Problem 48 - Calculus Volume 1

Question

The total cost \(C(x)\), in hundreds of dollars, to produce \(x\) jars of mayonnaise is given by \(C(x)=0.000003 x^{3}+4 x+300\) a. Calculate the average cost per jar over the following intervals: i. [100,100.1] ii. [100,100.01] iii. [100,100.001] iv. [100,100.0001] b. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.

Step-by-Step Solution

Answer

The estimated average cost to produce 100 jars is approximately $400.00.

Step 1: Understand the Problem

The cost function given is \( C(x) = 0.000003x^3 + 4x + 300 \). We need to calculate the average cost per jar over specific intervals and use this information to estimate the average cost of producing 100 jars.

Step 2: Formula for Average Cost Per Jar

The average cost per jar over an interval [a, b] is calculated using the formula \( \frac{C(b) - C(a)}{b - a} \). Apply this formula to each given interval.

Step 3: Calculate Average Cost for [100, 100.1]

First, calculate \( C(100.1) = 0.000003(100.1)^3 + 4(100.1) + 300 \). Then, find \( C(100) = 0.000003(100)^3 + 4(100) + 300 \). Use these values in the formula \( \frac{C(100.1) - C(100)}{0.1} \).

Step 4: Calculate Average Cost for [100, 100.01]

Calculate \( C(100.01) = 0.000003(100.01)^3 + 4(100.01) + 300 \). Then, find \( C(100) \). Use \( \frac{C(100.01) - C(100)}{0.01} \) to find the average cost for this interval.

Step 5: Calculate Average Cost for [100, 100.001]

Calculate \( C(100.001) = 0.000003(100.001)^3 + 4(100.001) + 300 \). Then, find \( C(100) \). Use \( \frac{C(100.001) - C(100)}{0.001} \) to find the average cost for this interval.

Step 6: Calculate Average Cost for [100, 100.0001]

Calculate \( C(100.0001) = 0.000003(100.0001)^3 + 4(100.0001) + 300 \). Then, find \( C(100) \). Use \( \frac{C(100.0001) - C(100)}{0.0001} \) to find the average cost for this interval.

Step 7: Estimate the Average Cost at 100 Jars

Use the results from Steps 3 to 6 to observe the trend in values as the interval becomes smaller, leading to a more precise estimate as we approach the point at 100 jars.

Key Concepts

Understanding Average Cost

The concept of average cost is important when you're examining how cost per unit changes with total production. In this context, average cost per jar is calculated over a certain interval

• Average cost helps in determining the cost efficiency of producing additional units.
• It tells you the cost incurred on average for each jar produced.

To find it over a given interval, we take the difference in total costs at the endpoints of the interval and divide by the change in quantity:\[\text{Average Cost Per Jar} = \frac{C(b) - C(a)}{b - a}\]Where \(b\) and \(a\) represent the upper and lower bounds of the interval respectively.This calculation offers insights into cost behavior as production scales, helping producers make informed decisions.

Exploring the Cost Function

A cost function is a mathematical representation that describes how total costs change with production levels. Here:

• The given cost function is a polynomial: \( C(x) = 0.000003x^3 + 4x + 300 \).
• This represents the total cost in hundreds of dollars to produce \(x\) jars.
• Each term in the polynomial contributes differently to costs depending on its degree.

- The cubic term \(0.000003x^3\) accounts for complex cost factors that grow significantly with more production.- The linear term \(4x\) represents variable costs per unit jar.- The constant term \(300\) may reflect fixed costs, such as rent, that do not change with production volumes.By analyzing this function, businesses can anticipate how increasing production will influence total and per-unit costs.

Interval Estimation Explained

Interval estimation involves calculating values over increasingly smaller intervals to approximate a value more accurately. In this problem, we use intervals around 100 jars to estimate the average cost. As we decrease the size of these intervals:

• We achieve a more precise understanding of the average cost near a specific production level.
• We use the principle of comparing costs at very close quantities to smooth out any outliers or minor irregularities.

The main goal is to observe trends in average costs over these intervals: - Start with a broader interval like [100, 100.1] and gradually shrink it to [100, 100.0001]. - Careful examination of these provides an estimate for the average cost when producing 100 jars directly. This interval approach aids in capturing subtle variations in cost at a detailed level, necessary for precise decision making.

The Role of Polynomial Functions

Polynomial functions, as used in the cost function here, are widely utilized in modeling real-world situations because they offer flexibility in depicting various cost behaviors depending on their degree and coefficients. In our example:

• The cubic polynomial \( C(x) = 0.000003x^3 + 4x + 300 \) reflects nuanced cost dynamics.
• Higher-degree terms capture non-linear cost increases that small models might miss.
• The simpler linear and constant terms provide baseline cost characteristics.

- Polynomials can model both economies and diseconomies of scale, showing how costs can decrease with scaled up production, or increase exponentially due to constraints.- In the manufacturing setting, a polynomial may show how initial cost savings can reverse as capacity is stretched.This makes polynomial cost functions invaluable for realistic financial planning and operational strategies.