Problem 8 - Algebra and Trigonometry

Question

A new car worth \(\$ 45,000\) is depreciating in value by \(\$ 5000\) 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 \(\$ 10,000\). 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 model representing the car's value is \( y = 45000 - 5000x \). b) It will take 7 years for the car's value to be \$10,000.

Step 1: Model the Car's Value

The value of the car decreases by \$5000 each year. This means the value after x years, y can be given by the equation \( y = 45000 - 5000x \).

Step 2: Determine the Years for Car's Value to be \$10,000

To find out when the car's value will be \$10,000, replace y in the above equation with 10000 and solve for x. \[ 10000 = 45000 - 5000x \] \[ 5000x = 45000 - 10000 \] \[ 5000x = 35000 \] \[ x = \frac{35000}{5000} = 7 \]. Therefore, it will take 7 years for the car's value to be \$10,000.

Step 3: Graphing the Linear Equation

On a rectangular coordinate system, each point \( (x, y) \) can be represented with x as the years and y as the car’s value. Plot the y-intercept (0, 45000). Also, plot a point for 7 years as x and \$10,000 as y. Then, draw the line that connects these points. The downward slope represents the constant depreciation of the car's value each year.