Question

(8) Home COMPUTER SALES The sales \(S\) (in millions of dollars) of home computers in the United States from 1988 to 1995 can be modeled by \(S=145.63 t^{2}+3327.56,\) where \(t\) is the number of years since \(1988 .\) Use this model to estimate the year in which sales of home computers will be 36,000 million dollars.

Short answer

To get a short answer, first solve the equation after substituting \(S\) in the sales model. Then add this value to 1988 to get the exact year.

Step-by-step solution

Understanding the problem and equation

The first step is to ensure comprehension of the given equation. Here, \(S = 145.63 t^{2} + 3327.56\) is the sales model, where \(S\) represents the sales (in millions of dollars) and \(t\) represents the numbers of years since 1988.

Substituting the target sales

Since the model is required to estimate the year for sales to reach 36,000 million dollars, replace \(S\) with 36000 in the equation. The updated equation is: 36000 = 145.63 t^{2} + 3327.56.

Calculating the years since 1988

Solve for \(t\) by subtracting 3327.56 from both sides of the equation and then dividing by 145.63. This provides the value of \(t\), which represents the years since 1988 when the sales will reach 36,000 million dollars.

Calculating the exact year

Once \(t\) is calculated, add this value to 1988 to get the exact year when the sales will reach the target amount.

Key concepts

Algebra 1

Algebra 1 serves as the cornerstone of high school mathematics, laying the groundwork for more advanced topics. In Algebra 1, we learn about variable expressions, equations, and how to use them to describe relationships and solve problems. A quintessential example is the quadratic equation, which depicts relationships that involve squared terms. In real-world scenarios, like modeling sales data, we apply quadratic equations because sales trends can often follow a path that accelerates or decelerates over time. Understanding the properties of Algebra 1 will ultimately empower students to tackle complex problems across various fields including economics, engineering, and social sciences. By wrapping our heads around Algebra 1, we can describe and predict crucial business metrics such as sales growth, maximum profit points, and optimal resource allocation.

Modeling Sales Data

When it comes to business, sales data can tell a vivid story about performance. To transform this data into meaningful insights, we often create mathematical models. For instance, the problem presents a quadratic model in the form of the equation S = 145.63t^2 + 3327.56 to represent sales over time. This sales model is instrumental in painting a picture of how sales might grow in a non-linear fashion, considering factors such as market trends, consumer behavior, and product lifecycle. Using this model, businesses can make estimations and projections that guide crucial decisions like budget allocation and strategy planning. For students, understanding how to manipulate and extract information from such models is a practical skill, extending far beyond the classroom into the real business world.

Solving Quadratic Equations

The process of solving quadratic equations is pivotal, not only in Algebra 1 but also in numerous practical applications. A quadratic equation typically takes the form ax^2 + bx + c = 0 and solutions can be found using a variety of methods such as factoring, completing the square, or applying the quadratic formula. In our example, the equation provided to model computer sales is a quadratic equation and we are tasked with finding when sales will reach a particular milestone. Solving such equations involves isolating the variable and performing arithmetic operations to find its value. Once the value for t (years since 1988) is calculated, it is straightforward to determine the year when sales will hit 36,000 million dollars. This ability to solve quadratic equations allows for predictions and informed decision-making in scenarios ranging from physics to finance.