Problem 33 - Calculus and its applications

Question

Suppose Fast Trends determines that the cost, in dollars, of producing \(x\) iPod holders is given by $$C(x)=-0.05 x^{2}+50 x$$Find \(\frac{C(305)-C(300)}{305-300},\) and interpret the significance of this result to the company.

Step-by-Step Solution

Answer

The average cost increase for each of the additional 5 iPod holders is $19.75.

Step 1: Define the Cost Function

The cost function is given by \( C(x) = -0.05x^2 + 50x \). This represents the cost in dollars to produce \( x \) iPod holders.

Step 2: Calculate C(305)

Substitute \( x = 305 \) into the cost function: \[ C(305) = -0.05(305)^2 + 50(305) \] Calculate this using a calculator.

Step 3: Calculate the Square of 305

Calculate \( 305^2 \) to proceed with the substitution: \[ 305^2 = 93025 \].

Step 4: Substitute and Simplify for C(305)

Now substitute \( 305^2 = 93025 \) into the equation: \[ C(305) = -0.05(93025) + 50(305) \].Calculate these separately as \(-0.05 \times 93025\) and \(50 \times 305\).

Step 5: Calculate -0.05 * 93025

Calculate \( -0.05 \times 93025 = -4651.25 \).

Step 6: Calculate 50 * 305

Calculate \( 50 \times 305 = 15250 \).

Step 7: Sum to Get C(305)

Combine the results obtained: \[ C(305) = -4651.25 + 15250 = 10598.75 \].

Step 8: Calculate C(300)

Substitute \( x = 300 \) into the cost function: \[ C(300) = -0.05(300)^2 + 50(300) \] Calculate this using a calculator.

Step 9: Calculate the Square of 300

Calculate \( 300^2 = 90000 \) to use in the substitution for \( C(300) \).

Step 10: Substitute and Simplify for C(300)

Now substitute \( 300^2 = 90000 \) into the equation: \[ C(300) = -0.05(90000) + 50(300) \].Calculate these separately as \(-0.05 \times 90000\) and \(50 \times 300\).

Step 11: Calculate -0.05 * 90000

Calculate \( -0.05 \times 90000 = -4500 \).

Step 12: Calculate 50 * 300

Calculate \( 50 \times 300 = 15000 \).

Step 13: Sum to Get C(300)

Combine the results obtained: \[ C(300) = -4500 + 15000 = 10500 \].

Step 14: Calculate the Average Rate of Change

Substitute \( C(305) = 10598.75 \) and \( C(300) = 10500 \) into the expression: \[ \frac{C(305) - C(300)}{305 - 300} = \frac{10598.75 - 10500}{5} \].

Step 15: Simplify the Expression

Calculate the numerator: \( 10598.75 - 10500 = 98.75 \).Then divide by 5: \[ \frac{98.75}{5} = 19.75 \].

Step 16: Interpret the Result

The result \( 19.75 \) represents the average additional cost per iPod holder for increasing production from 300 to 305 holders. This indicates increased expenses as production increases.

Key Concepts

Cost Function in Economics

In the realm of economics, a cost function serves as a critical tool for businesses to determine the costs associated with producing a certain number of goods. The function typically represents the total cost as a mathematical expression concerning the output quantity. Here, the cost function is given by \[ C(x) = -0.05x^2 + 50x \],where \( C(x) \) denotes the total cost in dollars for producing \( x \) iPod holders.The cost function is invaluable to businesses as it:

• Summarizes all costs involved in production.
• Helps in predicting future costs as production scales.
• Guides decision-making on production volumes and pricing strategies.

In this equation, the term \(-0.05x^2\) captures the increasing marginal cost if present, indicating inefficiencies or constraints that become more pronounced as production ramps up. Meanwhile, the linear term \(50x\) could relate to variable costs like materials and labor that change in direct proportion to the number of units produced.

Quadratic Functions in Business

Quadratic functions often arise in business scenarios, particularly where the relationships are not purely linear. This type of function is represented by an equation of the form \[ f(x) = ax^2 + bx + c \],where \(a\), \(b\), and \(c\) are constants.They are fundamentally used in business to:

• Model profitability and cost scenarios over varying output levels.
• Analyze marginal costs and benefits.
• Optimize production and profit margins.

In our given cost function, \(-0.05x^2 + 50x\), the parameter \(-0.05\) serves as the coefficient of \(x^2\), suggesting a curvature in the graph, reflecting diminishing returns or increasing inefficiencies as production increases. This helps managers see when the cost per additional unit starts rising, helping them plan production volumes to avoid hitting high-cost zones.

Production Cost Analysis

Production cost analysis focuses on understanding the various costs that come into play in manufacturing goods, guiding managers to make economically sound decisions. It involves examining fixed costs, variable costs, and the way these change with output levels.Analyzing the cost function\[ C(x) = -0.05x^2 + 50x \]involves steps such as calculating specific output levels, like \( C(300) \) and \( C(305) \), to determine how costs evolve with slight increases in production. As seen in the solution, these calculations reveal the total cost for producing 300 versus 305 units, which forms the basis for calculating the average rate of change.This average rate of change, calculated as \[ \frac{C(305) - C(300)}{305 - 300} = 19.75 \]dollars per unit, reflects the **marginal cost**, indicating the cost increase for producing each additional unit beyond 300. This metric is crucial for businesses aiming to balance production costs with sales pricing, achieving optimal profitability.