Problem 59 - Algebra and Trigonometry

Question

Production Cost The cost \(C\) in dollars of producing \(x\) yards of a certain fabric is given by the function $$ C(x)=1500+3 x+0.02 x^{2}+0.0001 x^{3} $$ (a) Find \(C(10)\) and \(C(100) .\) (b) What do your answers in part (a) represent? (c) Find \(C(0)\) . (This number represents the fixed costs.)

Step-by-Step Solution

Answer

(a) \( C(10) = 1532.1 \), \( C(100) = 3000 \). (b) They represent production costs for 10 and 100 yards. (c) Fixed cost \( C(0) = 1500 \).

Step 1: Understanding the Cost Function

The cost function is given by \( C(x) = 1500 + 3x + 0.02x^2 + 0.0001x^3 \), where \( x \) is the number of yards of fabric produced. Each term of the function represents different kinds of cost: constant cost, linear cost, quadratic cost, and cubic cost.

Step 2: Calculate C(10)

Substitute \( x = 10 \) into the cost function: \( C(10) = 1500 + 3(10) + 0.02(10)^2 + 0.0001(10)^3 \). Compute each term: \( 3(10) = 30 \), \( 0.02(10)^2 = 0.02(100) = 2 \), and \( 0.0001(10)^3 = 0.0001(1000) = 0.1 \). Sum these results: \( C(10) = 1500 + 30 + 2 + 0.1 = 1532.1 \).

Step 3: Calculate C(100)

Substitute \( x = 100 \) into the cost function: \( C(100) = 1500 + 3(100) + 0.02(100)^2 + 0.0001(100)^3 \). Compute each term: \( 3(100) = 300 \), \( 0.02(100)^2 = 200 \), and \( 0.0001(100)^3 = 1000 \). Sum these results: \( C(100) = 1500 + 300 + 200 + 1000 = 3000 \).

Step 4: Interpret Part (a) Answers

The answer to part (a) represents the total production cost of fabric for different amounts of yardage. \( C(10) = 1532.1 \) represents the cost to produce 10 yards, and \( C(100) = 3000 \) represents the cost to produce 100 yards.

Step 5: Calculate C(0) for Fixed Costs

Substitute \( x = 0 \) into the cost function to find the fixed costs: \( C(0) = 1500 + 3(0) + 0.02(0)^2 + 0.0001(0)^3 = 1500 \). Hence, the fixed cost, which is the cost even when no fabric is produced, is \( 1500 \).

Key Concepts

Production Cost

In economics, the production cost is the total expense incurred to manufacture or produce goods. It encompasses all costs from raw materials to labor and equipment use. Understanding production cost allows businesses to determine pricing strategies, profitability, and budget allocation.

The cost function given in the exercise represents a company's production cost for fabric. The cost function is a mathematical formula that describes how costs change with different levels of output and is crucial for strategic planning in manufacturing. By substituting different values of output, businesses can predict future expenses. This information is crucial when deciding on pricing and production levels.

Fixed Costs

Fixed costs are expenses that do not change with the level of production. They remain constant regardless of how much or how little is produced. Examples of fixed costs include rent, salaries, and insurance.

In the given exercise, the fixed cost can be determined by setting the production level to zero, which means finding the value of the cost function when no fabric is produced. This is done by substituting 0 into the cost function, leading to the equation:
\( C(0) = 1500 \).
This shows that even without producing any fabric, the company's baseline cost is $1500 due to necessary upkeep and administrative expenses.

Quadratic Function

A quadratic function is a polynomial function of degree two. It is generally represented as \( ax^2 + bx + c \). The graph of a quadratic function is a parabola. The quadratic component in a cost function typically represents aspects of production that increase costs at an accelerated rate with increased production.

In the cost function provided:
\( C(x) = 1500 + 3x + 0.02x^2 + 0.0001x^3 \),
the term \( 0.02x^2 \) is the quadratic component. It shows how certain costs may rise disproportionately as production scales, perhaps due to inefficiencies or increased material costs as larger batches are produced.

Cubic Function

Cubic functions are polynomial functions of degree three and have the general form \( ax^3 + bx^2 + cx + d \). These functions can model more complex scenarios where costs might escalate at an even higher rate due to increased production.

In the given cost function:
\( C(x) = 1500 + 3x + 0.02x^2 + 0.0001x^3 \),
the term \( 0.0001x^3 \) represents the cubic component. This term may reflect the costs associated with managing and scaling production, such as requiring additional infrastructure, management, or dealing with capacity issues as the production output expands.
In practical terms, these higher-degree terms highlight that production does not always scale linearly with output.