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Chapter 0: Problem 66

Cost The weekly cost \(C\) of producing \(x\) units in a manufacturing process is given by $$ C(x)=60 x+750 $$ The number of units \(x\) produced in \(t\) hours is given by \(x(t)=50 t\) (a) Find and interpret \((C \circ x)(t)\). (b) Find the time that must elapse in order for the cost to increase to \(\$ 15,000\).

Short Answer

Expert verified
The function \((C \circ x)(t) = 3000t + 750\) represents the weekly cost as a function of time. The cost increases by $3000 for each hour of production, with an additional fixed cost of $750. For the cost to increase to $15,000, approximately 4.75 hours must elapse.

Step by step solution

01

Calculate the composition of functions

We first get the composition of \(C\) and \(x\), denoted by \(C \circ x\). To find the composite function \((C \circ x)(t)\), we substitute \(x(t) = 50t\) into \(C(x)\). So we have: \[(C \circ x)(t) = C(x(t)) = C(50t) = 60 \cdot (50t) + 750 = 3000t + 750\]
02

Interpret the composite function

The composite function \((C \circ x)(t) = 3000t + 750\), is the cost \(C\) as a function of time \(t\). In words, it means that for every hour of production, the cost increases by $3000. An additional $750 is the fixed cost that does not depend upon the time.
03

Solve for the time when cost increases to $15,000

To find the time, we set \(C(t) = 15000\) and solve for \(t\). Like so: \[3000t + 750 = 15000\] Subtract 750 from both sides: \[3000t = 14250\] Then divide by 3000 to isolate \(t\): \[t = \frac{14250}{3000} = 4.75 \, \text{hours}\]

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Key Concepts

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

Cost Function in Manufacturing
In manufacturing, the cost function represents the total expenses incurred in production. It is a mathematical representation used to evaluate how costs change relative to different levels of output. In the exercise, the cost function is given as:\[C(x) = 60x + 750\] This equation suggests two components of the cost:
  • Variable Cost (\(60x\)): This part of the cost function represents the cost that varies with the production level. For each additional unit produced, there is an additional cost of $60.
  • Fixed Cost (750): This is a constant part of the cost function, representing costs that do not change with the level of output. These could include rent, salaries, or machinery depreciation.
Understanding the cost function helps businesses plan and strategize on production quantities and pricing to cover costs and achieve profitability.
Unit Production Calculation
Unit production calculation is vital for assessing output and production efficiency. In this scenario, the number of units produced, \(x\), varies with time, \(t\), and is described by:\[x(t) = 50t\] This linear function indicates:
  • For every hour spent producing, 50 units are manufactured. The coefficient 50 denotes production capacity per hour.
  • This straightforward relationship allows companies to predict output by simply measuring time, facilitating workforce management and resource allocation.
Understanding unit production calculations assists managers in decision-making regarding shift schedules and budgets based on anticipated production targets.
Time and Cost Relationship
The relationship between time and cost is fundamental in assessing the financial implications of production planning. By composing the functions, we find the cost as a function of time:\[(C \circ x)(t) = 3000t + 750\] Here's what this means:
  • The cost increases by \(3000 for every hour of production due to the variable costs associated with producing additional units.
  • The fixed cost is \)750, which remains constant irrespective of the time spent producing.
This information is critical when planning production schedules and budgeting. For example, determining at what point the total cost reaches $15,000 helps the company decide on production goals and timelines. Consequently, the company can strategize to optimize both time and cost resources effectively.

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Most popular questions from this chapter

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3}+1 $$

Writing In your own words, explain the meanings of domain and range.

The estimated revenues \(r\) (in billions of dollars) from sales of digital music from 2002 to 2007 can be approximated by the model \(r=15.639 t^{3}-104.75 t^{2}+303.5 t-301, \quad 2 \leq t \leq 7\) where \(t\) represents the year, with \(t=2\) corresponding to 2002. (Source: Fortune) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 2002 to 2007 . Interpret your answer in the context of the problem.

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is dropped from a height of 120 feet. $$ t_{1}=0, t_{2}=2 $$

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from ground level at a velocity of 120 feet per second. $$ t_{1}=3, t_{2}=5 $$
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