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.

Accepted Answer

Step 1: To model the car's value, we recognize that the car's value decreases linearly over time. The initial value of the car is $$45,000, and it depreciates by 5$$,000 per year. The formula for the car's value, y, after x years can be written as a linear equation: y = 45000 - 5000x. Step 2: To determine after how many years the car's value will be $$10,000, substitute y = 10,000 into the formula from part (a). This gives the equation: 10000 = 45000 - 5000x. Solve for x by isolating the variable. Step 3: Rearrange the equation from Step 2 to isolate x. Subtract 45,000 from both sides: 10000 - 45000 = -5000x. Simplify the left-hand side to get: -35000 = -5000x. Then divide both sides by -5000 to solve for x. Step 4: For part (c), graph the formula y = 45000 - 5000x in the first quadrant of a rectangular coordinate system. The y-intercept is 45,000 (when x = 0), and the slope is -5,000, indicating the car's value decreases by 5$$,000 per year. Plot the line using these points and extend it until the value reaches $$10,000. Step 5: On the graph, locate the point where the line intersects y = 10,000. This x-coordinate represents the number of years it takes for the car's value to depreciate to 10$$,000. Verify that this x-coordinate matches the solution from part (b).

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Key Concepts

Linear Functions

Depreciation

Graphing Linear Equations