Question

1. suppose the average cost function for a particular company is given by $a(x) = \frac{600 + 3.5x}{x}$, where $x$ is the number of units produced.

a. what is $a(30)$? what is the practical meaning?
b. what is the domain of this function? what is its practical domain?
c. what is the horizontal asymptote of this function? what is its practical meaning?
d. find the horizontal intercept of the function. does it have a practical meaning?

Explanation:

Step1: Calculate \(A(30)\)

Substitute \(x = 30\) into \(A(x)=\frac{600 + 3.5x}{x}\).

$$
LATEXBLOCK0
$$

The practical meaning is that when \(30\) units are produced, the average cost per unit is \(23.5\).

Step2: Find the domain of \(A(x)\)

For the function \(A(x)=\frac{600 + 3.5x}{x}\), the denominator \(x
eq0\). So the domain is \(\{x|x\in R,x
eq0\}\).
In a practical sense (since \(x\) represents the number of units produced), \(x>0\). So the practical domain is \(\{x|x > 0,x\in R\}\).

Step3: Determine the horizontal asymptote

We can rewrite \(A(x)=\frac{600+3.5x}{x}=\frac{600}{x}+3.5\).
As \(x
ightarrow\pm\infty\), \(\frac{600}{x}
ightarrow0\). So \(y = 3.5\) is the horizontal asymptote.
The practical meaning is that as the number of units produced \(x\) gets very large, the average cost per unit approaches \(3.5\).

Step4: Find the horizontal intercept

Set \(A(x)=0\), then \(\frac{600+3.5x}{x}=0\). Multiply both sides by \(x\) (assuming \(x
eq0\)) gives \(600 + 3.5x=0\), so \(x=-\frac{600}{3.5}\approx - 171.43\).
Since \(x\) represents the number of units produced (\(x>0\) in practical terms), it has no practical meaning.

Answer:

a. \(A(30) = 23.5\), meaning when \(30\) units are produced, the average cost per unit is \(23.5\).
b. Domain: \(\{x|x\in R,x
eq0\}\), practical domain: \(\{x|x > 0,x\in R\}\).
c. Horizontal asymptote \(y = 3.5\), meaning as \(x\) (number of units) gets very large, average cost per unit approaches \(3.5\).
d. Horizontal intercept \(x\approx - 171.43\), has no practical meaning (since \(x>0\) for number of units produced).