Problem 66 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

DEPRECIATION The resale value of a certain industrial machine decreases at a rate that changes with time. When the machine is \(t\) years old, the rate at which its value is changing is \(200(t-6)\) dollars per year. If the machine was bought new for \(\$ 12,000\), how much will it be worth 10 years later?

Step-by-Step Solution

Answer

The machine will be worth \$10,000\ after 10 years.

Step 1: Understand the Given Information

The initial value of the machine is \( \$12,000 \). The rate of value depreciation is given by \( 200(t-6) \) dollars per year, where \( t \) is the age of the machine in years.

Step 2: Set Up the Integral for Depreciation

To find the total depreciation over 10 years, integrate the rate of depreciation function with respect to time from \( t = 0 \) to \( t = 10 \). We need to calculate \[ \int_{0}^{10} 200(t-6) \, dt \].

Step 3: Integrate the Function

Compute the integral \[ \int_{0}^{10} 200(t-6) \, dt \]: \[ \int_{0}^{10} 200t \, dt - \int_{0}^{10} 1200 \, dt = 200 \left[ \frac{t^2}{2} \right]_0^{10} - 1200 \left[ t \right]_0^{10} \].

Step 4: Evaluate the Integral

Evaluate the definite integrals: \[ 200 \left( \frac{10^2}{2} - \frac{0^2}{2} \right) - 1200 (10 - 0) = 200 \left( 50 - 0 \right) - 1200 \times 10 = 10000 - 12000 = -2000 \].

Step 5: Calculate the Resale Value after 10 Years

Subtract the total depreciation from the initial value: \( 12000 - 2000 = 10000 \). Hence, the machine's value after 10 years is \$10,000\.

Key Concepts

Resale Value

When discussing the resale value of an item, we refer to the amount of money the item can be sold for after it has been used for a certain period. For industrial machines, the resale value decreases over time due to wear and tear. This decrease is known as depreciation.
In our example, the initial value of the machine was \(12,000. After 10 years, we calculated that the machine's resale value dropped to \)10,000 due to the rate of depreciation provided.

Rate of Depreciation

The rate of depreciation is how quickly an item's value decreases over time. It is often expressed as a function of time.
In this problem, the rate of depreciation is given as a function: \(200(t-6)\), where \(t\) represents the number of years. This rate means the machine's value decreases by an amount that changes each year.
Understanding this rate can help in predicting the machine's future value and making financial decisions accordingly.

Definite Integral

A definite integral is a tool from calculus used to calculate the total change in a quantity over a specified interval. In our context, we used the definite integral to sum up the total depreciation over 10 years.
We analyzed the integral: \[ \int_{0}^{10} 200(t-6) \, dt \]
This integral computes the total depreciation from year 0 to year 10 by summing the continuous changes in value.

Initial Value

The initial value is the starting point value of the item before any depreciation occurs. For the machine in our example, the initial value was given as $12,000.
This initial value is crucial for determining how much the value has decreased over time and for calculating the current resale value after accounting for depreciation.

Integrating Functions

Integrating a function involves finding the integral of that function over a particular interval. This process helps calculate quantities like total depreciation.
When we integrated the depreciation rate \(200(t-6)\), we split it into two simpler integrals: \[ \int_{0}^{10} 200(t-6) \, dt = 200 \int_{0}^{10} t \, dt - 1200 \int_{0}^{10} 1 \, dt \]
These integrals were then evaluated to find the overall change in the machine's value, leading to the final resale value after 10 years.