If the total revenue function for a computer is R(x) = 1600x − 26x2 − x3, find the level of sales, x, that maximizes revenue and find the maximum revenue in dollars.

Question

x= ________ computers

R(x) = $ ______ ?

Answer 1

Since R(x) = 1600x − 26x² − x³, to find the maximum vale, take the derivative ad set it equal to zero.

R'(x) = 1600 - 52x - 3x²

0 = 1600 - 52x - 3x²

0 = (100 + 3x)(16 - x)

Set each factor equal to zero and solve:

100 + 3x = 0 or x = -100/3

And 16 - x = 0 or x = 16

We would not expect negative computers to be sold so we can throw that answer out.

So, the sales level that maximizes revenue is 16 computers and that maximum value will be R(16):

R(x) = 1600x − 26x² − x³

R(16) = 1600(16) − 26(16)² − (16)³ = 25600 - 6656 - 4096 = 14848

So the maximum revenue is $14,848

Be careful on this because questions like this often refer to "x" in thousands or revenue in thousands or some such thing.