Question

The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, produced by Phonola Media, is related to the price per compact disc. The equation
p = −0.00051x + 5 (0 ≤ x ≤ 12,000)
where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by
C(x) = 600 + 2x − 0.00002x2 (0 ≤ x ≤ 20,000).
Hint: The revenue is
R(x) = px,
and the profit is
P(x) = R(x) − C(x).
Find the revenue function,
R(x) = px.
R(x) =

Answer 1

Answer:

$R(x) = -0.00051x^2 + 5x$

$P(x) = -0.00049x^2 + 3x-600$

Step-by-step explanation:

Given

$p = -0.00051x + 5$ $\to$ $(0 \le x \le 12,000)$

$C(x) = 600 + 2x - 0.00002x^2$ $\to$ $(0 \le x \le 20,000)$

Solving (a): The revenue function

We have:

$R(x) = x * p$

Substitute $p = -0.00051x + 5$

$R(x) = x * (-0.00051x + 5)$

Open bracket

$R(x) = -0.00051x^2 + 5x$

Solving (b): The profit function

This is calculated as:

We have:

$P(x) = R(x) - C(x)$

So, we have:

$P(x) =-0.00051x^2 + 5x - (600 + 2x - 0.00002x^2)$

Open bracket

$P(x) =-0.00051x^2 + 5x -600 - 2x +0.00002x^2$

Collect like terms

$P(x) = 0.00002x^2-0.00051x^2 + 5x - 2x-600$

$P(x) = -0.00049x^2 + 3x-600$