Question

we found that the marketing research department for the company that manufactures and sells memory chips for microcomputers established the following price - demand and revenue functions
$p(x)=95 - 4x$ price - demand function
$r(x)=xp(x)=x(95 - 4x)$ revenue function
where $p(x)$ is the wholesale price in dollars at which $x$ million chips can be sold, and $r(x)$ is in millions of dollars. both functions have domain $1\leq x\leq19$

(a) which of the following is the graph of the revenue function?
a. graph with x - axis from 0 to 20, y - axis from 0 to 600, a curve starting at (0,0), rising to a peak around x = 10, then falling
b. graph with x - axis from 0 to 20, y - axis from - 600 to 0, a curve starting at (0,0), falling to a trough around x = 10, then rising
c. partial graph with x - axis from - 20 to 5, y - axis with some values

(b) find the output that will produce the maximum revenue.
\boxed{} million chips (type an integer or a fraction. simplify your answer)

Explanation:

Step1: Expand the revenue function

Given \( R(x)=x(95 - 4x)=95x-4x^{2}\). This is a quadratic function of the form \(y = ax^{2}+bx + c\), where \(a=-4\), \(b = 95\), \(c = 0\).

Step2: Use the formula for the vertex of a quadratic function

For a quadratic function \(y=ax^{2}+bx + c\), the \(x\) - coordinate of the vertex (which gives the maximum for \(a<0\)) is \(x=-\frac{b}{2a}\).
Substitute \(a=-4\) and \(b = 95\) into the formula: \(x=-\frac{95}{2\times(-4)}=\frac{95}{8}=11.875\)

Answer:

\(11.875\)