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Chapter 5: Problem 44

BUSINESS: Cost A company's marginal cost function is \(M C=21 x^{4 / 3}-6 x^{1 / 2}+50\), where \(x\) is the number of units, and fixed costs are \(\$ 3000\). Find the cost function.

Short Answer

Expert verified
The cost function is \( C(x) = 9x^{7/3} - 4x^{3/2} + 50x + 3000 \).

Step by step solution

01

Understanding Marginal Cost

The marginal cost (MC) function, denoted by \( MC = 21x^{4/3} - 6x^{1/2} + 50 \), represents the rate of change of the total cost with respect to the number of units, \( x \). To find the cost function, you will need to integrate the marginal cost function with respect to \( x \).
02

Integrate the Marginal Cost Function

To find the cost function from the marginal cost function, integrate each term of \( MC \) with respect to \( x \):\[\int MC \, dx = \int (21x^{4/3} - 6x^{1/2} + 50) \, dx\]This gives:\[\int 21x^{4/3} \, dx = \frac{21}{(4/3)+1} x^{(4/3)+1} = \frac{21}{7/3} x^{7/3} = 9x^{7/3}\]\[\int -6x^{1/2} \, dx = \frac{-6}{(1/2)+1} x^{(1/2)+1} = \frac{-6}{3/2} x^{3/2} = -4x^{3/2}\]\[\int 50 \, dx = 50x\]
03

Construct the General Cost Function

Combine the integrated terms to form the general cost function, \( C(x) \):\[ C(x) = 9x^{7/3} - 4x^{3/2} + 50x + C_0 \]where \( C_0 \) is the constant of integration determined by the fixed costs.
04

Determine the Constant of Integration

Use the fixed costs to determine \( C_0 \). Given that the fixed costs are \$3000, when \( x = 0 \), the cost function is equal to these fixed costs:\[ C(0) = 9(0)^{7/3} - 4(0)^{3/2} + 50(0) + C_0 = 3000 \]Thus, \( C_0 = 3000 \).
05

Write the Complete Cost Function

Substitute \( C_0 \) back into the cost function:\[ C(x) = 9x^{7/3} - 4x^{3/2} + 50x + 3000 \]

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Key Concepts

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

Marginal Cost
The marginal cost is a fundamental concept in economics, representing the cost of producing one additional unit of a good. It's vital for businesses to understand how costs change with varying levels of production.
  • The marginal cost function in our problem is given by: \( MC = 21x^{4/3} - 6x^{1/2} + 50 \).
  • This function shows the rate at which the total cost changes as more units are produced.
  • Knowing the marginal cost helps businesses set optimal production levels to maximize profits.
Marginal cost can be plotted on a graph to analyze cost behavior over different production levels, helping businesses make informed decisions.
Integration
Integration is a mathematical process used to find the antiderivative or the general form of a function. In the context of finding a cost function, integration is used to determine the total cost from the marginal cost function.
  • To achieve this, we integrate the marginal cost function \( MC = 21x^{4/3} - 6x^{1/2} + 50 \) with respect to \( x \). This provides us the total cost function \( C(x) \).
  • The integration of each term separately yields:\[\int 21x^{4/3} \, dx = 9x^{7/3}, \\int -6x^{1/2} \, dx = -4x^{3/2}, \\int 50 \, dx = 50x\]
Integration accumulates all the small changes in cost with each additional unit, giving a comprehensive view of overall cost as production increases.
Fixed Costs
Fixed costs are expenses that do not change with the level of goods or services produced. These costs are incurred even when production is zero.
  • Examples include rent, salaries, and insurance.
  • In our problem, the fixed costs are given as \$3000.
  • These fixed costs help determine the constant of integration \( C_0 \) when integrating the marginal cost function.
  • This constant \( C_0 \) ensures that the cost function incorporates all components of production cost, both variable and fixed.
A firm must cover its fixed costs to stay in business, so understanding and accounting for them is crucial in financial planning.
Calculus in Business
Calculus is a powerful mathematical tool in business, especially in analyzing changes over time. It helps businesses determine cost efficiency, profit, and optimal production levels.
  • Through calculus, businesses can find the cost function by integrating the marginal cost, as demonstrated here.
  • It helps in identifying how costs evolve as production scales, providing strategic insights into production planning.
  • Optimization techniques in calculus aid in maximizing profits by finding the production level where costs are minimized.
  • Knowledge of differential calculus enables businesses to determine the break-even points and other key metrics essential for financial health.
By using calculus, businesses can make data-driven decisions that enhance operational efficiency and profitability, making it an invaluable part of the business toolset.

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Most popular questions from this chapter

Find the area bounded by the given curves. $$ y=x^{n} \text { and } y=x^{n-1}(\text { for } n>1) $$

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67-68. BUSINESS: Cumulative Profit A company's marginal revenue function is \(M R(x)=700 x^{-1}\) and its marginal cost function is \(M C(x)=500 x^{-1}\) (both in thousands of dollars), where \(x\) is the number of units \((x>1)\). Find the total profit from $$ x=100 \text { to } x=200 $$

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BIOMEDICAL: Bacteria A colony of bacteria is of size \(S(t)=300 e^{0.1 t}\) after \(t\) hours. Find the average size during the first 12 hours (that is, from time 0 to time 12).
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