Problem 35 - Algebra 2

Question

Business A computer valued at \(\$ 6500\) depreciates at the rate of 14.3\(\%\) per year. a. Write a function that models the value of the computer. b. Find the value of the computer after three years.

Step-by-Step Solution

Answer

The function that models the value of the computer is \( A(t) = 6500 * (0.857)^t \). After 3 years, the computer's value will be approximately \$3724.84.

Step 1: Write the Depreciation Function

So the value of the computer decreases by 14.3% per year, which means it retains 100%-14.3% = 85.7% of its value each year. Depreciation, a form of exponential decay, can be modeled with the following equation where \( P \) is the initial value, \( r \) is the rate of decay (expressed as a decimal), \( t \) is the time (in years), and \( A(t) \) is the amount left after depreciation:\n\( A(t) = P(1 - r)^t \)\nWe can plug the given values into this equation to form the depreciation function:\n\( A(t) = 6500 * (1 - 0.143)^t \)

Step 2: Simplify the Function

First convert 14.3% (or 0.143) to its equivalent decimal form by subtracting it from 1, which equals 0.857. So, the function simplifies to:\n\( A(t) = 6500 * (0.857)^t \)

Step 3: Calculate the Value After Three Years

To find the value of the computer after three years (t=3), replace \( t \) in the function from Step 2 with 3 and calculate the result:\n\( A(3) = 6500 * (0.857)^3 \)

Key Concepts

Depreciation Function

In finance and business, depreciation reflects how a valuable asset like a computer loses its value over time.This process is often used in accounting to allocate the cost of an asset over its useful life.
For example, if a computer originally valued at $6500 depreciates at 14.3% annually, we can model this using a depreciation function.The depreciation function is a special form of an exponential decay function. It helps determine the remaining value of an asset after each year.To write a depreciation function, you typically use the format:\[A(t) = P(1 - r)^t\]Here:

• \(A(t)\) represents the asset's value after \(t\) years.
• \(P\) is the initial value or purchase price of the asset.
• \(r\) is the depreciation rate, expressed as a decimal.
• \(t\) is the time in years.

By plugging in the numbers for our computer example, we get the function: \[A(t) = 6500 \times (0.857)^t\]This equation helps predict the computer's value as time progresses and the asset´s value diminishes.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent.They are expressed as \( f(x) = a \cdot b^x \), where:

• \(a\) is a constant.
• \(b\) is the base.
• \(x\) is the exponent.

Exponential functions can describe rapid growth or decay occurrences.
When \(b > 1\), the function represents growth, such as population growth or investment returns.
However, when \(0 < b < 1\), it signifies decay, similar to our depreciation example.Like in the case of our computer, if we set \(b = 0.857\) (indicating a 14.3% loss), it helps to model how swiftly an asset decreases in value over time.
These functions can help businesses and individuals calculate values whether they are increasing or decreasing.

Mathematical Modeling

Mathematical modeling is the process of using mathematical structures, like equations, to represent real-world scenarios.This approach simplifies complex systems to help in analyzing and predicting behaviors in various fields.In the context of depreciation, mathematical modeling allows businesses to forecast the future value of their assets.
This is helpful for planning investments, managing inventories, or preparing financial statements.Constructing a model requires:

• Identifying the problem or scenario to model (e.g., value depreciation).
• Deriving the mathematical equation that best fits the situation.
• Plugging in specific values to solve the equation.

Having created a model like the depreciation function \(A(t) = 6500 \times (0.857)^t\), the next step involves using it to calculate needed outcomes.
In our exercise, inserting \(t = 3\) predicts the computer's value three years in the future.This process displays how mathematical modeling simplifies real-world problem-solving by translating it into comprehensible mathematical formats.