An automobile purchased for \( \$ 22,000 \) is worth \( \$ 2400 \) after 7 years. Assuming that the car's value depreciated steadily from year to year, what was it worth at the end of the third year?

To find the value of the automobile at the end of the third year, we first need to calculate the total depreciation over the 7 years. The initial value of the car was \( \$ 22,000 \), and after 7 years, it is worth \( \$ 2400 \). The total depreciation can be calculated as follows: \[ \text{Total Depreciation} = \text{Initial Value} - \text{Final Value} = 22000 - 2400 = 19600 \] Next, we find the annual depreciation by dividing the total depreciation by the number of years: \[ \text{Annual Depreciation} = \frac{\text{Total Depreciation}}{\text{Number of Years}} = \frac{19600}{7} = 2800 \] Now, we can determine the car's value at the end of the third year by calculating the depreciation for those 3 years: \[ \text{Depreciation after 3 Years} = \text{Annual Depreciation} \times 3 = 2800 \times 3 = 8400 \] Finally, we subtract the depreciation from the initial value to find the car's worth at the end of the third year: \[ \text{Value after 3 Years} = \text{Initial Value} - \text{Depreciation after 3 Years} = 22000 - 8400 = 13600 \] Therefore, the automobile was worth \( \$ 13,600 \) at the end of the third year.