Problem 76 - Calculus an Applied Approach

Question

Depreciation The value \(V\) of a machine \(t\) years after it is purchased is inversely proportional to the square root of \(t+1 .\) The initial value of the machine is \(\$ 10,000 .\) (a) Write \(V\) as a function of \(t .\) (b) Find the rate of depreciation when \(t=1\) (c) Find the rate of depreciation when \(t=3\)

Step-by-Step Solution

Answer

The function for \(V\) in terms of \(t\) is \(V(t) = \frac{10000} {\sqrt{ t + 1 }}\). The rate of depreciation at \(t=1\) is approximately $-1767.77 per year, and the rate of depreciation at \(t=3\) is $-625 per year.

Step 1: Formulate the Function

Given that the value \(V\) is inversely proportional to the square root of \(t+1\), the function can be expressed as \(V = \frac{k} {\sqrt{ t + 1 }}\), where \(k\) is a constant of proportionality. Since the initial value of the machine (\(t=0\)) is $10,000, you can substitute these values into the equation to solve for \(k\): \[k = V(0)\times \sqrt{t+1}\] \[k = 10000 \times \sqrt{0+1} = 10000\] Thus, the function for \(V\) as a function of \(t\) is: \[V(t) = \frac{10000} {\sqrt{ t + 1 }}\]

Step 2: Compute Depreciation Rate at \(t=1\)

The rate of depreciation is given by the derivative of \(V(t)\) with respect to time. So, first differentiate \(V(t) = \frac{10000} {\sqrt{ t + 1 }}\) to get \[V'(t) = -\frac{5000}{(t+1)^{\frac{3}{2}}}\] Then, substitute \(t=1\) into the equation: \[V'(1) = -\frac{5000} {(1+1)^{\frac{3}{2}}} = -\frac{5000}{\sqrt{8}} \approx -1767.77\$ per year.\] This means the value of the machine decreases by about $1767.77 per year at \(t=1\).

Step 3: Compute Depreciation Rate at \(t=3\)

Follow the same process as in Step 2, but now substitute \(t=3\) into the equation: \[V'(3) = -\frac{5000} {(3+1)^{\frac{3}{2}}} = -\frac{5000}{8} = -625\$ per year.\] This means the value of the machine decreases by $625 per year at \(t=3\).

Key Concepts

Rate of Depreciation

Understanding the rate of depreciation is crucial for businesses and individuals to manage their assets effectively. Depreciation refers to the decrease in the value of an asset over time. It is quantified as the annual or periodic rate at which this value decline occurs. For a machine or any tangible asset, factors such as usage, wear and tear, and technological obsolescence contribute to depreciation.

For example, if a machine is worth \(10,000 and decreases in value by \)625 per year, its rate of depreciation is 6.25% annually. It's not only important for accounting and tax purposes, but also for making informed decisions about when to replace or upgrade assets.

Inverse Proportionality

Inverse proportionality is a concept where one value decreases as another value increases, and vice versa. In the context of our exercise, the value of the machinery, denoted by V, diminishes as time increases. This relationship is particularly important when dealing with depreciation because it models scenarios where assets deteriorate more quickly early in their life and then gradually slow down.

To describe this mathematically, if one quantity is inversely proportional to another, we can write it as: \( y = \frac{k}{x} \) where \( k \) is a constant, and \( x \) and \( y \) are the variables. In the example of the machine with an initial value of $10,000, a starting point is used to determine the constant for the inverse relationship.

Derivative of a Function

The derivative of a function is a fundamental concept in calculus that represents the rate of change. It's like capturing a snapshot of how fast something is moving at a specific moment. In terms of calculating deprecation, the derivative tells you how quickly the value of machinery declines over time.

Mathematically, if V(t) is the value of the machinery at time t, then its derivative, V'(t), is the rate of depreciation. The solution steps illustrate this by calculating the derivative to find out how quickly the machine loses value at year 1 and year 3.

Time-Value of Machinery

The time-value concept reflects the notion that money available today is worth more than the same amount in the future due to its potential earning capacity. This principle also applies to machinery and other assets through depreciation. As time progresses, machinery often loses value, not just in terms of its future revenue-generating ability but also because of increased maintenance costs, potential downtime, and the likelihood of technology becoming outdated.

Understanding how the value of machinery is expected to drop over time not only affects accounting but also strategic business decisions, such as budgeting for replacements or investing in new technology. In our previous steps, calculating the initial value and depreciation rate over time encapsulates this decline and assists in plotting out the financial timeline for company assets.