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

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

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

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`