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Chapter 4: Problem 69

A company estimates that the marginal cost (in dollars per item) of producing \(x\) items is \(1.92-0.002 x .\) If the cost of producing one item is \(\$ 562,\) find the cost of producing 100 items.

Short Answer

Expert verified
The cost of producing 100 items is $742.08.

Step by step solution

01

Write the Marginal Cost Function

The marginal cost (MC) function is given as \(MC(x) = 1.92 - 0.002x\). This represents the additional cost of producing one more item.
02

Define the Total Cost Function

The total cost function \(C(x)\) can be found by integrating the marginal cost function, since the integral of the marginal cost gives the total cost. We have \(C(x) = \int (1.92 - 0.002x)\, dx\).
03

Integrate the Marginal Cost

Integrate \(1.92 - 0.002x\) with respect to \(x\):\[C(x) = \int (1.92 - 0.002x)\, dx = 1.92x - 0.001x^2 + C\]where \(C\) is the constant of integration.
04

Apply Initial Condition

We know that the cost of producing one item is \$562. Therefore, \(C(1) = 562\). Substitute this into the total cost function:\[562 = 1.92(1) - 0.001(1)^2 + C\]\[562 = 1.92 - 0.001 + C\]\[562 = 1.919 + C\] Solve for \(C\):\[C = 562 - 1.919 = 560.081\].
05

Write the Total Cost Function Including the Constant

Substitute \(C = 560.081\) back into the total cost function to get:\[C(x) = 1.92x - 0.001x^2 + 560.081\]
06

Calculate the Cost of Producing 100 Items

Substitute \(x = 100\) into the total cost function to find the cost of producing 100 items:\[C(100) = 1.92(100) - 0.001(100)^2 + 560.081\]\[C(100) = 192 - 10 + 560.081\]\[C(100) = 742.081\]

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Key Concepts

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

Marginal Cost
In calculus, the concept of marginal cost is a crucial tool to understand how increasing production impacts costs. The marginal cost function, given as \(MC(x) = 1.92 - 0.002x\), tells us the cost to produce one additional unit.
This function allows companies to predict how their expenses will change when they produce more items.
  • Makes budgeting more accurate.
  • Helps in pricing strategy.
  • Aids in determining the optimal level of production.
The beauty of marginal cost lies in its simplicity yet profound impact on decision-making. It simply gives a linearly decreasing relation with more production due to the -0.002 coefficient with \(x\).
Total Cost Function
To find out the overall expense of producing any number of items, the total cost function \(C(x)\) is needed. This function can be found by integrating the marginal cost.
The exercise involves calculating \(C(x)\) by integrating \(1.92 - 0.002x\), resulting in:\[C(x) = 1.92x - 0.001x^2 + C\]This equation represents the sum of all costs incurred up to the production of \(x\) items.
  • Reflects both fixed and variable costs.
  • Allows for calculation of total production cost for any number of items.
The main aim is to determine how much the whole process costs rather than focusing only on one additional unit.
Constant of Integration
When we integrate a function, we end up with a general solution plus a constant of integration \(C\). Here in the total cost equation, \(C\) represents the constant of integration.
In this problem, the constant is needed to account for the initial condition given by the cost of producing one item.
  • Used to adjust the indefinite integral to a specific curve.
  • It helps eliminate any constant discrepancies in the equation.
For example, in \[C(x) = 1.92x - 0.001x^2 + 560.081\], \(560.081\) is adjusted using the cost at \(x = 1\) to ensure the function truly replicates the situation.
Initial Condition
The initial condition is vital as it helps to find the precise total cost function by determining the constant of integration. Here, when the cost of producing one item is \(\\(562\), it forms our initial condition.
We apply this to find \(C\) which adjusts the total cost function to:\[562 = 1.92(1) - 0.001(1)^2 + C\]Solving this yields \(C = 560.081\).
  • Provides a specific point on the total cost curve.
  • Prevents any inaccuracies in calculating total costs.
This correction ensures the calculated cost functions give reliable answers for any production level, like for 100 items which costs \(\\)742.081\). This showcases practical use of initial conditions in real-life cost management.

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Most popular questions from this chapter

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x)=\frac{1}{\left(1-x^{2}\right)^{2}+c x^{2}}\)

Use a computer algebra system to graph \(f\) and to find \(f^{\prime}\) and \(f^{\prime \prime} .\) Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x)=\sqrt{x+5 \sin x}, \quad x \leqslant 20\)

Use a computer algebra system to graph \(f\) and to find \(f^{\prime}\) and \(f^{\prime \prime} .\) Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x)=\frac{1-e^{1 / x}}{1+e^{1 / x}}\)

Let \(v _ { 1 }\) be the velocity of light in air and \(v _ { 2 }\) the velocity of light in water. According to Fermat's Principle, a ray of light will travel from a point \(A\) in the air to a point \(B\) in the water by a path \(A C B\) that minimizes the time taken. Show that $$\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { v _ { 1 } } { v _ { 2 } }$$ where \(\theta _ { 1 }\) (the angle of incidence) and \(\theta _ { 2 }\) (the angle of refraction) are as shown. This equation is known as Snell's Law.

\(23-28\) Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. \(3 \sin \left(x^{2}\right)=2 x\)
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