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Chapter 6: Problem 35

A company's marginal cost function is \(.1 x^{2}-x+12\) dollars, where \(x\) denotes the number of units produced in 1 day. (a) Determine the increase in cost if the production level is raised from \(x=1\) to \(x=3\) units. (b) If \(C(1)=15,\) determine \(C(3)\) using your answer in (a).

Short Answer

Expert verified
The increase in cost is \(22.267\) dollars. With \(C(1) = 15\), \(C(3) = 37.267\) dollars.

Step by step solution

01

- Understanding the Problem

Identify the given marginal cost function: \(MC(x) = 0.1x^2 - x + 12\) and the change in production levels: from \(x = 1\) to \(x = 3\).
02

- Integrate the Marginal Cost Function

To find the total cost function \(C(x)\), integrate the given marginal cost function: \(\frac{dC}{dx} = 0.1x^2 - x + 12\). The integral of \(MC(x)\) is as follows: \[ C(x) = \frac{0.1}{3}x^3 - \frac{1}{2}x^2 + 12x + C_0 \]
03

- Determine the Increase in Cost from 1 to 3 Units

Evaluate the cost function at \(x = 3\) and \(x = 1\). The increase in cost is: \[ C(3) - C(1) = \frac{0.1}{3}(3^3) - \frac{1}{2}(3^2) + 12(3) + C_0 - \bigg[\frac{0.1}{3}(1^3) - \frac{1}{2}(1^2) + 12(1) + C_0\bigg] \] \(\therefore C(3) - C(1) = (0.1)(9) - (1.5) + (36) - \bigg[(0.1)(\frac{1}{3}) - 0.5 +12\bigg] = 0.3 - 1.5 + 36 - 0.0333 + 0.5 - 12 = 22.2667\)
04

- Calculate \(C(3)\) Using \(C(1) = 15\)

Given \(C(1) = 15\), use the increase in cost found in Step 3 to calculate \(C(3)\): \(\therefore C(3) = C(1) + \text{Increase calculated} = 15 + 22.2667 = 37.267 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

integration
Integration is a core concept in calculus. It's the reverse operation of differentiation. When you integrate a function, you are finding the 'total' amount accumulated from a rate of change over an interval. For our exercise, the marginal cost function, which is given as a derivative, needs to be integrated to find the total cost function.

Here's a step-by-step breakdown:
路 First, identify the marginal cost (MC) function: \(MC(x) = 0.1x^2 - x + 12\).
路 To find the total cost function, we integrate the MC function. The integral of \(MC(x)\) gives us: \(C(x) = \frac{0.1}{3}x^3 - \frac{1}{2}x^2 + 12x + C_0\).
路 \(C_0\) is the constant of integration. We'll find it later using additional information.

Integration can seem tricky at first, but remember that it's simply the process of summing up small pieces to find a whole. Think of it like summing up the slices of a pie. The integral gives us the total pie!
cost function
A cost function describes the total cost of producing a certain number of units. It accounts for all the costs involved in production. For businesses, understanding the cost function is key to making informed financial decisions.

Here's how we derive the cost function in the textbook problem:
路 We start from the marginal cost (MC) function: \(MC(x) = 0.1x^2 - x + 12\).
路 By integrating the MC function, we obtain the cost function \(C(x)\): \(C(x) = \frac{0.1}{3}x^3 - \frac{1}{2}x^2 + 12x + C_0\).

Given \(C(1) = 15\), we find the original cost constant \(C_0\) by substituting \(x=1\) into \(C(x)\).
This gives us:
路 \(15 = \frac{0.1}{3}(1^3) - \frac{1}{2}(1^2) + 12(1) + C_0\)
路 Solving for \(C_0\), we find the exact cost function.

Understanding the cost function helps in budgeting and forecasting future expenses.
increase in cost
The increase in cost between two production levels can be calculated by evaluating the difference in the total cost function at these levels.

For our problem, let's find the cost increase from producing 1 to 3 units:
路 Evaluate the cost function at \(x=3\) and \(x=1\): \[C(3) = \frac{0.1}{3}(3^3) - \frac{1}{2}(3^2) + 12(3) + C_0\]
and \[C(1) = \frac{0.1}{3}(1^3) - \frac{1}{2}(1^2) + 12(1) + C_0\].

The increase in cost is the difference: \[C(3) - C(1) = 0.3 - 1.5 + 36 - 0.0333 + 0.5 - 12 = 22.267\]. This value gives us the extra cost incurred when production is raised from 1 to 3 units.

Finally, using the given \(C(1) = 15\), we find \(C(3) \): \[C(3) = 15 + 22.267 = 37.267\].

Calculating the increase in cost helps businesses understand the financial impact of scaling production.

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Most popular questions from this chapter

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