Problem 55 - Algebra 1: Concepts and Skills

Question

In 1995 you purchase a parcel of land for \(8000. The value of the land depreciates by 4% every year. What will the approximate value of the land be in 2002? $$(A) 224 \quad dollar$$ $$(B) 5760 \quad dollar$$ $$(C) 6012 \quad dollar$$ $$(D) 7999\) \quad dollar$$

Step-by-Step Solution

Answer

The approximate value of the land in 2002 will be 5760 dollars.

Step 1: Identify key variables

First, the key information from the problem must be identified. The initial value, \( P_0 \), is 8000 dollars, the rate of depreciation, \( r \), is 4% (or 0.04 when expressed as a decimal), and the time, \( t \), is 2002 - 1995 = 7 years.

Step 2: Apply the formula for exponential decay

The formula for exponential decay is \( P = P_0 \times (1 - r)^t \), where \( P \) is the value at time \( t \), \( P_0 \) is the initial value, \( r \) is the rate of decay, and \( t \) is the time. Plug the identified values into the formula and simplify: \( P = 8000 \times (1 - 0.04)^7 \).

Step 3: Calculate the value

Use the calculator to compute the value of the formula, rounding to nearest whole number to match the answer choices provided in the exercise. \( P = 8000 \times (0.96)^7 \approx 5760 \) dollars. Keep in mind that due to the continuous depreciation, the property's value should naturally decrease, so this answer makes sense.

Key Concepts

Depreciation

Depreciation is an essential concept in finance and economics. It refers to the gradual reduction in the value of an asset over time. In the context of our exercise, the land purchased depreciates by 4% annually.

Understanding depreciation is crucial because it helps individuals and businesses forecast the future value of their assets. By knowing how quickly an asset loses value, one can make informed decisions about investing or selling.

Main points to remember about depreciation:

• Assets such as vehicles, buildings, and land often lose value over their useful life.
• Depreciation can be calculated using various methods, like straight-line or declining balance. In this case, we use exponential decay, which models a consistent percentage decrease.
• It's essential to know the percentage rate and the time frame to apply the correct depreciation calculation.

Exponential Functions

Exponential functions are vital in modeling growth and decay processes. In the exercise, the land's value decreases exponentially over time. This type of function is characterized by a constant percentage change per unit time.

For exponential decay, the basic formula is \[ P = P_0 \times (1 - r)^t \] where:

• \( P \) is the current value of the asset,
• \( P_0 \) is the initial value,
• \( r \) is the rate of decay (expressed as a decimal), and
• \( t \) is the time elapsed.

Exponential functions are widely used in various fields due to their ability to represent continuous growth or decay. Recognizing these functions in problems helps solve them effectively and predict future outcomes accurately.

In our example, using the exponential decay formula allows us to see how quickly the property loses value. The same concept applies when modeling population decline or radioactive decay.

Algebraic Problem Solving

Algebraic problem solving involves strategically identifying, representing, and manipulating equations to find solutions. In the exercise, we begin by identifying the necessary values to insert into the exponential decay formula. This step-by-step approach ensures clarity and accuracy throughout the calculation process.

Key strategies for effective algebraic problem solving include:

• Clearly identifying each variable's role, such as initial value, rate of change, and time.
• Using appropriate formulas to describe the mathematical relationships between these variables.
• Performing calculations deliberately and checking results for plausibility.

Let's break down the process:

• First, identify the known quantities, like the initial price \( P_0 \) (8000 dollars), rate of depreciation (4% or 0.04 as a decimal), and time span (7 years).
• Plug these values into the exponential decay formula.
• Compute the result using precise arithmetic operations.

This approach not only solves the problem at hand but also enhances your skills for dealing with similar exercises in the future. Remember, practice and familiarity with algebraic expressions are crucial for mastering problem solving.