The revenue function for a sporting goods company is given by R(x)=x⋅p(x) dollars where x is the number of units sold and p(x)=400−0.25x is the unit price. Find the maximum revenue.

Answer 1

Hi Macy, this problem is straightforward with calculus, but tricky using only algebra. The tricks are: (1) complete the square, and (2) realize that a squared value cannot be negative. Here is how to do it.

R(x) = x(400 - 0.25x)

= -0.25(x^2 - 1600x)

= -0.25(x^2 - 1600x + 640000) + 160000 (I completed the square)

= -0.25(x - 800)^2 + 160000

Now, this is equal to 160000 minus a square, so the total is, at most, 160000.

So the maximum revenue is $160,000.

I hope this is helpful. Contact me for more help!