Question

You buy a used car for \(\$ 7000\). The car depreciates at the rate of \(6 \%\) per year. Find the value of the car in the given years. 2 years

Short answer

The value of the car after 2 years of depreciation is \$6185.2

Step-by-step solution

Understand the depreciation rate

In the first step, it's crucial to understand what a 6% depreciation rate means. Each year, the car's value is reduced by 6%. This implies that the car maintains \(94 \% \) (or \(0.94\)) of its value from the previous year. This calculation is made by subtracting the depreciation rate from 100%.

Apply the depreciation for the first year

By the end of the first year, the car has depreciated 6%. This means the car's value is now \(94 \% \) of its original value. This is calculated as follows: \(0.94 * \$7000 = \$6580\). So, after the first year, the car is worth \$6580.

Apply the depreciation for the second year

Again, the car's value will reduce by 6%. Therefore, its worth by the end of the second year will be \(94 \% \) of \$6580. This calculation goes as: \(0.94 * \$6580 = \$6185.2\). So, after the second year, the car is worth \$6185.2.

Key concepts

Depreciation Rate

When discussing assets like vehicles, the term depreciation rate refers to the percentage by which an asset's value decreases over a specific period of time. In the context of the exercise provided, a used car is purchased for \(\$7000\) and depreciates at a rate of 6% annually. This percentage represents a fixed part of the car's value that is lost each year due to factors such as wear and tear and obsolescence.

To ascertain the car's worth after one year, we calculate 6% of \(\$7000\) and subtract it from the original price. However, it might be simpler to think of the remaining value - 100% minus the 6% depreciation rate, which equals 94%. Hence, the car retains 94% of its value each year, expressed as a decimal (0.94).

Percent Decrease

The concept of percent decrease reflects the proportion by which a number has been reduced from its original amount. It is crucial in evaluating the devaluation of items over time. In our exemplar, a used car's value depreciates by 6% each year; this 6% is the percent decrease of the car's worth annually. To apply this to our problem, the value after one year can be found by calculating 94% of the initial \(\$7000\) value. In mathematical terms, the formula for percent decrease is as follows: \(\text{Percent Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100\%\text{.}\)

To visualize this, after the first year the car's value drops to \(\$6580\), which is \(0.94 \times \$7000\). The percent decrease from the second year is calculated using \(\$6580\) as the original value, mirroring the procedure of the first year to find the new value for that year, which is \(\$6185.2\). This iterative calculation is critical for understanding the declining value of the car over time.

Exponential Decay

Exponential decay is a term used in mathematics to describe processes that decrease at a rate proportional to their current value. This type of decay is characterized by a constant percentage decrease over time. In the car depreciation problem, this equates to the car's value reducing by a steady 6% each year, which is an example of exponential decay.

To model exponential decay in our example, we can use the formula: \(\text{Final Value} = \text{Original Value} \times (1 - \text{Depreciation Rate})^{\text{number of years}}\). Thus, calculating the car's value after two years involves the original price of \(\$7000\) being multiplied by \(0.94\) raised to the power of 2 (since we are looking two years into the future). The calculation reveals the car's value to be \(\$6185.2\) after two years. It's important to understand that the value doesn't decrease by the same amount each year but rather by the same percentage, resulting in a smaller numerical decrease each subsequent year.