not equal to zero. Without a solution to these, finding specific information about various models, like the given alligator population, would be impossible.

Solving them can be approached in different ways, such as factoring, completing the square, or utilizing the Quadratic Formula, which provides a reliable method for finding both real and complex roots of the equation. To employ this formula successfully, one must identify the coefficients (a, b, and c) from the equation and plug them in accordingly.

Quadratic Formula

The Quadratic Formula, \( x = \frac{-b\pm \sqrt{b^2-4ac}}{2a} \), is a critical tool for algebra students. This formula is derived from the process of completing the square and allows us to solve any quadratic equation, regardless of whether it can be factored easily.

Practical Application

When applied to our alligator population model, we substitute the values \( a=-10, b=475, c=-3750 \) into the formula to find the number of years, \( x \), after which the population reaches 7250. The formula simplifies and solves the problem, yielding the time during which the protection program effectively increases the alligator population to the desired count.

Algebraic Models

Algebraic models use mathematical equations to represent real-world scenarios. These models can encapsulate anything from simple movements to complex behaviors like population growth, financial trends, or even weather patterns.

Insights and Predictions

An algebraic model, like our alligator population function, helps in making predictions and understanding dynamics of change over time. By setting up equations that depict the relationship between different variables, analysts can make informed predictions. For example, predicting the alligator population relies on understanding how populations change over time due to factors modeled in the equation.

Population Growth Models

Population growth models are algebraic models that describe how a population changes over time. These can be simple linear models or more complex ones like quadratic equations that account for various factors affecting growth rates. The quadratic nature allows for the inclusion of factors that cause the population to increase up to a point and then decrease, reflecting limits on growth due to resources or other constraints.

Fitting the Real-World Scenario

In our alligator population case, the quadratic equation models the effect of a protection program over time. The coefficients \( a, b, \) and \( c \) capture the complex interaction of the factors impacting growth. By solving the quadratic equation, we can identify specific points in time, like reaching a population maximum or a specific target like 7250, as dictated by the model.