Problem 5 - Algebra and Trigonometry

Question

In \(2014,\) the average price of a new car was \(\$ 37,600 .\) For the period shown, new-car prices increased by approximately \(\$ 1250\) per year. If this trend continues, how many years after 2014 will the price of a new car average \(\$ 46,350 ?\) In which year will this occur?

Step-by-Step Solution

Answer

The price of a new car will average $46,350 seven years after 2014, which is in the year 2021.

Step 1: Setup the linear equation

First, construct a linear equation to represent the situation. The formula for a linear equation is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In our case, the slope \(m\) is the annual increase in car price: $1250. The year 2014 should be thought as year 0, and in that year, the car price which is the y-intercept \(c\) is $37600.

Step 2: Solve for number of years

The problem asks for the number of years it takes for the price to reach $46350. Solve the equation obtained in step 1 for the value of x, which represents the number of years. The equation becomes: \(46350 = 1250x + 37600\). Rearranging the equation to solve for \(x\), we subtract 37600 on both sides of the equation to get \(8750 = 1250x\). Divide both sides of the equation by 1250 to isolate \(x\), gives us \(x = 7\).

Step 3: Determine the year

Since the year 2014 is treated as year 0 in this problem, add the number of years found in step 2 to 2014 to get the actual year this will occur. This gives \(2014 + 7 = 2021\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is an essential concept in linear equations, offering a simple yet powerful way to describe straight lines. Presented as \(y = mx + c\), this formula helps us understand the relationship between two variables, often visualized on a graph.
In this formula:

• \(y\) represents the dependent variable or the outcome we are interested in.
• \(x\) is the independent variable or the input that affects the outcome.
• \(m\) is the slope, indicating how much \(y\) changes for a unit increase in \(x\).
• \(c\) is the y-intercept, the point where the line crosses the y-axis, showing the value of \(y\) when \(x\) is zero.

Using this form helps simplify complex relationships into something more manageable, which is especially helpful in understanding trends and predicting future outcomes, like car prices over time. It serves as a fundamental tool in exploring various linear relationships in mathematics.

Problem Solving

Problem solving in mathematics is all about breaking down complex questions into manageable steps. Understanding the problem's context is vital, as seen with our exercise on car prices.
This particular problem focused on determining how many years it would take for car prices to increase to a given level, based on historical growth rates. Here's how we solve it systematically:

• Identify the given data, like the initial price and yearly increase, then translate it into a mathematical model using a linear equation.
• Set up the equation using the slope-intercept form, ensuring that the known values are correctly substituted.
• Apply algebraic techniques to solve the equation, isolating the unknown variable, which, in this case, was the number of years needed for the price to reach a particular value.

These steps showcase how math can be incorporated into real-world situations. Whether you're working on car prices or other problems, this methodical approach is key to finding accurate solutions.

Linear Growth

Linear growth describes a situation where a quantity increases by a constant amount over equal intervals. This concept frequently appears in financial forecasts, population studies, and price trends, as demonstrated by the car prices' problem.
Here are a few key aspects of linear growth to consider:

• The change or growth rate, indicated by the slope in the linear equation, stays constant over time.
• This predictability allows for predictable forecasting, useful in businesses and economic planning.
• The relationship between time and the growth quantity is direct and proportional, making calculations straightforward.

By understanding linear growth, we can predict how different variables might change over time, providing valuable insights and aiding decisions in various fields. The exercise highlighting the incremental price rise of cars over several years is an excellent application of this concept, illustrating potential future pricing with straightforward computation.