Problem 55 - College Algebra and Calculus: An Applied Approach

Question

The weekly cost \(C\) of producing \(x\) units in a manufacturing process is given by the function \(C(x)=50 x+495\) The number of units \(x\) produced in \(t\) hours is given by \(x(t)=30 t\) Find and interpret \((C \circ x)(t)\).

Step-by-Step Solution

Answer

The composed function \((C \circ x)(t) = 1500t + 495\) signifies the weekly cost of producing \(x(t)\) units. It interprets that for every t hours of work, the cost of production increases by 1500t and there is a fixed cost of 495 units regardless of the production time.

Step 1: Understanding the Functions

We have two functions here: \n1. The cost function, \(C(x)=50 x+495\), where x is the number of units manufactured.\n2. The production function, \(x(t)=30 t\), where t is time in hours and x(t) is the number of units produced in that time. The task is to compose these to find the cost of producing x(t) units.

Step 2: Generating the Composed Function

We want to find \((C \circ x)(t)\), which means we will plug the function \(x(t)\) into the function \(C(x)\). Doing this yields \(C(x(t)) = 50 * (30t) + 495\).

Step 3: Simplifying the Composed function

We then simplify this to: \((C \circ x)(t) = 50 * 30t + 495 = 1500t + 495\).

Step 4: Interpreting the Result

The function \((C \circ x)(t) = 1500t + 495\) tells us that for every t hours worked, the cost of production is increased by 1500t. The constant 495 is the fixed cost regardless of the time of production.

Key Concepts

Cost Function

A cost function is essential in determining how expenses accumulate in a production process. In our scenario, the cost function is given as \(C(x) = 50x + 495\). Here, \(x\) represents the number of units produced. The formula consists of two components:

• The variable cost, \(50x\), indicates that producing each additional unit costs \(50.
• The constant 495 represents fixed costs, which are unavoidable expenses incurred regardless of production levels.

Breaking it down:
- **Variable Costs**: These are expenses that change with the number of goods produced. Here, it's the \)50 tagged onto each unit manufactured.
- **Fixed Costs**: Such costs are consistent regardless of production levels, like rent and basic utilities. In this exercise, these costs amount to $495.
The entire function allows for predicting the total cost based on the number of units produced.

Production Function

The production function expresses the relationship between input, in this case, time, and output, the units produced. It's captured by the equation \(x(t) = 30t\). Here, \(x(t)\) is the number of units produced for \(t\) hours of work.

• 30t indicates the rate of production, here, 30 units per hour.
• Time, \(t\), acts as the input determining the output \(x(t)\).

Understanding this function helps estimate output over a given period.
For example, if \(t = 5\), substituting into the function yields \(x(5) = 150\), indicating 150 units are produced in 5 hours.
This relationship is crucial for planning production schedules and allocating resources effectively.

Time and Cost Relationship

Understanding the link between time and production cost helps in optimizing manufacturing processes. By combining the cost and production functions, we form the composed function \((C \circ x)(t)\).In this exercise, the composition is given as \((C \circ x)(t) = 1500t + 495\), indicating how costs accumulate over time:
- **1500t** reflects variable costs influenced by time, calculated from the production of 30 units per hour, each costing \(50.- **495** remains steadfast as fixed overhead costs that do not vary with time.This formula enables management to see that every hour of production incurs an additional \)1500 in costs. It's invaluable for decision-making regarding how many hours to operate, considering both cost-saving strategies and meeting production targets.