Problem 31 - Calculus and its applications

Question

Barton Novelties determines that its marginal cost, in dollars, for producing \(x\) keyholders is given by $$ C^{\prime}(x)=3,600,000 x^{-1.8}, \quad \text { where } x \geq 1 $$ Suppose it were possible for this company to make infinitely many keyholders. What would the total cost be?

Step-by-Step Solution

Answer

The total cost of producing infinitely many keyholders is $4,500,000.

Step 1: Understanding Marginal Cost

The marginal cost function, \( C'(x) = 3,600,000 x^{-1.8} \), represents the rate of change of the cost with respect to the number of keyholders produced. Our goal is to find the total cost of producing infinitely many keyholders.

Step 2: Setting Up the Total Cost Function

To find the total cost, we need to integrate the marginal cost function. This means evaluating the integral \( \int C'(x) \, dx \) from 1 to infinity, which will give us the total cost function \( C(x) \).

Step 3: Integrate the Marginal Cost Function

The integral of the marginal cost function is calculated as: \[ \int_{1}^{\infty} 3,600,000 x^{-1.8} \, dx \]. To integrate, we use the formula for integrating power functions, which is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \) when \( n eq -1 \).

Step 4: Compute the Indefinite Integral

Compute the indefinite integral: \[ \int 3,600,000 x^{-1.8} \, dx = 3,600,000 \cdot \frac{x^{-0.8}}{-0.8} = -4,500,000 x^{-0.8} + C \]. This helps us find the integral from 1 to infinity.

Step 5: Evaluate the Definite Integral

Since we are evaluating from 1 to infinity, calculate: \[ \lim_{{b \to \infty}} [-4,500,000 x^{-0.8}]_{1}^{b} = \lim_{{b \to \infty}} (0 - (-4,500,000 \cdot 1^{-0.8})) \]. This limit simplifies as \( b \to \infty \) reduces \( x^{-0.8} \) to 0.

Step 6: Solve the Limit

The expression simplifies to: \[ \lim_{{b \to \infty}} -4,500,000 (b^{-0.8}) + 4,500,000 = 4,500,000 \]. The term \( b^{-0.8} \to 0 \) as \( b \to \infty \), leaving us with only 4,500,000.

Key Concepts

Marginal Cost

Marginal cost is the additional cost incurred when producing one more unit of a product. In other words, it represents how much extra it costs to increase production by a single unit. Understanding marginal cost is crucial in economics because it helps businesses decide the optimal production level to maximize profit.

For Barton Novelties, the marginal cost of producing keyholders is given by the function \( C'(x) = 3,600,000 x^{-1.8} \). This function shows how the cost changes as more keyholders are produced. The negative exponent \(-1.8\) indicates that as production increases, the marginal cost per unit decreases. This is typical in manufacturing due to economies of scale. Lower marginal cost with increased output means the company gets more efficient.

• Understanding marginal cost helps in pricing decisions.
• Aids in analyzing profitability and cost structure.
• Provides insight into optimal production levels.

Definite Integral

A definite integral is a fundamental concept in calculus that calculates the accumulation of quantities, such as area under a curve, from one point to another. For Barton Novelties, we need the definite integral to determine the total cost of producing an infinite number of keyholders.

In mathematical terms, finding the definite integral of the marginal cost function \( C'(x) \) involves evaluating the integral \( \int_{1}^{\infty} C'(x) \, dx \). This definite integral gives the total change in cost from producing one keyholder to infinitely many keyholders. It answers the question: "How much would it cost to increase production to infinity?"

• A definite integral has limits of integration, in this case from 1 to infinity.
• It provides the total accumulation of a quantity across a specific range.
• Useful for finding total costs, areas, and other accumulations.

Improper Integral

Improper integrals are used when dealing with infinite limits or unbounded functions, like the marginal cost function in this scenario. These integrals help us handle situations where traditional definite integrals can't be applied directly due to infinity.

In the case of Barton Novelties, the improper integral \( \int_{1}^{\infty} 3,600,000 x^{-1.8} \, dx \) evaluates the total cost as the production volume approaches infinity. The integral from 1 to infinity is considered improper because the upper limit is infinite.

To solve this, we use a limit process. We replace infinity with a variable, often \( b \), and then calculate the limit as \( b \to \infty \). In our exercise, this process results in a finite total cost, showing that even though it seems production grows without bounds, the expense remains manageable due to decreasing marginal costs.

• Improper integrals help evaluate functions over infinite intervals.
• Useful for determining total costs, probabilities in infinite spaces, etc.
• The concept includes using limits to resolve infinities in integration.