Problem 7 - Algebra and Trigonometry

Question

A new car worth \(\$ 24,000\) is depreciating in value by \(\$ 3000\) per year. a. Write a formula that models the car's value, \(y,\) in dollars, after \(x\) years. b. Use the formula from part (a) to determine after how many years the car's value will be \(\$ 9000\). c. Graph the formula from part (a) in the first quadrant of a rectangular coordinate system. Then show your solution to part (b) on the graph.

Step-by-Step Solution

Answer

a. The car's value over time can be modelled by the equation \(y = -3000x + 24000\). b. It would take 5 years for the car's value to become \(\$ 9000\). c. The graph is a straight line starting from (0,24000) and passes through the points representing each year's value exerted by the equation, including the point (5,9000).

Step 1: Formulate the Linear Equation

The formula is going to be of the form \(y = mx + c\), where \(m\) is the slope (or rate of change) and \(c\) is the y-intercept (initial value). In this case, the car is worth \(\$ 24,000\) initially, so \(c = 24000\). Each year, the car's value decreases by \(\$ 3000\), so \(m = -3000\). Hence, the linear equation will be \(y = -3000x + 24000\).

Step 2: Determine the Number of Years

To find out how many years it will take for the car's value to reach \(\$ 9000\), we will substitute \(y = 9000\) in the equation and solve for \(x\): \(9000 = -3000x + 24000 \Rightarrow x = (24000 - 9000) / 3000 \Rightarrow x = 5\). So, it will take 5 years for the car's value to become \(\$ 9000\).

Step 3: Graph the Equation and The Solution

To graph the equation \(y = -3000x + 24000\) in the first quadrant: The y-intercept (0,24000) is a starting point. From this point, we move down 3000 units and right 1 unit as per the slope. Individual points can be marked, each a year apart (1,21000), (2,18000), and so on. For part-b solution, mark the point (5,9000). Draw a line through these points, representing the depreciation of the car's value over time.