Problem 25 - Algebra 1: Concepts and Skills

Question

You buy a used truck for 20,000 dollar. The truck depreciates 7% per year. Find the value of the truck after the given number of years. $$12 years$$

Step-by-Step Solution

Answer

The value of the truck after 12 years of depreciation is obtained by substituting the values into the formula and solving the equation.

Step 1: Understanding the problem

The truck was purchased for $20,000 and depreciates 7% each year. Therefore, the future value of the truck can be determined using the formula: \( FV = P (1 - r)^n \), where: \( P \) = initial price = $20,000, \( r \) = depreciation rate = 7% = 0.07, and \( n \) = number of years = 12.

Step 2: Substitute the values into the formula

Substituting the given values into the formula gives: \( FV = 20000 * (1 - 0.07)^{12} \)

Step 3: Calculate the future value of the truck

By calculating the values, the future value of the truck after 12 years can be obtained.

Key Concepts

Exponential Decay

Exponential decay is a pattern of change that occurs when a quantity decreases by a consistent percentage over equal time intervals. In this case, the concept applies to how the value of a used truck decreases each year. Each year, the truck's value reduces by 7% of its current value. This percentage decrease continues over the specified time frame of 12 years. Why exponential decay is important:

• It helps to forecast how much an asset will depreciate over time.
• This pattern is commonly seen in real-life scenarios like depreciation of vehicles, radioactive decay, and cooling of hot objects.
• A consistent percentage makes calculations predictable and steady over time.

Understanding exponential decay provides insights into the future valuation of investments or products you own. It allows you to plan financially by knowing how much an asset like a used truck will likely be worth in the future.

Mathematical Formulas

Mathematical formulas give structure to complex problems by offering a clear method to calculate outcomes. The depreciation of the truck is calculated using the formula:\[ FV = P (1 - r)^n \]Here's what each element represents:

• \( FV \) (Future Value): The estimated value of the truck after a certain period.
• \( P \) (Principal value): The initial cost of the truck, which is $20,000 in this exercise.
• \( r \) (Rate of depreciation): The percentage by which the truck decreases in value each year, equating to 0.07 in decimal form (7%).
• \( n \) (Number of years): The duration over which the depreciation is calculated, which is 12 years in our problem.

This formula is a powerful tool used not only in depreciation calculations but also in various fields needing exponential calculations, such as biology, physics, and finance. Using it effectively, you can address complex decay-based scenarios effortlessly.

Financial Mathematics

Financial mathematics uses mathematical techniques to solve problems related to finance. When dealing with depreciation, such as in the problem of the truck, it enables accurate tracking and prediction of asset value changes over time. Key aspects of financial mathematics in depreciation:

• Asset Planning: Understanding how assets like vehicles lose value each year helps in making smarter buying, selling, or investing decisions.
• Budgeting: Anticipating depreciation assists in creating more accurate long-term financial plans and budgets.
• Investment Strategies: Knowing how depreciation affects value helps construct strategies that maximize investment returns.

Financial mathematics is crucial as it provides the analytical base for economic decisions. It doesn’t just assist in personal financial planning but also plays a significant role in business and governmental planning.