Problem 21 - Algebra Form and Function

Question

Identify the initial value and the rate of change, and explain their meanings in practical terms. The total amount, \(C\), in dollars, spent by a company on a piece of heavy machinery after \(t\) years in service is given by \(C=20,000+1500 t\).

Step-by-Step Solution

Answer

Answer: The initial value is $20,000, which represents the initial amount spent by the company on the heavy machinery. The rate of change is $1500, which represents the increase in cost per year as the company maintains or upgrades the machinery.

Step 1: Identify the initial value

To find the initial value, simply plug in \(t=0\) into the equation and find the value of \(C\). \(C = 20,000 + 1500(0)\) \(C = 20,000\) The initial value is $20,000.

Step 2: Explain the initial value

The initial value is the amount the company spent initially on the heavy machinery. So, at the beginning (\(t=0\)), before any years have passed, the company has spent $20,000 on the machinery.

Step 3: Identify the rate of change

The rate of change in the given equation is the coefficient of the variable \(t\), which is \(1500\). This is the amount that the cost increases with each passing year.

Step 4: Explain the rate of change

The rate of change, \(1500\), represents the increase in cost per year. In practical terms, this means that the company is spending an additional \(1500 per year on the heavy machinery to maintain or upgrade it. So, as time goes by, the total cost for the company will increase by \)1500$ for each year the machinery is in service.

Key Concepts

Understanding Initial Value

When we talk about the initial value in the context of linear functions, we are referring to the starting point of a relationship. In this example, the given equation for the total cost of machinery is represented as:
\[ C = 20,000 + 1500t \]
The initial value here is \(20,000, which is the amount spent when the machinery was first acquired. It signifies that at time zero (\( t=0 \)), before any years have passed, the company has already invested \)20,000. This value is crucial as it sets the baseline of expenditure. Think of it as an upfront payment or the starting balance of a bank account, showing what you begin with before any transactions occur. Understanding the initial value helps in planning finances and budgeting accurately from the onset.

Rate of Change Explained

The rate of change in a linear function highlights how a quantity increases or decreases over time. In our machinery example, the rate of change is represented by the coefficient of the variable (\( t \))in the equation:
\[ C = 20,000 + 1500t \]
The slope, or rate of change here, is 1500. This signifies that for every additional year the machinery is in service, the cost increases by $1,500. This could represent maintenance costs, upgrades, or other continuous expenses.

• It is a linear rate, meaning that the cost expands evenly with each passing year.
• Understanding this helps in anticipating future expenses and making informed financial decisions.

By observing the rate of change, companies can forecast how their costs will evolve over time, allowing them to plan accordingly.

Real-World Applications of Linear Functions

Linear functions and their components, such as initial value and rate of change, have practical significance in various sectors. They allow businesses to model costs, income, and expenses over time, providing a framework for reliable predictions.
In the case of the machinery cost equation, this linear function can help the company:

• Plan budgets: By understanding both the upfront cost and yearly increments, better budget allocation can occur.
• Evaluate profitability: See how ongoing costs affect the return on investment.
• Schedule maintenance: Coordinate timing for regular maintenance or upgrades based on cost increases.

Beyond business, linear functions also apply to personal finance, scientific measurements, and anywhere growth or decrease needs to be tracked over time. The simplicity of a linear relationship allows for clear and straightforward analysis, valuable across all facets of planning and strategy.