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. 5 years

Short answer

The value of the car after 5 years is Aprroximately USD 5244.15

Step-by-step solution

Identify the Given Variables

Firstly, recognize the variables given in the problem. The initial price of the car (P) is \$7000, the annual depreciation rate (r) is 6% or 0.06 when expressed as a decimal, and the number of years (n) is 5.

Formula Substitution

Now, substitute these values into the depreciation formula. This gives \( V = 7000(1 - 0.06)^5 \).

Carry Out Calculation

Calculating this gives a final result. Ensure to preserve the order of operations (PEMDAS/BODMAS rule), subtract the rate from one before raising to power 5, and finally multiply by 7000.

Key concepts

Understanding the Depreciation Formula

Depreciation refers to the decrease in the value of an asset over time. In the case of a car, its value diminishes each year due to wear and tear as well as the introduction of newer models. The depreciation formula is a mathematical expression used to calculate this reduction in value. The standard form of the formula is:

\[ V = P(1 - r)^n \]
where:

• \(V\) represents the future value of the asset after depreciation.
• \(P\) is the initial purchase price of the asset.
• \(r\) is the annual depreciation rate, as a decimal.
• \(n\) is the number of years of depreciation.

To solve the problem given, we substitute the respective values into the formula: \(P = \$7000\), \(r = 0.06\), and \(n = 5\). Using this formula allows us to predict the expected value of the car after 5 years. Always remember to convert percentage rates into decimals by dividing by 100 before using the formula.

Exponential Decay and Asset Value

Exponential decay is a concept that describes how a quantity decreases at a rate proportional to its current value. This concept is very closely related to the depreciation of assets. It's termed 'exponential' because the value decreases exponentially, meaning it reduces by a consistent percentage over equal time periods.

In the context of depreciation, the formula for exponential decay helps to understand how the value of assets like cars or machinery depreciates over time. The car's value is not simply reduced by a fixed amount each year; instead, the reduction is a fixed percentage of its value at the start of each year. The value shrinks more slowly as the time goes on because each year, it's a smaller percentage of a decreasing amount. This calculation is very powerful because it gives a more accurate representation of the quite complex real-world process of depreciation.

Executing Mathematical Operations Correctly

To get the accurate value of the car after 5 years, it is essential to follow the correct order of mathematical operations. This is sometimes remembered by the acronym PEMDAS (stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Following this order:

• We first subtract the depreciation rate from one: \(1 - 0.06 = 0.94\).
• Next, we apply the exponent to this result: \(0.94^5\).
• Finally, we multiply this by the initial value of the car: \(7000 \times 0.94^5\).

It is important to carry out these operations correctly to avoid any mistakes in the calculation. Misordering these steps could lead to a significant error in the final valuation of the car.