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Chapter 4: Problem 40

Economies of Scale In 1955 Surdis \(^{87}\) obtained records from a utility company regarding its trench digging operations. The records show that the unit cost \(C(n)\) per foot of earth removed by the mechanical trencher is given approximately by $$ C(n)=\frac{15.04+0.74 n}{25 n} $$ where \(n\) is the number of hours worked per day. a. Graph. Find values for \(C^{\prime}(n)\) at \(x=2,4,6,\) and \(8 .\) Interpret what these numbers mean. What is happening? Units costs for hand digging was found to be \(\$ 0.60 .\) b. Approximate the number of hours worked at which using the trench digging machinery is more cost effective than hand digging.

Short Answer

Expert verified
Find \( C'(n) = \frac{-376}{625n^2} \) and evaluate at given \(x\). Trench digging is cheaper for \( n > 5 \) hours.

Step by step solution

01

Differentiate the Cost Function

To find the derivative of the function \( C(n) = \frac{15.04 + 0.74n}{25n} \), we apply the quotient rule. The quotient rule states that for \( f(n) = \frac{u(n)}{v(n)} \), \( f'(n) = \frac{u'(n)v(n) - u(n)v'(n)}{[v(n)]^2} \). In this case, set \( u(n) = 15.04 + 0.74n \) and \( v(n) = 25n \). Then, \( u'(n) = 0.74 \) and \( v'(n) = 25 \).Substituting these into the quotient rule gives:\[ C'(n) = \frac{(0.74)(25n) - (15.04 + 0.74n)(25)}{(25n)^2} = \frac{18.5n - 25(15.04 + 0.74n)}{625n^2} \].This simplifies to:\[ C'(n) = \frac{18.5n - (376 + 18.5n)}{625n^2} = \frac{-376}{625n^2} \].

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Key Concepts

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

Economies of Scale
When companies produce goods or perform services like trench digging, they often experience advantages called economies of scale. This concept implies that increasing the scale of operation reduces the cost per unit. When workers or machines operate for more hours per day ( ), they might do so more efficiently, lowering the unit cost.
In the context of trench digging, if more hours are worked, the cost per foot of trench can decrease. Mechanizing tasks stops being cost-effective at a certain point. Observing how unit costs change with different operational hours can help companies decide the optimal working hours to minimize costs.
  • If hours worked increase but unit cost goes down, the company is enjoying economies of scale.
  • Important considerations include changes in efficiency and fixed costs distribution over more hours.
Derivative
A derivative in calculus helps us understand how a function changes at any given point. It represents the rate of change or slope of the function at that point. In the case of the cost function for trench digging, the derivative tells us how the unit cost changes as the number of hours worked () varies.
To find the derivative of the function given by \(C(n) = \frac{15.04 + 0.74n}{25n}\), we use the quotient rule of differentiation. This rule is vital when differentiating a function that is a ratio of two other functions.
  • The quotient rule states: \(f'(n) = \frac{u'(n)v(n) - u(n)v'(n)}{[v(n)]^2}\).
  • This tells us how changes in the numerator and denominator influence the overall function.
Understanding derivatives helps predict how costs react to operational adjustments, informing business decisions.
Cost Function
A cost function like \(C(n) = \frac{15.04 + 0.74n}{25n}\) is a mathematical representation of how costs evolve depending on a variable, in this case, the number of hours worked per day. It shows us the relationship and dependencies of various elements that contribute to cost.
The cost function's components are crucial:
  • The numerator \((15.04 + 0.74n)\) includes fixed costs (\(15.04\)) and variable costs depending on hours worked (\(0.74n\)).
  • The denominator \((25n)\) helps distribute these costs over working hours.
The goal is to utilize this function to find optimal operational strategies where costs are kept to a minimum, enabling better decisions on the trench digging operations.
Mathematical Modeling
Mathematical modeling creates equations to simulate real-world processes and predict outcomes. In the context of the provided exercise, the cost function \( \frac{15.04 + 0.74n}{25n} \) acts as a model for trench digging costs.
By modeling costs, businesses can simulate different scenarios and understand how various factors affect outcomes. With trench digging, the model predicts costs associated with varying hours worked, allowing for strategic decision-making.
  • Mathematical models simplify complex operations into understandable formulas.
  • The prediction afforded by models helps in determining the most economical way to utilize resources like labor and machinery.
Through understanding and utilizing such models, businesses can optimize operations, ensuring cost efficiency while maintaining productivity.

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Most popular questions from this chapter

Biology Talekar and coworkers \(^{91}\) showed that the number \(y\) of the eggs of \(O\). furnacalis per mung bean plant was approximated by \(y=f(x)=-57.40+63.77 x-\) \(20.83 x^{2}+2.36 x^{3},\) where \(x\) is the age of the mung bean plant and \(2 \leq x \leq 7\). Find \(f^{\prime}(x),\) and explain what this means. Use the quadratic formula to show that \(f^{\prime}(x)>0\) for \(2 \leq x \leq 7\). What does this say about the preference of O. furnacalis to laying eggs on older plants?

Economies of Scale in Food Retailing Using data from the files of the National Commission on Food Retail- ing concerning the operating costs of thousands of stores, Smith \(^{24}\) showed that the sales expense as a percent of sales \(S\) was approximated by the equation \(S(x)=0.4781 x^{2}-\) \(5.4311 x+16.5795,\) where \(x\) is in sales per square foot and \(x \leq 7\) a. Find \(S^{\prime}(x)\) for any \(x\). b. Find \(S^{\prime}(4), S^{\prime}(5), S^{\prime}(6),\) and \(S^{\prime}(7)\). Interpret what is happening. c. Graph the cost function on a screen with dimensions [0,9.4] by \([0,12] .\) Also graph the tangent lines at the points where \(x\) is \(4,5,6,\) and 7 . Observe how the slope of the tangent line is changing, and relate this to the observations made above concerning the rates of change. d. Use the available operations on your computer or graphing calculator to the find where the function attains a minimum.

Take \(x>0\) and find the derivatives. \(\ln \left(x^{3}\right)\)

Protein in Milk Crocker and coworkers \(^{12}\) studied the northern elephant seal in Ano Nuevo State Reserve, California. They created a mathematical model given by the equation \(W(D)=62.3-2.68 D+0.06 D^{2},\) where \(D\) is days postpartum and \(W\) is the percentage of water in the milk. Find \(W^{\prime}(D)\). Find \(W^{\prime}(10)\). Give units.

Take \(x>0\) and find the derivatives. \(\left(x^{3}+2\right) \ln x\)
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