Question

The management of Acrosonic plans to market the ElectroStat, an electrostatic speaker system. The marketing department has determined that the demand for these speakers is represented by the following function, where p denotes the speaker's unit price (in dollars) and x denotes the quantity demanded. Find the following functions (in dollars), find the value (in dollars) and interpret your results.

p = −0.02x + 610 (0 ≤ x ≤ 20,000)

a. Find the revenue function R.
b. Find the marginal revenue function R'(x).
c. Compute the following value.
R'(5,400) = _______________

Answer 1

(a)
`R(x)=-0.02x^2+610x`

(b)
`R'(x)=-0.04x+610`

(c)R'(5400)=$394

Explanation:

Given that x is the quantity demanded and the speaker's unit price (in dollars) is p where:

p = −0.02x + 610 (0 ≤ x ≤ 20,000)

(a)Revenue function R.

Revenue = Price X Quantity Demanded

Therefore:

R(x)=xp

`=x(-0.02x + 610)\\R(x)=-0.02x^2+610x`

(b)Marginal revenue function R'(x)

If
`R(x)=-0.02x^2+610x`

Then, the marginal revenue function

`R'(x)=-0.04x+610`

(c)We want to compute R'(5,400)

`R'(5400)=-0.04(5400)+610\\R'(5400)=394`

From the above, we can infer that the revenue that will be generated on the sales of the 5401st item is $394.