Problem 255 - Calculus Volume 1

Question

[T] The total cost to produce \(x\) boxes of Thin Mint Girl Scout cookies is \(C\) dollars, where \(C=0.0001 x^{3}-0.02 x^{2}+3 x+300 .\) In \(t\) weeks production is estimated to be \(x=1600+100 t\) boxes. a. Find the marginal cost \(C^{\prime}(x)\) . b. Use Leibniz's notation for the chain rule, \(\frac{d C}{d t}=\frac{d C}{d x} \cdot \frac{d x}{d t}, \quad\) to find the rate with respect to time \(t\) that the cost is changing. c. Use b. to determine how fast costs are increasing when \(t=2\) weeks. Include units with the answer.

Step-by-Step Solution

Answer

The cost increases at 90300 dollars per week when \(t=2\) weeks.

Step 1: Find the Marginal Cost

The marginal cost, denoted as \(C'(x)\), is the derivative of the cost function with respect to \(x\). The given cost function is \(C(x) = 0.0001x^3 - 0.02x^2 + 3x + 300\). To find \(C'(x)\), differentiate each term:\[\frac{d}{dx}(0.0001x^3) = 0.0003x^2\]\[\frac{d}{dx}(-0.02x^2) = -0.04x\]\[\frac{d}{dx}(3x) = 3\]The derivative of the constant 300 is 0.Thus, \(C'(x) = 0.0003x^2 - 0.04x + 3\).

Step 2: Apply the Chain Rule

To find the rate of change of cost with respect to time \(t\), we use the chain rule: \(\frac{dC}{dt} = \frac{dC}{dx} \cdot \frac{dx}{dt}\).From Step 1, \(\frac{dC}{dx} = 0.0003x^2 - 0.04x + 3\).Given \(x = 1600 + 100t\), the derivative \(\frac{dx}{dt} = 100\).Therefore, \(\frac{dC}{dt} = \left(0.0003x^2 - 0.04x + 3\right) \cdot 100\).

Step 3: Evaluate Rate of Cost Increase at Specific Time

Now, we find \(\frac{dC}{dt}\) when \(t=2\) weeks. Compute \(x\) at \(t=2\):\[x = 1600 + 100(2) = 1800\]Substitute \(x = 1800\) into \(C'(x)\):\[C'(1800) = 0.0003(1800)^2 - 0.04(1800) + 3\]Calculate each term:\[0.0003(3240000) = 972\]\[-0.04(1800) = -72\]The sum is \(972 - 72 + 3 = 903\).Then, \(\frac{dC}{dt} = 903 \times 100 = 90300\).Thus, the cost is increasing at a rate of 90300 dollars per week at \(t=2\) weeks.

Key Concepts

Chain Rule

The chain rule is a crucial tool in calculus, especially when dealing with composite functions. When you need to find the derivative of a function that is composed of other functions, the chain rule comes in handy. Here, we apply it in the context of differentiating costs with respect to time.Imagine you have a function that depends on another variable, which, in turn, depends on a third variable. The chain rule allows you to find the rate of change of the first variable concerning the third one by linking the derivatives step-by-step. Here's the rule using Leibniz notation:

• \(\frac{dC}{dt} = \frac{dC}{dx} \cdot \frac{dx}{dt}\)

In our example, we find the initial derivative \(\frac{dC}{dx}\), which represents how the cost \(C\) changes with the number of boxes \(x\). Then, we determine \(\frac{dx}{dt}\), which is the rate at which the number of boxes changes over time. Multiplying these gives you \(\frac{dC}{dt}\), the rate of cost change over time, which is essential for understanding dynamic production scenarios.

Derivatives

Understanding derivatives is fundamental in calculus. A derivative represents the instantaneous rate of change of a function with respect to one of its variables. In simpler terms, it tells you how quickly something is changing.When we talk about derivatives in the context of costs, we often refer to the marginal cost. This is the derivative of the cost function concerning the quantity produced, here represented by \(x\). The marginal cost tells us how the total cost will change if one more box is produced. For any polynomial function, like our cost function \(C(x) = 0.0001x^3 - 0.02x^2 + 3x + 300\), you find the derivative by:

• Calculating the derivative of each term individually.
• Applying the power rule, which suggests multiplying the term by its exponent and reducing the exponent by one.

Thus, for each term in our example, you differentiate to get terms like \(0.0003x^2\) and \(-0.04x\), resulting in the marginal cost \(C'(x)\). These steps are critical in any differential calculus problems involving practical applications like cost analysis.

Cost Function

A cost function is a mathematical model that represents the total cost incurred by a company to produce a certain quantity of products. It typically includes a fixed cost independent of the production level and variable costs that vary with the number of goods produced.In this exercise, our cost function is given by \(C(x) = 0.0001x^3 - 0.02x^2 + 3x + 300\). Each component of this function reflects aspects of cost:

• The term \(0.0001x^3\) could represent increasing marginal costs at higher production levels due to inefficiencies.
• The term \(-0.02x^2\) may account for decreasing marginal costs at moderate levels of production.
• The linear term \(3x\) relates directly to variable costs per unit.
• The constant term \(300\) reflects fixed costs, such as rent or salaries, that don't change with production volume.

By analyzing and differentiating the cost function, you can make strategic decisions about scaling production, predicting future costs, and optimizing profit margins. Understanding how these functions work helps businesses manage and control costs effectively.