Question

The cost function for a certain model of personal digital assistant (PDA) is given by $C=13.50x+45,750$, where $C$ is measured in dollars and $x$ is the number of PDAs produced. (a) Find the average cost function $\overline{C}$. (b) Find $\overline{C}$ when $x=100$ and $x=1000$. (c) Determine the limit of the average cost function as $x$ approaches infinity. Interpret the limit in the context of the problem.

Short answer

The average cost function in this case is $\overline{C}=\frac{13.50x+45,750}{x}$. When $x=100$ and $x=1000$, the average costs are calculated by substituting these values into the average cost function. The limit of the average cost function as $x$ approaches infinity is $13.50, which represents the minimum average cost for producing a PDA as production quantity increases indefinitely.

Step-by-step solution

Find the average cost function

The average cost function, denoted by $\overline{C}$, is obtained by dividing the total cost function, $C$, by the quantity produced, $x$. So, $\overline{C}=\frac{C}{x}$ substituting $C$ from the question yields $\overline{C}=\frac{13.50x+45,750}{x}$.

Find average cost $\overline{C}$ when $x=100$ and $x=1000$

Substitute $x=100$ and $x=1000$ into the formula $\overline{C}=\frac{13.50x+45,750}{x}$ to get the average cost when producing 100 and 1000 PDAs respectively. It results in $\overline{C}(100)=\frac{13.50\ast 100+45,750}{100}$ and $\overline{C}(1000)=\frac{13.50\ast 1000+45,750}{1000}$.

Determine the limit of average cost function

The limit of the average cost function as $x$ approaches infinity is denoted by $\underset{x\to \mathrm{\infty }}{lim}\overline{C}(x)$. To find this, substitute $x=\mathrm{\infty }$ into $\overline{C}=\frac{13.50x+45,750}{x}$, this simplifies to $\underset{x\to \mathrm{\infty }}{lim}\overline{C}(x)=13.50$.

Interpretation of the limit

The limit value gives the minimum possible average cost. This means, as the production quantity approaches infinity or in other words, as you produce more and more units, the average cost of producing a PDA tends to $13.50.

Key concepts

Cost Function

Understanding the cost function is crucial for businesses to make informed financial decisions. A cost function, in mathematical terms, is an equation that summarizes the costs incurred by a company to produce a certain quantity of goods or services. In our exercise's context, the cost function for producing a personal digital assistant (PDA) is expressed as $C=13.50x+45,750$, where $C$ represents the total cost in dollars and $x$ represents the number of PDAs produced.

In real-world scenarios, the fixed cost, such as manufacturing setup or administrative expenses, is represented by the constant term (\