Problem 26 - Calculus Early Transcendentals

Question

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. A large die-casting machine used to make automobile engine blocks is purchased for \(\$ 2.5\) million. For tax purposes, the value of the machine can be depreciated by \(6.8 \%\) of its current value each year. a. What is the value of the machine after 10 years? b. After how many years is the value of the machine \(10 \%\) of its original value?

Step-by-Step Solution

Answer

#Short Answer# To find the value of the machine after 10 years, we first need to determine the decay constant k, which is found by taking the natural logarithm of 0.932, the machine's retention rate per year. The exponential decay function representing the value of the machine over time is: \(V(t) = 2.5 e^{-kt}\). To find the value after 10 years, plug in \(t=10\) and calculate: \(V(10) = 2.5 e^{-10k}\). To find how many years it takes for the machine's value to be 10% of its original value, set the function equal to 0.25 (10% of the initial value): \(0.25 = 2.5 e^{-kt}\). Then, solve for t to determine the number of years required for the machine's value to decrease to 10% of its original value.

Step 1: Identify the given information

The machine's initial value (\(V_0\)) is \(2.5\) million and the annual depreciation rate is \(6.8 \%\).

Step 2: Convert the depreciation rate to a decimal

To use the rate in the exponential decay function, we need it in decimal form: \(0.068 = 6.8 \%\). Remember that the value of the machine is depreciated by \(6.8 \%\) of its current value each year. So each year, the machine retains \((100 - 6.8)\% = 93.2\%\) of its value.

Step 3: Determine the decay constant k

Since the machine retains \(93.2\%\) of its value each year, we have: \(k = - \ln(0.932)\), where \(\ln(0.932)\) is the natural logarithm of 0.932.

Step 4: Write the exponential decay function for the machine's value

With the decay constant, the exponential decay function for the machine's value is: \(V(t) = 2.5 e^{-kt}\), where \(t\) is in years.

Step 5: Find the value of the machine after 10 years (Part a)

To find the value of the machine after 10 years, plug in \(t=10\) into the function and calculate: \(V(10) = 2.5 e^{-10k}\).

Step 6: Find the time it takes for the machine's value to be 10% of its original value (Part b)

Let the machine's value be \(10\%\) of its original value, which is \(0.1 * 2.5 = 0.25\) million. Set the function equal to this value and solve for t: \(0.25 = 2.5 e^{-kt}\). Finally, solve for t to find the number of years it takes for the machine's value to be \(10\%\) of its original value.

Key Concepts

Depreciation

Depreciation is a term commonly used in accounting to describe the gradual reduction in the value of an asset over time. In the context of our example with the die-casting machine:

• The depreciation rate helps calculate how much value the machine loses each year.
• It is expressed as a percentage of the current value that the asset is expected to depreciate each year.

For the die-casting machine, the value loses 6.8% of its current worth annually.
This means the machine retains 93.2% of its value each year after depreciation.

This concept is crucial in determining the declining value of any fixed asset, such as machinery, vehicles, or even intangible assets like patents and software.
Understanding depreciation ensures that the financial statements accurately reflect the asset's true value over time.

Exponential Function

An exponential function is a mathematical function that grows or decays at a rate proportional to its current value.
In simpler terms, it's the kind of function you'd use when a quantity decreases (or increases) at a constant percentage rate per time period.
It follows the general form:

• \[V(t) = V_0 e^{-kt}\]where:
• \(V(t)\) is the value at time \(t\).
• \(V_0\) is the initial value at time \(t=0\).
• \(k\) is the decay constant.

In our problem, the exponential function models the decay of the machine's value due to depreciation.

Using exponential functions allows us to predict future values effectively and efficiently over continuous time frames.
They are used in various fields beyond depreciation, including population growth, radioactive decay, and finance.

Decay Constant

The decay constant \(k\) is a key parameter in an exponential decay equation that represents the rate of decay.
In a financial context like with depreciation:

• The decay constant defines how quickly the value is dropping over a given time period.
• It is derived from the percentage of value retained each period (or lost, in the case of depreciation).
  

For our exercise, since the machine retains 93.2% of its original value every year, the decay constant is calculated using the natural logarithm:

• \[k = - \ln(0.932)\]

This constant is vital for constructing an accurate model of exponential decay.It ensures calculations remain consistent and valid when trying to determine future values, making mathematical modeling of depreciation reliable.