Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Given that the value of the machine \(t\) years after it is purchased is inversely proportional to the cube root of \(t+1\).

Step-by-Step Solution

Answer

The value of the machine after \(t\) years, \(V\), is given by the equation \(V = \frac{k}{(t+1)^\frac{1}{3}}\), where \(k\) is a constant that depends on the initial cost of the machine.

Step 1: Make Assumptions And Setup

Let's denote the value of the machine after \(t\) years as \(V\), and let \(k\) be the constant of variation. Because \(V\) and \((t+1)^\frac{1}{3}\) are inversely proportional, there exists a constant \(k\) such that \(V = \frac{k}{(t+1)^\frac{1}{3}}\).

Step 2: Rearrange The Equation

We can rewrite the equation as \(V(t+1)^\frac{1}{3} = k\). Note that the equation remains the same, it's just a different way of representing the same relationship.

Step 3: Establish The Constant \(k\)

The constant \(k\) can only be established if we know of a certain time \(t\) and its corresponding value \(V\). Without any additional information provided, at this point we can only say that the value at any time \(t\) is given by \(V = \frac{k}{(t+1)^\frac{1}{3}}\).

Key Concepts

Constant of Variation

The concept of the constant of variation is central whenever we talk about proportional relationships. Inverse proportionality means that one quantity increases as another quantity decreases.
The constant of variation, denoted as \( k \), is the number that remains unchanged even though the quantities themselves vary. It acts like a bridge, connecting the two reversing sides of the equation. In our specific problem, the equation is formulated as \( V = \frac{k}{(t+1)^{\frac{1}{3}}} \). Here, \( V \) is the value of the machine, and \((t+1)^{\frac{1}{3}}\) represents the cube root aspect we'll discuss soon.

To find the actual number for \( k \), we need known values for both \( V \) and \( t \). This might be the initial condition given or measured values during use. Once we know \( k \), any change in \( V \) or \( t \) can be easily calculated, showcasing the strength of understanding proportionality relationships.

Cube Root

The cube root is a mathematical function that essentially reverses the process of cubing a number. Given a value, let's call it \( x \), the cube root operation asks "what number, when raised to the power of 3, gives me back \( x \)?" This is denoted as \( x^{\frac{1}{3}} \).
In the context of our machine's value problem, we examine the cube root of \( t+1 \), which indicates how the passage of time, \( t \), impacts the machine's value via a third root operation. By using \( (t+1)^{\frac{1}{3}} \), we ensure that even as \( t \) grows larger, the rate at which it affects value is gradually scaling down, making it more realistic for machine depreciation.

Understanding cube roots is crucial in contexts where changes occur at diminishing rates, reflecting gradual processes rather than abrupt changes.

Value of Machine

The value of the machine is the central focus in this problem. Symbolically represented by \( V \), it helps us understand how a machine's worth decreases over time. With the value inversely proportional to the cube root of \( t+1 \), we deduce that immediate changes in time have a more significant effect, which decrease as time goes on.
Practically, this would mean new machines depreciate rapidly in early stages (as represented by the large initial change with small \( t \)) while later in their life, this depreciation would slow down, reflecting the typical curve of depreciation across time.

By understanding the relationship \( V = \frac{k}{(t+1)^{\frac{1}{3}}} \), we get a structured way to calculate or predict value changes. This is crucial in economics and business, where asset valuation affects decisions.