The annual total revenue for a product is given by

Upper R left parenthesis x right parenthesis equals 10 comma 000 x minus 2 x squaredR(x)=10,000x−2x2

dollars, where x is the number of units sold. To maximize revenue, how many units must besold? What is the maximum possible annual revenue?

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1 Answer

R(x)=10000x-2x² can be written R(x)=2(5000x-x²)=-2(x²-5000x+2500²-2500²).

R(x)=2(2500)²-2(x-2500)². When x=2500 the second term becomes zero and R(2500)=2(2500)² which is the maximum revenue. 2×2,500×2,500=2×6,250,000=12,500,000.

The maximum revenue is $12,500,000 when 2,500 units are sold.

The problem can also be solved through calculus:

Differentiate R(x): R'(x)=10000-4x=0 when the revenue is maximum. So 4x=10000, x=2500 units. And R(2500)=$12,500,000.

by Top Rated User (695k points)

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