Problem 55 - Algebra 1

Question

Use the following information. The number of recreational vehicles (RVs) sold in the United States from 1985 to 1991 can be modeled by \(N=-9.5 t^{2}+48.9 t+343.5,\) where \(N\) represents the number of vehicles sold (in thousands) and \(t\) represents the number of years since 1985. According to the model, in what year will the number of RVs sold in the United States drop to \(0 ?\)

Step-by-Step Solution

Answer

The year when the RVs sold drop to 0 is obtained after the calculations. However, beware of multiple or negative solutions for the year. The year will be 1985 plus the calculated \(t\) value.

Step 1: Write down the given equation

The given equation is: \(N = -9.5t^2 + 48.9t + 343.5\) Since we want to find when the number of RVs sold will drop to 0, we set \(N = 0\).

Step 2: Set the given equation to zero

We substitute \(N = 0\) to get: \(0 = -9.5t^2 + 48.9t + 343.5\) This is the quadratic equation we need to solve for \(t\).

Step 3: Solve the quadratic equation

Now solve the equation for \(t\). Since it is a quadratic equation, we can use the quadratic formula to solve it which is \(t = \frac{-b ± √(b^2 - 4ac)}{2a}\). Substitute \(a = -9.5\), \(b = 48.9\), and \(c = 343.5\) into the quadratic formula.

Step 4: Interpret the solution

The solutions for \(t\) will give the years after 1985 when no RVs are sold. However, it's important to note that if we have two values for \(t\), or if \(t\) is negative, we should discard it because time cannot be negative or double.

Key Concepts

Solving Quadratic Equations

Quadratic equations, which take the form of ax^2 + bx + c = 0, where a, b, and c are constants, are fundamental to algebra and can model a variety of real-world scenarios. Solving such equations requires finding the value(s) of x that make the equation true. There are various methods to tackle these problems, including factoring, completing the square, graphing, and using the quadratic formula. For complex or non-factorable equations, the quadratic formula provides a reliable solution method.

The steps to solve a quadratic equation using the quadratic formula include identifying the coefficients a, b, and c from the equation and then substituting them into the formula x = (-b ± √(b^2 - 4ac)) / (2a). After simplification, you'll obtain two solutions since the ± symbol indicates two possible values. It's crucial to consider the context of the problem, as sometimes only one solution may be valid based on real-world constraints.

Modeling with Quadratic Functions

Quadratic functions, represented as f(x) = ax^2 + bx + c, are powerful tools for modeling situations where the relationship between two quantities is not linear but parabolic, such as the path of a projectile, area optimization, or, as in the textbook exercise, the sales of recreational vehicles over time. The ability to translate real-world problems into quadratic functions allows us to predict trends and outcomes.

In the exercise regarding RV sales, a quadratic function models the number of vehicles sold over a period. The variable t represents time, and the equation provides a way to estimate sales in any given year after 1985. When modeling with quadratic functions, it is essential to understand the role of each term: the t^2 term determines the curvature of the parabola, t influences the direction and steepness, and the constant term c shows the starting point or initial value of the function. By analyzing the function, one can draw conclusions about sales trends, such as when sales might increase, reach a peak, or decline to zero.

Quadratic Formula

The quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), is a pivotal tool in algebra that allows us to solve any quadratic equation, even when other methods fail or are impractical. The formula is derived from the process of completing the square and provides the solutions by accounting for all coefficients of the quadratic equation.

The '±' sign in the formula indicates two possible solutions or roots of the equation: one by adding the square root term and one by subtracting it. When applying the formula, you must calculate the discriminant D = b^2 - 4ac first. The discriminant tells us about the nature of the roots: if D is positive, there are two distinct real roots; if D is zero, there is one real root (the parabola touches the x-axis); and if D is negative, the roots are complex. In any practical scenario, such as the RV sales question, we consider only real, positive solutions since negative time or negative sales do not make sense in the given context.