Problem 94 - College Algebra and Calculus: An Applied Approach

Question

A company purchases a new machine for which the rate of depreciation can be modeled by \(\frac{d V}{d t}=10,000(t-6), \quad 0 \leq t \leq 5\) where \(V\) is the value of the machine after \(t\) years. Set up and evaluate the definite integral that yields the total loss of value of the machine over the first 3 years.

Step-by-Step Solution

Answer

The overall loss of the value of the machine over the first 3 years is $27,000.

Step 1: Establish the integral

The rate of depreciation is given by \( \frac{dV}{dt} = 10000(t-6)\). To find the total value lost, we need to integrate this function with respect to time from 0 to 3 years. So, we set up the integral like this: \(\int_{0}^{3} 10000(t-6) dt\).

Step 2: Evaluate the integral

The next step is to evaluate the integral. To do that, we first calculate the anti-derivative of the integrand, and then substitute the limits of 0 to 3. The antiderivative of the function \(10000(t-6)\) is \(5000t^2 - 60000t\). Therefore, the total value lost is given by \([5000*(3)^2 - 60000*3] - [5000*(0)^2 - 60000*0]\).

Step 3: Calculate the overall loss

Finally, we substitute the values of \( t = 3 \) and \( t = 0 \) into the equation obtained above. We get \([-27,000] - [0]\), which simplifies to an overall loss of $-27000 over the first three years.

Key Concepts

Rate of Depreciation

The rate of depreciation describes how much the value of an asset decreases over time. In this exercise, we have an equation that shows the rate of depreciation for a machine. It is given as \( \frac{dV}{dt} = 10,000(t-6) \). This equation tells us how the machine's value diminishes with each passing year.

• It depends on the variable \(t\), which represents time measured in years.
• The expression \((t-6)\) in the formula indicates that the rate of depreciation changes as the machine ages.

Understanding the rate of depreciation is crucial because it helps the business to estimate how quickly the value of their machinery will decrease. This insight aids in planning for future financial decisions, like when to replace old machines.

Anti-Derivative

The anti-derivative, also known as the integral, is the reverse process of differentiation. It allows us to recover a function from its derivative. In this context, we need to find the anti-derivative of the depreciation rate function \(10000(t-6)\) to determine the total depreciation over a period of time.

• To integrate \(10000(t-6)\), we apply basic integration rules.
• The anti-derivative calculation results in \(5000t^2 - 60000t\).

This step transforms the rate of depreciation into a formula that represents the total depreciation, giving us a clear picture of how much value is lost over a given timeframe. It’s like turning a speedometer reading (instantaneous speed) into an odometer reading (total distance traveled).

Machine Value

Machine value represents the worth of the machine at any given time. Initially, when a machine is purchased, it has a full value. But over time, due to factors such as use, wear and tear, and technological advancement, the value tends to drop. We use the concept of depreciation to express this change in value:

• Depreciation affects the machine’s market and book value.
• The initial value minus the total depreciation gives the machine's current value after a certain period.

Knowing the current value of a machine is essential for business accounting, resale, and when determining insurance costs. The machine's value is constantly decreasing due to depreciation, and the understanding of this can impact various financial calculations and decisions.

Loss of Value

Loss of value refers to the total decrease in the machine's worth over a specified time. By integrating the depreciation rate over a given time period, we calculate this total loss. In the exercise, this is achieved by evaluating the definite integral:

• The integration bounds are set from \(t = 0\) to \(t = 3\) years.
• The evaluation of this integral gives us the total loss, which is \(-27000\) over the first three years.

The negative sign indicates a decrease in value, meaning the machine is worth less than it was bought for. Calculating the loss of value helps businesses assess whether machinery turns into liabilities and when they might need to reinvest in new equipment. Understanding this helps in budget planning and capital management.