Question

A plant has the capacity to produce from 0 to \(100 \mathrm{com}\) puters per day. The daily overhead for the plant is \(\$ 5000\), and the direct cost (labor and materials) of producing one computer is \(\$ 805 .\) Write a formula for \(T(x)\), the total cost of producing \(x\) computers in one day, and also for the unit cost \(u(x)\) (average cost per computer). What are the domains of these functions?

Short answer

\(T(x) = 5000 + 805x\) with domain \([0, 100]\); \(u(x) = \frac{5000 + 805x}{x}\) with domain \((0, 100]\).

Step-by-step solution

Understand the Problem

We need to create two formulas: one for the total cost \(T(x)\) of producing \(x\) computers, and another for the unit cost \(u(x)\) which is the total cost per computer. Additionally, we need to identify the domains of these functions.

Formula for Total Cost \(T(x)\)

The total cost \(T(x)\) consists of the fixed overhead cost and the variable cost, which depends on the number of computers produced. The overhead cost is \(5000, and the cost per computer is \)805. Thus, \(T(x) = 5000 + 805x\).

Formula for Unit Cost \(u(x)\)

The unit cost \(u(x)\) is the average cost per computer produced. It is calculated by dividing the total cost \(T(x)\) by the number of computers \(x\). Therefore, \(u(x) = \frac{T(x)}{x} = \frac{5000 + 805x}{x}\), assuming \(x eq 0\).

Determine the Domains

The domain for \(T(x)\) is from 0 to 100 computers, inclusive, as the plant has the capacity to produce between these amounts, so the domain is \([0, 100]\). The domain for \(u(x)\) is \(0 < x \leq 100\), as unit cost is undefined for \(x = 0\).

Key concepts

Total Cost Formula

Understanding the total cost formula is essential in the field of manufacturing as it helps in determining the expenses involved in producing goods. The total cost, denoted as \( T(x) \), includes both fixed and variable costs. In the context of our example, the plant incurs a fixed overhead cost of \(5000 daily, regardless of the number of computers produced. This is the cost associated with maintaining the plant's operations, including rent, utilities, and salaries.

Next, we have the variable cost, which changes with the production level, stemming from the direct cost of labor and materials required to make each computer. Each computer incurs a direct cost of \)805. Thus, the more computers produced, the higher the variable cost. Therefore, the total cost formula combines these elements as \( T(x) = 5000 + 805x \).

In essence, \( T(x) \) provides a linear relationship where the fixed cost remains constant, and the variable cost scales linearly with production. This formula helps managers and accountants to budget and plan for financial requirements accordingly.

Unit Cost Calculation

Unit cost, or average cost per item, is a critical concept in manufacturing for pricing and profitability analysis. It's referred to as \( u(x) \) in our problem and represents how much it costs, on average, to produce a single item - in this case, a computer.

To compute the unit cost, you divide the total production cost by the number of computers produced. This is done using the formula \( u(x) = \frac{T(x)}{x} = \frac{5000 + 805x}{x} \). The formula indicates how the average cost per computer behaves as production levels change.

An essential note is that \( x \) cannot be zero because you cannot divide by zero. Therefore, this formula applies only when at least one computer is produced. Understanding how unit cost behaves helps businesses determine pricing strategies, ensuring they cover costs and achieve desired profit margins.

Function Domains

Function domains provide vital insights into the applicable range for which a function is valid. Knowing a function's domain ensures that calculations are meaningful and references realistic production scenarios. In our case, we analyze the domains for both the total cost \( T(x) \) and unit cost \( u(x) \) functions.

For the total cost function \( T(x) = 5000 + 805x \), the domain is the range of production from 0 to 100 computers. This reflects the plant's physical capability to produce this number, supported by its maximum production capacity. Thus, \( T(x) \) is valid for \( x \) ranging from 0 to 100, inclusive, represented by \([0, 100]\).

Contrarily, the unit cost \( u(x) = \frac{5000 + 805x}{x} \) requires a slightly different approach. Since dividing by zero is undefined, \( x \) must be greater than 0. Therefore, even though the maximum remains at 100, the domain for \( u(x) \) is \( 0 < x \leq 100 \). Understanding these domains ensures the manufacturing calculations remain within practical limits.